Chowla–Selberg period constants

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Numbers

$D$

$b_D$

-3:

4.932127815881101425466867117189975491238048870011176402782347120795278124422015762097676583146796766

comment: The field $\mathbb{Q}(\sqrt{-3})$ has $h_D=1$ and $w_D=6$. Its Gamma expression is (5).

-4:

5.244115108584239620929679179782238827365509902863246325633643407601581174140828500460591065922858187

comment: The Gaussian field $\mathbb{Q}(i)$ has $h_D=1$ and $w_D=4$. The square-lattice identities are (6).
equals: Values_of_the_Beta_function_at_pairs_of_rational_numbers#1/4,1/2

-7:

5.883156652850391944549245050252011991018522700207855109114298661432078754756876154977801206134065455

-8:

5.991051932477664176790681726352437979184227265814619469308370436830936054966300439419011255927179638

-11:

6.184341053563215130336290612794849162331118567185679931274166371045957960356907593237123164871611674

-15:

6.763877182970595474715873644373609964290943784317287120176274799200962128400640325362278729846287738

-19:

6.186491653192832733710687884439930920263698945569972782485082785411644034548354788774689670531524256

-20:

6.949491778175185172294226970483838715503444676713965143204702990300086615727153439100276801256389925

-23:

7.322646326097958231969481975959173912037026727591888246837247826817117140586053219187464920790250775

-24:

6.991364502549040267124114805122013882424284030695564055763799039446314257572359459119624619754774330

-31:

7.429732717328923183019510895281225861989889228470670667488111199613277446558254974678731053392350054

-35:

7.104705926031802724991863121048238504106229554657088494422011058390637154462555061265936199984696744

-39:

7.841881478935116949629588252114858350020702440125849894028003088986533988371076746023396159992156221

-40:

6.824945802758021030806208779219339381355297441883646596192199947530243090979781714405291726431324256

-43:

5.233970380542507142534185218479640145838648699997119902706281556440078389023828589370962042382955009

-47:

8.170179410146150090220808826996899661461295377525762095277805820417098807262208686131867924511966740

-51:

6.935918298889648550929018769619362628328984324832514378054352788316303258350830557633118674717726523

-52:

6.538801594598620246094066013338427578931885304811878464810624299038555321856975912225638573718436287

-55:

7.824934850444684019137729858848304391475624690314259782065322231786919309295629152365290073410364717

-56:

8.017948982314032522592521077993997559449018698996729885723145541764438357690800509924522677122862809

-59:

7.740581062975736160544848865125554771975074086895549418970086444961526382815268323718455723895663878

-67:

4.266225341160705155802669336279169781508967755305428245504615841490912073064282047423012106975688730

-68:

7.928565927474570895832291266375494015066438866436233708019952094367712787583209546270083458587612023

-71:

8.768790986844776921607049002203831134496930449351318034279811049730449654860274647926587412665743842

-79:

8.134801283448532797805763551930129396394659558148762744170345053399001386047805229482513364317239282

-83:

7.458937014561877942453445080345029985197631708681269690614984783299071471003880092783823555229087051

-84:

7.870265886810997372369383242459191816686696730358104621614870008990925940602682250505190057139243302

-87:

8.443035779687197451326781605668338723146353520909898128111920595937740002816865150570081725561185493

-88:

5.555357225181844081071460336384404905301462806593103060085725473540991605759275975329912112135734019

-91:

6.338916461683070807345407584921896896287693399307718865660447546742782545656967760165675291660166290

-95:

8.997275312631892565281139308927608001763992583819454913087884055521019894023405651635352947043250233

-103:

7.881263456811906438724206721370940716780810320042122011138959813834790643904125781497408694978990663

-104:

8.698866449452180874266833943814444132011806625357917647301996954561767970750060482853065975650797703

-107:

7.078167994329979426626019480122844415145844445841552859648092097333157516898346374677159150461073926

-111:

9.039761222717322606897436476787285823180288480018370799624718561838497719641328934229925177526673632

-115:

5.918609957531616080722853906920386820034306005907856267286187895876919571153985550504915672375444713

-116:

8.606971194548780426159057503219403163173108725432057319956530508166478467952301737052206278882300494

