Critical values of Rankin-Selberg L-functions of level-one cusp forms

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Numbers

$k$

$i$

$\ell$, $j$

$s$

$L(f_{k,i}\otimes f_{\ell,j},s)$

12, 1

12:

1.764790264111159589525680806569476356180861793197522430986097214994965967507062908824912966606643003e-1

12, 1

13:

4.466097886582346732956292974976054891286031294102911115704005508452412740053850379238900986149227348e-1

12, 1

14:

6.781326054939848077418159788406417165230996865699562287825546140909601681084947210022002242584395294e-1

12, 1

15:

8.286448297159346761607758126580157677554507138161197632807947135848968239032800023794648586848325008e-1

12, 1

12:

7.120697594231555484207254022999195859767158673424945088748248180652066674524985036866005365593045565e-1

12, 1

13:

1.229873195509947938637908761219050651594710884529156651831332637037445792056212409889981875026131054e0

12, 1

14:

1.333885923427122569731199244049956739738853850530981316905585406830473995681079351983491505825829196e0

12, 1

15:

1.253802512419759249598637213389765464896878455477575438696333181720432367011547238072675498023442762e0

12, 1

16:

1.154956580836328770266686430891076847269347296859164456138047101651305542323244262505556148914217776e0

12, 1

17:

1.085615671741842668246128270197130834749771485839619648299008996094208211482315768345028918343433874e0

12, 1

12:

5.380003562880316956827779273601915299106576448831298289712299199202412886772263471256873014161606863e0

12, 1

13:

5.618889612918519170485291332591258584374476213460572668631612709574554895161562268416393793779090545e0

12, 1

14:

3.513063561721912183912475754842380556933325828381652994569066236496695793000756616539868785192561381e0

12, 1

15:

1.981288433718699550641790846063984466229282347782770946024944338345867899960175535520877452493749042e0

12, 1

16:

1.303635536350501271930719751168943802808263223316969349523750633623472478482748851283507641498447439e0

12, 1

17:

1.072197252248449491916280925225401361266740308077370577030117746699718371280375147927565440074277510e0

12, 1

18:

1.007825020916877553944971232332446159988141137535884052276641904653022423812070971514750994588520274e0

12, 1

19:

9.946834261969189613489377166947575275311582740153590998192822532528941633994618234735995757253562767e-1

12, 1

12:

9.514822365567510798862666025154094496687222456903499368310691334728797574995449222919458908763963518e1

12, 1

13:

6.886552397617828234061592982285970412536675856470898239808064661358245150185579906270509217456514013e1

12, 1

14:

2.751722585095602925738414486728140101405908415695605915982381524664308452982753807952706057108759813e1

12, 1

15:

8.621718519503666594043222222131163824264183675098653517198887290869073533092617721986565921625133083e0

12, 1

16:

2.836431701501567233702942220951151587956354955793896750943116529086101719170321079590364654137402504e0

12, 1

17:

1.399722940226462294870824968502621677716951301087918313037068646949277788077808984341149208116730458e0

12, 1

18:

1.083506799307045451828019083080164184158834953197935289208294479301726498411713199287952556113998202e0

12, 1

19:

1.018455569052481925024067452289986191036726965662364337147151221881102821117064431226954048217412954e0

12, 1

20:

1.005175356663434279018531351848027565169539028300129555196734644181446958477640907634321201461508611e0

12, 1

21:

1.002089204443636032074087914898548566318833114296618256040026576086354788535879469379789970046459500e0

12, 1

12:

3.135288133159181321527794216251230216971650153878468169743777111919566235887424421011590484779725304e3

12, 1

13:

1.706055861945800013451486248212643626324734401376158261945490931894645908876286723405596048178744566e3

12, 1

14:

5.019983601909643452535138305893083929380638286923897884845358400932580619790259611703399029715115116e2

12, 1

15:

1.110300767323345885118817630494399308689463602853271970474854498161192411243853821291151474566158514e2

12, 1

16:

2.259091508555605454695439630870223371273527057274952775124383097913730773456464838770022161584781089e1

12, 1

17:

5.441980408141968265333217181287514977282459274969012229825054760681347384502461945268490443093255150e0

12, 1

18:

2.106282109287844473620822820706310682645756514428003425600654264147103440074840456585765367809360326e0

12, 1

19:

1.351924160362957667130643346006030017339541331251871098485340372265620668560604436565737668655374644e0

12, 1

20:

1.135491222293512683054016826431793016173672067468116543312226297594148977833413021684100767975008029e0

12, 1

21:

1.058383639268302896656192413357504921487941075906800896112883151502856407071463801809433200015030566e0

12, 1

22:

1.026740982647085063076392564193525996653977407847225976724299804388896923195519618127327412980788141e0

12, 1

23:

1.012670060130833295710994869929643236897203328428848283565390049932500375901598148270208506434992457e0

12, 1

12:

3.050125073221586177915046098756800410524651228538680002881627442839483028150874636372290601979346119e3

12, 1

13:

1.611732791520152264610078031588459200794659770376074728774561986042593522192254432354256571495080399e3

12, 1

14:

4.453238474366547738408984723306210223580531635345644265746929326663611373366398141094905940853373314e2

12, 1

15:

8.563612328412571976038310458104929833540619737456645279079601030087588522296475953576125300868725284e1

12, 1

16:

1.269289782118402903660042707186894863250135136903121233059946669700857185653566354234713102965297966e1

12, 1

17:

1.725242522991605968162614535679889841413952062952600641661258867590577929918061598551080836160419563e0

12, 1

18:

6.677435763868416956252811293320978222896607493945082417094259195426152430352921468388971278050328798e-1

12, 1

19:

7.595900902858295932327562312396947830009805944923671253143480009229557697194849077136049085769675027e-1

12, 1

20:

8.757903188231476333113173962679559525398408200758264654335588463450922691829915470059250375044766219e-1

12, 1

21:

9.388944500210594628862080936751907660293769946745148380552732758504220171549986739993308121005297656e-1

12, 1

22:

9.699753372920348573275508246408794646074673987570240259990055606554880642774451878479107678149172486e-1

12, 1

23:

9.851631525149674054556502460964153150726682232506572165528677010222856655746432255527149574784257204e-1

12, 1

12:

1.710738400971672751756376412156252382685672019559611355251403218930710819281506485024996772131440831e5

12, 1

13:

7.122746687837443200149723448173327466174916454848962064365837922635677235371723068512955259116873158e4

12, 1

14:

1.548137055328998835052628000346343660791830536057743399979023702485939431346603116878435369172788077e4

12, 1

15:

2.381220825158120539385452315216337495745844387997416005735242168696860348123851871621092445586846834e3

12, 1

16:

2.984343814088408965140725160104611565025553423985336824832539060798437786951055617911130772791250726e2

12, 1

17:

3.436334164887255772792744613830928755607017386090814273900724786352209325833787230896596712968718901e1

12, 1

18:

4.750036246132350884985438030303838438020294741628100175005027861553833509412581723956665166653972518e0

12, 1

19:

1.488285036195613530199894641950979553449230928807927133533087477657695245527690495484098532205204915e0

12, 1

20:

1.082331492665069882407974202845099319580109967884735064657430259845992714690587742711134361327955018e0

12, 1

21:

1.017350825089979864542087461044893272134815559283918102479517776804507424991804597858502790438196652e0

12, 1

22:

1.004085018075995555668614068885306747805769273713237824319818552858291151287728174795189629606780639e0

12, 1

23:

1.001002213481892317079906040844902109015958292237788728236746439942755710428877459128431390182861477e0

12, 1

24:

1.000253719195842083238021155753367303609373264961863470387915887068892053231682750429139684311386309e0

12, 1

25:

1.000068879367134261580444229068647710552493108208848371810270472327518135506419154628980388456417229e0

Definition

For normalized level-one cuspidal Hecke eigenforms $f_{k,i}$ and $f_{\ell,j}$ of even weights $k>\ell$, this table gives the uncompleted arithmetic Rankin–Selberg value $L(f_{k,i}\otimes f_{\ell,j},s)$ at its critical integers [1].

Parameters

$k$

— larger weight ($k>\ell$ is even and $\dim S_k(\mathrm{SL}_2(\mathbb Z))>0$)

$i$

— embedded eigenform of weight $k$ ($1\leq i\leq\dim S_k(\mathrm{SL}_2(\mathbb Z))$; forms are ordered lexicographically by their real Fourier coefficients $(a_2,a_3,\ldots)$)

$\ell$

— smaller weight ($\ell<k$ is even and $\dim S_\ell(\mathrm{SL}_2(\mathbb Z))>0$)

$j$

— embedded eigenform of weight $\ell$ ($1\leq j\leq\dim S_\ell(\mathrm{SL}_2(\mathbb Z))$; forms use the same lexicographic Fourier-coefficient ordering)

$s$

— critical integer ($\ell\leq s\leq k-1$)

Formulas

(1)

For a prime $p$, let $\alpha_p,\beta_p$ be the roots of $X^2-a_p(f)X+p^{k-1}$ and $\gamma_p,\delta_p$ the roots of $X^2-a_p(g)X+p^{\ell-1}$, where $f=f_{k,i}$ and $g=f_{\ell,j}$. For $\operatorname{Re}s>(k+\ell)/2$, $L(f\otimes g,s)=\prod_p\prod_{u\in\{\alpha_p,\beta_p\}}\prod_{v\in\{\gamma_p,\delta_p\}}(1-uvp^{-s})^{-1}$ [1], Proposition 4.1.2. Analytic continuation defines its other values.