-119:

9.455614188360612141705603949415025910481334327035633360714932058494617471970724345676635300206878719

-120:

7.400502169561486886011908868426536898234492905103809022964587240499321353960216339330296511137812319

-123:

5.403721278306448394372915346291880174864464727803612854742905744822465731750475233995145956077301751

-127:

7.555403856135309470501808244801665273737289048710664085301841040467773846372852232248541928491747797

-131:

8.575862020298787233272725730662008936041439781217967841911660740657050474619491308721118618103996298

-132:

7.257636607436713194455403389962301164253680193538602119367402870962381018729654709595887603019300686

-136:

7.449990781896666741749074795478864138365702940508324503283545252721930858697048665549775280072230238

-139:

7.250044971384241869009823188173609570680812215622229603553384358816779030798768541692720674182967903

-143:

9.379725571875022822362264168432884323552874880011033533163510507697353586833311564163597261733023385

-148:

4.170048068878650752026651241347582259106702061387383713357835355548235893855101998888622340539509928

-151:

8.501897584204166606630318979136113235332535662111569339244429247125723604056624438701073687150374936

-152:

8.326243671421959156440008866248581466307906208223933155328757182718648153158908941149995650064791903

-155:

7.510786194903699603333870774919057712613653316998438886420841383050025052684032797895392091408511419

-159:

9.442452137285466790951947575423755678492532493755779985054829631918899272555339972192642394299202158

-163:

2.005207895553038685960947236172789488291495333192442304667045812074565604266178581853806912748144496

-164:

9.206603493720630780560806345127088124162805184588206857534061731320049944595075502136845826123985320

-167:

9.550841779413574047195468803890235141116393336869146690285190256025658869530671833947365395846127844

-168:

6.740443107061824801341598930916030646162058330555744586059497555163224046507640908936937609844315056

-179:

8.129380394991043993943131629625136443423756975147387998922445877141880058987057132463476348252533399

-183:

8.513271956814975138382371300429222849177121043107175275057601850570292262451149093195549953160466032

-184:

6.842136203643471625048536034668595876040135050275552844736975989361938581925716042492392172636564013

-187:

4.826691648188408300127392728662975349965399346480347603301922085212488149271870413417336579607582616

-191:

9.927544999286701992639137286900900533143350002526050269668642258213654989280384182775513238655192765

-195:

7.488081136818254823307179342480404849090088633770966434643356015754266211454685191923311255497972851

-199:

9.070826528054084953420655691976970936353805075789016534031962405500677105910006760859573176236793036

-203:

6.982705854806180446907688687516061790409608160053790795660296441727780812705666470837362772256144822

-211:

6.398937889315356871497140762324635895881438921734134247109370270117806683680003342520948175431167921

-212:

7.703458777985518895371809421834718452976284697944636173613138893619793460844283153663928917734847943

-215:

10.10313884330393173380728994653016251580988669024124186265788363750226754211533602519073476010106163

-219:

7.150811854470486853926172145185016667555343859109786897227455158646789865089558798827613014111896106

-223:

7.872547764863025364877086795446643217219920253564069405574644432944760185130461943015350482487906932

-227:

7.808829008333693092131375120457010022384705504828458856606738198019725407532982562563786884000323836

-228:

5.939817757829401110870485115336624730423545690154688725187634142211703723771663152744091478039728316

-231:

9.690853176688668461898879298860112914664176757517305392312012295326026054660668316678101437341260846

-232:

2.859590010784716858398624467076199938024788395458011475399622699237141821477790467126224906666616660

-235:

4.099303892034959467961852573610492007402239322423625872833529513677067165190210720802840756514650197

-239:

10.21331847078768001100491814825809426859577961126989258069575837019337665200815710638002268222937079

-244:

8.169557574208848795207334330008218483000020537695233761228315649731638063054890097695812267265725124

-247:

6.860502311085408973271141741850484090240086446470641964980859087396522808760905634205913402291368811

-248:

8.708816296913917387225495598078149949076959221879162412680828854295416972582457371027748500012748872

-251:

8.979274214380381609531753932176667719671425493111421988109060460187239616475886155984860057323482003

-255:

9.598391595989483421539334466592335511260426388308415329967748210852465537871364931281612002832156662

-259:

7.384147093449548795063378528455258302035281006776494945378179754586921941115997361973275980451769088

-260:

8.555185395892571113197681743019824364010857385233666720129647197512561955768052361340236479424896325

-263:

9.620818032413392548941533524128564250285742018745576105153832564754136784599422226232545965964481485

-264:

8.871908270526949381647913838166665113307408658477441333059776705663806313461479336519378663521954807

-267:

3.184780193498973756476700267445461162110831011809750157402601466083450151665437563900638743047190795

-271:

9.444604939474416602216876131623879896469076072256223011089826622658455386910983972331524931955516918

-276:

8.783171661655503290513142125910151729820416970599504311308131172512057329707437064005014694519371755

-280:

5.807083532302079664916491148969852643641030961584354324559702541914246497696158645799518727160957433

-283:

5.919997191772598964041584040019489869063887192146858232555193006636361362049659374167109748396093614

-287:

9.785782349395764873132048252366635784971600881817583817229124694701373143979378217478515321690786676

-291:

6.379152871478361301399413456335349171027382159157976741552101998175619330719107124290854580692560507

-292:

5.809575239760805906524711449230901242264924392006245950637278348799567967001375442691871538804662218

-295:

7.818313444674528379513846379271599273707331840386900237561776526873634064767686477439098720553673455

-296:

9.269849460750747839222812476148606535169457893614650881967172461000043196537720193240747239846434877

-299:

9.285889888904954473874362295294417084083980759852047388469447479580332981635481404836547745316110511

-303:

8.512683727509546398071687055368980316442074910427617882465766871609108788779689593230539632229240341

-307:

5.695807319363485283746483934270615763988818163757858813794162223427263199360570649384841151808609199

-308:

8.255803114061307705998229342952393726463492746346891695249527839701802701728317101718145421451886179

-311:

10.81117895459745023767492645313982997079732428106197959036899273548163782251974358945594509628433351

-312:

4.966262176149639656999189412577049798869691738598331646256669962339675378322691977798849449497474033

-319:

8.693969923828348712064316486965409902347423222542164996867382112128502057039152805582664754323404232

-323:

5.690368249791946714143455842616186572472008643494522260279343789643149950684266224557145461373945686

-327:

9.224916919089320311154080809695655711624841557175140421686682353171840002195189118653379911643303871

-328:

5.464451991999868126626201121539159321235832892406325288034665372015110152700411231256432384111447018

-331:

5.119946788992590663976353741307090915269668413893534277345054731260204342984320595751892952573824509

-335:

10.55945127886416162786152437816597791493109069873253032470139727547170881254102218880406628427584255

-339:

8.151877385822734551492549892549708122794987276707074393062543500973088363252136175490705516549564150

-340:

5.255836226556148421565151378940803965451335097163142948786066278027254900685073001615119920216898589

-344:

8.945308052064548483655896586929418124897173763071211917929071297945039208731680218346283183468592547

-347:

6.789499561003482116091207684369305916686962950011904338503673473361830096400703232011877479617274199

-355:

6.736387187033196160878555654410291169638355066793252511438586886914290732321922452558101605146945576

-356:

9.742692982098369529915984754254174857186862899611142822745264119756262776549483063862307160514583931

-359:

10.65148124060668324675388907458468489514537083350704140477240292924236813467827100043120566413278731

-367:

7.958652943695445943033716512189384032688199370118485159264893184394104163649924951979700259617073299

-371:

8.863856935851770927844477724505243216156441714061053514572014249299093213846880603335694101574453142

-372:

4.384735184386221755648159561090240805774696679957277351400822004547232885175818426336162307354386777

-376:

8.737180876010672430922470728955041849379825897739543526282594473015825358475511821778998164622064536

-379:

4.688588682767068609483547069554787073436946275740967790192838184081794560186376501075709474280818371

-383:

10.15435029728101374544384167636923753029712095161504319158304789915078353654957422303348002724399680

-388:

4.935380751744305832651501268397279425647989345837468867583690839515488002295833523616257954882043482

-391:

9.942004871504460696190552565940764926146794694259902239452223952304383209932839661280226633458252701