(2)

Put $\Gamma_{\mathbb C}(u)=2(2\pi)^{-u}\Gamma(u)$. Then $\Lambda_\otimes(f,g,s)=\Gamma_{\mathbb C}(s)\Gamma_{\mathbb C}(s-\ell+1)L(f\otimes g,s)$ satisfies $\Lambda_\otimes(f,g,s)=\Lambda_\otimes(f,g,k+\ell-1-s)$, with conductor and root number both $1$.

(3)

For $\operatorname{Re}s>(k+\ell)/2$, $L(f\otimes g,s)=\zeta(2s-k-\ell+2)\sum_{n\geq1}a_n(f)a_n(g)n^{-s}$. Equivalently, its Dirichlet coefficient is $c_n=\sum_{d^2\mid n}d^{k+\ell-2}a_{n/d^2}(f)a_{n/d^2}(g)$. Here $\zeta$ is the Riemann zeta function.

(4)

For $k-\ell\geq4$, let $E_{k-\ell}$ be the constant-one Eisenstein series and set $\langle f,h\rangle=\int_{\mathrm{SL}_2(\mathbb Z)\backslash\mathbb H}f(z)\overline{h(z)}y^k\,dx\,dy/y^2$. Rankin unfolding gives $L(f\otimes g,k-1)=\zeta(k-\ell)(4\pi)^{k-1}\langle f,gE_{k-\ell}\rangle/(k-2)!$. The integral convention agrees with the Petersson self-products.

(5)

For even $h\geq4$, $E_h=1-\dfrac{2h}{B_h}\sum_{n\geq1}\sigma_{h-1}(n)q^n$, where $B_h$ is the Bernoulli number with $B_2=1/6$ and $\sigma_r(n)=\sum_{d\mid n}d^r$.

Comments

(6)

Write $q=e^{2\pi iz}$ and $f_{k,i}(z)=\sum_{n\geq1}a_n(f_{k,i})q^n$, with $a_1=1$. Every finite Euler factor is included. Level one has trivial character and no ramified finite factors. The arithmetic normalization has centre $(k+\ell-1)/2$.

(7)

The indices $i$ and $j$ count all real embeddings of all normalized eigenforms in their respective weights. In the selected weights the $a_2$ values are distinct, so lexicographic Fourier-coefficient ordering agrees with the increasing-$a_2$ convention of the ordinary critical-value table, the normalized Hecke-eigenvalue table and the Petersson table. The roots of the Hecke polynomial $\chi_{k,2}$ identify the weight-$k$ forms.

(8)

An integer is critical when the gamma factors in (2) are regular both at $s$ and at $k+\ell-1-s$. This gives exactly $\ell\leq s\leq k-1$. Equal-weight pairs have no critical integers in this sense and are outside the table's domain.

(9)

Write $\Delta$ for the modular discriminant, and $E_4$ and $E_6$ for the constant-one Eisenstein series of weights four and six. The selected higher-weight forms are $\Delta E_4$, $\Delta E_6$, $\Delta E_4^2$, $\Delta E_4E_6$, $\Delta E_4^3+(a_2-696)\Delta^2$, and $\Delta E_4^2E_6$ at weights $16,18,20,22,24,26$, respectively. Each is paired with $f_{12,1}=\Delta$. In weight $24$, $a_2$ runs through both roots of $X^2-1080X-20468736$, isolated by $(-4017,-4016)$ for $i=1$ and $(5096,5097)$ for $i=2$.

(10)

The convention $\Gamma_{\mathbb C}(u)=2(2\pi)^{-u}\Gamma(u)$ in (2) includes a factor $2$ absent from [1], Proposition 4.1.5. Our completion is therefore four times theirs. The uncompleted $L$-values are identical.