-395:

8.847139101038000312146873121525525126424175702103385988223007768139930919465434494421539118147944632

-399:

10.07861602500382763773611135502908100247611868227739602150244390097744975157506956412058020687904811

-403:

2.793598503822872472397832685914128479100910901366781786018121756387140769885975351056234797165911363

-404:

10.27530206497906807348160988334104689143589402055538186310727334838550551147693751597755836447033439

-407:

9.776746838545938256355162855795011506703563060581276215161134613221049946195458597754394117115911669

-408:

4.075283141743591687030074586886895680699627526824178487730399903180812527961517908071395825067191389

-411:

7.617598151827488384886932505572147813016113879471959444570254852079913428508365641589551301665784495

-415:

8.155769430649311443809409140183818004375647161029229657998662648258073630545409152275269332006261773

-419:

9.172575978863120661192380155497285848211637416419387096608672853280427050624351010714335004811524939

-420:

7.948802634500821219647014643458268593345065969129453273263017519522829175533342986093167749656868656

-424:

6.816288289500230085676050405638067646089742149882636886301610086368977740468414267597365102431817129

-427:

2.565075311950082392341330528887589079877053224696415703618906959076129676250656161477729626457933072

-431:

10.81925085064963644914605162558325136294307084562871384781532949611878201816547108367671893072768845

-435:

5.333881096209198052033560163488763612825717559197413214352941818966776314054070836189625061693833115

-436:

6.728199642851991203147462929859946721180780639884888680908678993604626240724298120264160376509099137

-439:

10.06994054306043666130326761369146450326536288010150512176569806685195627256705679279842446541197203

-440:

9.420415680212143037366226382948341703206578131316988520791431081531094029917830480266853202245895362

-443:

6.033726979613868988888650026408914650017401577597985840629592781785790687309798714130075012531364250

-447:

9.326442641599608354537945325546411360487412444562003409267095706143530110620870240229091898743743005

-451:

8.241534858893721408388697621583088068700419477318015486578722695711200623723867654228540655351423376

-452:

7.205576336239125653886467952767047391701663052196057848718755581950308144782481649833445869426591631

-455:

10.55759035112117346954835285976316414569505949851444031265897763136432226061897037835759129605464108

-456:

7.562127015626818816468606290097901037458340131632334088457307065987871818464701932847520133412216433

-463:

5.772416214767836056865365237091751556978500834034209519763107170304674210906606326776198630781975685

-467:

7.828721785798495568188708892828197017581957116017644039284125849762573808239410438208093937581787146

-471:

9.744340912789366296814376104369677879999844804898210882768758542634338451841986700292501956892456358

-472:

6.849614435155626564918878442561516752723643027932003660245100693968020190151801728426452961468706932

-479:

11.42848883977331027745050101011517393717802605322351447185303689892913609285066103204962121823491465

-483:

5.186206539252300365584901077211528653106340635728262448322469718022051824719279921078416186846122950

-487:

5.593754666722267059679411723358245394773135636739194950464085877533508957873563153957633200468858232

-488:

8.238656437017826115588008965965109207991116677221356898242467572553647253921163593536128114314252277

-491:

8.902621678370781490625370593062923856235272489364812432362736665662937732815313928830955127392392903

-499:

3.788564542006635999580245301542717485533862936051934205154910648362264117835392911712857052222897001

-503:

10.69772648123435060240458253119040924437777249120669433176488286780499025170160049984777994711415031

-511:

9.374174950132873339228067807033935834590912414919423437083006434781864599247852028400492109882101472

-515:

6.433988988691926063792423374209007759152894859037225892244094857028071101693219600210385176493927490

-516:

9.555630697755849909467675133931299011865602490755022973002238624876422123286232206898079995595431110

-519:

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Definition

The Chowla–Selberg period constant $b_D$ of the imaginary quadratic field $K_D=\mathbb Q(\sqrt D)$ is the positive Gamma product (1), in the normalization of [1], equation (2.1).

Parameters

$D$

— fundamental field discriminant ($D<0$ is a fundamental discriminant)

Formulas

(1)

$b_D=\sqrt\pi\displaystyle\prod_{a=1}^{d-1}\Gamma(a/d)^{w_D\chi_D(a)/(4h_D)}$, where $\Gamma$ is Euler’s Gamma function [1].