Programs

(P1)

PARI/GP

default(realbitprecision, 750);
k=16; s=15; h=(k+10)/2;
f=mfeigenbasis(mfinit([1,k],0))[1]; g=mfDelta();
RS(n)={my(a=mfcoefs(f,n),b=mfcoefs(g,n),c=vector(n)); for(d=1,sqrtint(n),for(m=1,n\d^2,c[d^2*m]+=d^(k+10)*a[m+1]*b[m+1])); vector(n,j,c[j]/j^h)};
L=lfuncreate([RS,0,[h,h+1,h-11,h-10],1,1,1]);
lfun(L,s-h)  \\ L(f_(16,1) tensor Delta,15) = 0.828644829715934676...

References

[1]

A. Lei, D. Loeffler and S. L. Zerbes, Euler systems for Rankin–Selberg convolutions of modular forms, Annals of Mathematics 180 (2014), 653–771, sections 4.1 and 4.2. (doi)

Links

[2]

PARI/GP: L-functions and numerical accuracy

[3]

PARI/GP: modular forms and Petersson products

[4]

SageMath: Dokchitser's L-function calculator

Similar tables

Critical values of symmetric-square L-functions — degree-three symmetric-square values for the same eigenform roster; this table pairs each higher-weight form with Delta

Fourier coefficients of the modular discriminant — the coefficients of the fixed weight-twelve factor Delta

Hecke polynomials of level-one cusp forms — the roots of the weight-k Hecke polynomial at prime 2 identify and order the embedded eigenforms

Normalized Hecke eigenvalues — stores $a_p/p^{(k-1)/2}$; multiplying by $p^{(k-1)/2}$ recovers the arithmetic coefficients in (1)

Ordinary critical L-values — degree-two values of the individual factors, with the same Fourier normalization and embedding order

Petersson norms of level-one cusp forms — the self-products enter the endpoint relation (4) when $gE_{k-\ell}=f$

Critical L-values of higher-level newforms — ordinary critical values with additional levels, characters and coefficient fields

Petersson norms of higher-level newforms — Petersson self-products beyond level one, where tensor constructions require additional ramified-factor conventions

Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

Table is complete: no (it holds all $66$ critical values for $f_{k,i}\otimes\Delta$ in every nonempty even-weight cusp space with $12<k\leq26$, including both embeddings in weight $24$. The anchor $\Delta=f_{12,1}$ is the unique minimal-weight cusp form and defines a reference slice of tensor products. The noncanonical weight bound includes all intervening spaces, the first quadratic coefficient field and the next one-dimensional space, balancing weight and coefficient-field coverage before extending to all pairs. Every integer $12\leq s\leq k-1$ is included for each selected form)

How they were obtained:

Values are recorded to $100$ significant digits after fresh PARI/GP calculations at $160$ and $220$ requested working decimal digits, using explicit bit precisions and exact coefficient data [2]. The maximum relative difference between these calculations is less than $1.3\times10^{-163}$. These are agreement checks, not certified analytic error bounds.

The generator builds exact Fourier coefficients from the divisor-sum expansions of $E_4$ and $E_6$ and $\Delta$, using $\Delta=(E_4^3-E_6^2)/1728$, then forms reciprocal quartic Euler factors and uses multiplicativity. It evaluates in analytic coordinates with $h=(k+\ell-2)/2$, coefficients $c_n/n^h$ and argument $s-h$, preserving the arithmetic values stored here. The quadratic embedding is refined inside each coefficient callback at the precision PARI requests. An initial unshifted weight-sixteen experiment retained only about $105$ agreeing digits; the exact coordinate shift removed that measured loss.

An independent computation constructs Sage modular-symbol eigenforms, forms the square-divisor coefficients in (3), and evaluates every critical argument using Dokchitser's L-function calculator at $700$ bits [4]. Every Fourier coefficient used by that computation agrees exactly with the generator, beyond each Sturm bound. The independent computation checks all reflected values and uses the nonzero control $\zeta(2)=\pi^2/6$ before comparing results. Its cancellation warnings are retained in the attached evidence.

Every right-endpoint value is additionally checked by (4), using PARI's independent Haberland period-pairing algorithm for $\langle f,\Delta E_{k-12}\rangle$ [3]. This includes both weight-twenty-four embeddings, without replacing the product by a self-product. Known initial coefficients of $\Delta$ control the modular-form construction. The exact dimension roster includes every embedding through weight $26$.

The attached generator, exact roster, argument-selection rationale, numerical evidence and checks record the software versions, source hashes and supported precision.

Every independent L-value and endpoint pairing agrees to at least $130$ relative decimal digits. The attached evidence records each measured difference. Dokchitser reports a maximum loss of 6 decimal digits from cancellation.