(2)

$\log b_D=\tfrac12\log\pi+\dfrac{w_D}{4h_D}\sum_{a=1}^{d-1}\chi_D(a)\log\Gamma(a/d)$. All logarithms are natural.

(3)

$\displaystyle\sum_{a=1}^{d-1}a\chi_D(a)=-\frac{2dh_D}{w_D}$, the odd quadratic-character class number formula [2], equation (5.8).

(4)

$\displaystyle\log b_D=\log(2\pi)+\tfrac14\log d+\frac1{h_D}\sum_{[a,b,c]}\log F(\tau_{a,b,c})$, where the sum runs over the reduced forms specified in (12). This is the conductor-one case of [2], Theorem 5.1, rearranged in the present normalization.

(5)

$b_{-3}=(3/4)^{3/4}\Gamma(1/3)^3/\pi$ and $b_{-4}=\Gamma(1/4)^2/\sqrt{2\pi}$ [1], §2.1–2.2.

(6)

$b_{-4}=2\sqrt2\,K(1/2)=2\pi\sqrt2\,\eta(i)^2=B(1/4,1/2)$. Here $K(1/2)$ uses elliptic parameter $m=1/2$, $\eta(i)$ is the square-lattice eta value, and $B(1/4,1/2)$ uses Euler’s beta normalization.

(7)

$\gamma_{K_D}=2\gamma+\log(2\pi^2)-2\log b_D$, where $\gamma$ is Euler’s constant and $\gamma_{K_D}$ is the Euler–Kronecker constant. This follows by substituting (3) into the imaginary-quadratic logarithmic Gamma expression for $\gamma_{K_D}$.

(8)

$h_{\mathrm{st}}(D)=\tfrac14\log d-\log b_D$, in the maximal-order convention specified in (13), by [2], Theorem 1.3.

Comments

(9)

Write $d=|D|$, let $\chi_D(a)=(D/a)$ be the primitive Kronecker character, and let $h_D$ be the ideal class number of the maximal order. The number of roots of unity in $K_D$ is $w_D$. All logarithms are natural and all real powers of positive quantities take the positive branch.

(10)

The unit count is $w_{-3}=6$, $w_{-4}=4$, and $w_D=2$ otherwise. The normalization fixes a positive number, rather than a period specified only up to an algebraic factor. For a CM elliptic curve defined over $\overline{\mathbb Q}$ with a nonzero differential defined over $\overline{\mathbb Q}$, its nonzero periods are algebraic multiples of $b_D$ [1].

(11)

The index specifies a field, not an order or a chosen elliptic curve. Passing to a nonmaximal order in the same field leaves this $b_D$ unchanged. Hilbert class polynomials distinguish the orders and their singular moduli.

(12)

Write $F(\tau)=\sqrt{\operatorname{Im}\tau}\,|\eta(\tau)|^2$, where $\eta(\tau)=e^{\pi i\tau/12}\prod_{n\geq1}(1-e^{2\pi i n\tau})$ [3]. For a reduced primitive positive definite form $[a,b,c]$ of discriminant $D$, set $\tau_{a,b,c}=(-b+i\sqrt d)/(2a)$. The representatives satisfy $|b|\leq a\leq c$, $\gcd(a,b,c)=1$, and $b\geq0$ when $|b|=a$ or $a=c$. Their number is $h_D$. Rectangular CM eta values cover individual points; the class product in (4) uses all ideal classes.

(13)

For the maximal order, $h_{\mathrm{st}}(D)$ denotes the stable Faltings height of a CM elliptic curve with endomorphism ring $\mathcal O_{K_D}$, using the metric $(i/2)\int\omega\wedge\overline\omega$ [2]. This convention agrees with the rational-curve height table. Nonmaximal orders require a conductor correction and are not additional rows of this field-indexed table.

(14)

Holomorphic periods of Fermat curves retain the complex phase of a specified differential and oriented cycle. Their low-degree Gamma factors give geometric context, while $b_D$ is a positive constant normalized by its field.

Programs

(P1)

PARI/GP

\p 120
D = -23;
d = abs(D); h = qfbclassno(D);
w = if(D == -3, 6, if(D == -4, 4, 2));
sqrt(Pi)*exp(w/(4*h)*sum(a=1,d-1,kronecker(D,a)*lngamma(a/d)))

References

[1]

W.-C. W. Li, L. Long and F.-T. Tu, Computing Special L-Values of Certain Modular Forms with Complex Multiplication, SIGMA 14 (2018), 090, §2.1, equation (2.1). (arXiv) (doi)

[2]

A. Barquero-Sanchez, L. Cadwallader, O. Cannon, T. Genao and R. Masri, Faltings heights of CM elliptic curves and special Gamma values, Research in Number Theory 3 (2017), 13, Theorems 1.3 and 5.1. (doi)

Links

[3]

NIST DLMF: Dedekind eta product, equation 23.17.8

Similar tables

Values_of_the_Gamma_function_at_rational_numbers — provides individual factors of the defining character-weighted product

Complete_elliptic_integral_of_the_first_kind_at_the_singular_values — contains $K(1/2)=b_{-4}/(2\sqrt2)$ at $r=1$

Dedekind_eta_values_at_rectangular_CM_points — gives individual rectangular CM values of the eta function used in the full class product

Euler-Kronecker_constants_of_quadratic_fields — holds $\gamma_{K_D}=2\gamma+\log(2\pi^2)-2\log b_D$ for the same imaginary fields and also for real quadratic fields

Faltings_heights_of_elliptic_curves_over_Q — contains the same stable-height normalization for rational CM curves among its rational-curve entries

Hilbert_class_polynomials — encodes the singular moduli of imaginary quadratic orders, including the maximal orders indexing these periods

Values_of_the_Beta_function_at_pairs_of_rational_numbers — contains the equal value $B(1/4,1/2)=b_{-4}$

Holomorphic_periods_of_Fermat_curves — holds marked complex periods whose low-degree Gamma factors are related to CM period formulas

Stable Faltings heights of CM elliptic curves — For maximal orders the stable height satisfies $h_{\mathrm{st}}(D)=\tfrac14\log|D|-\log b_D$. For nonmaximal orders of discriminant $\Delta=f^2D$, the height includes the conductor correction, while the field constant $b_D$ remains unchanged as $f$ varies.

Data properties

Entries are of type: real number

digits: 100 (Unknown key)

How well the digits are known: proven

Table is complete: no (it holds every negative fundamental field discriminant with $|D|\leq1000$, ordered by increasing $|D|$. This arithmetic family includes both discriminant parities and all class numbers in the interval, so the class-product structure is represented beyond the class-number-one cases. The noncanonical cutoff keeps each Gamma product under 1000 factors and permits an independent eta-class-product check for every field. The attached argument-selection.md explains this choice)

How they were obtained:

The attached generator evaluates the character-weighted logarithmic Gamma sum in Arb real balls at 429 bits for 100 significant digits. The class number is obtained from the exact finite character sum. The smallest measured relative accuracy is 416 bits; no precision escalation was needed. Every resulting ball lies inside the interval represented by its written decimal, whose uncertainty is one unit in its final place.

All 305 values were independently compared with the eta class product using reduced primitive forms, 512-bit real balls and 80 product terms per form. If $r=|e^{2\pi i\tau}|$, the omitted part of $\log F(\tau)$ has absolute value at most $2r^{81}/((1-r)(1-r^{81}))$. The largest such bound is less than $8\cdot10^{-192}$. The eta computations retain at least 502 relative accuracy bits and their entire enclosures also lie inside the written decimal intervals. Every comparison requires finite, narrow enclosures. A square-lattice control uses the arithmetic-geometric mean computation of $K(1/2)$; perturbing each period by $10^{-90}$ is rejected.

Independent squarefree-radicand enumeration gives the same 305 discriminants. Reduced-form counts agree with both the character sums and PARI class numbers throughout. The Euler–Kronecker identity is checked against all 305 imaginary-field entries in T230, the square-lattice identities against T296, T506 and T177, and the stable-height normalization against ten maximal-order CM rows of T211. The attached checks can repeat the enclosure and identity tests on an authenticated export of the stored draft.