Arithmetic factors $A_k$ in the moments of quadratic Dirichlet $L$-functions

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Numbers

$k$

$A_k$

1/2:

0.8710745340511250504612054830175426266491034345278751608021447745799814511498354321815954299628373618

0.7044422009991655927366033503266372101885864314170980494142268425910970566820067785368082441456931338

comment: The prime product in (5) is tabulated as OEIS A065463 [4].

3/2:

0.5043014421159503784723366758859157308177514033585388231226261749702333441296864007109085000867797962

0.2972100247430660325729962454715592316932240554398697196905975958942245449583116081861878885544967115

5/2:

0.1343688716977946081778496936553101357677673924039421741466629016187329357767240909652329103841681889

0.04401723165932221944195956633753720410465301049754961438022678528262944033571060021620761107819094436

7/2:

0.01000655278207438504541922220846843177512989904066291656706501259853346193710464622870072433779456086

0.001528096549969385816749073171853640003879973727900383860958775398340903993796790265487847973557403654

9/2:

0.0001528725716115276747837765883721545091433992176489383085442091898152974521822228538368309322849910503

0.000009821300391317444150950178539036899751700031021149773080529566426506043819651913824586533806025973247

11/2:

3.986485284837906287516270927723672975818015988624202465703098619519257367275005212718604437159890790e-7

1.008439217511471660151574457020410448759783924984909996311378509661855375088230473396348927813261668e-8

1.492591666178780773025660265676484107529027345208209472951191698304351656116128123090990156803059064e-12

2.946142164836727464282113082496374007380516592210189352735069642041560297082979909788460783574862679e-17

7.287794129204276455297119551302394051175570180566225039850256954767221295960746873189532692672819885e-23

10:

2.145868290621392510673045320906844538364582409217245220075182885641924382688309518516276323087347813e-29

11:

7.198716771628843709140982974369123109683029357703679292650295760628759670257638414358551514055888711e-37

12:

2.648872344204725620050326669196614837042129484866034199096247945233299797314097392592021634687395512e-45

Definition

For $k>0$ rational, $A_k$ is the positive real Euler product in (3). It is the arithmetic factor in (1) for central values of quadratic Dirichlet $L$-functions [1] [2].

Parameters

$k$

— moment parameter ($k>0$)

Formulas

(1)

Let $D(X)$ be the set of fundamental discriminants $d$ with $|d|\leq X$, and let $\chi_d(n)=\left(\frac{d}{n}\right)$ be the Kronecker character. Conjecturally, $\frac{1}{|D(X)|}\sum_{d\in D(X)} L(1/2,\chi_d)^k \sim A_k f_{USp}(k)(\log X)^{k(k+1)/2}$, where $f_{USp}(k)$ is defined in (2).

(2)

Let $\Lambda_A(z)$ be the characteristic polynomial of $A$, and let $X$ be the conductor-normalised matrix-size variable. The symplectic random-matrix factor is defined by $\langle\Lambda_A(1)^k\rangle_{USp}\sim f_{USp}(k)X^{k(k+1)/2}$ [1]. For positive integer $k$, the factorial normalisation is $g_{USp}(k)=\left(k(k+1)/2\right)!\,f_{USp}(k)$.

(3)

$A_k=\prod_p (1-p^{-1})^{k(k+1)/2} \frac{\frac12\left((1-p^{-1/2})^{-k}+(1+p^{-1/2})^{-k}\right)+p^{-1}} {1+p^{-1}}$, where the product runs over rational primes.

(4)

If $k$ is a positive integer, the local factor in (3) is $(1-p^{-1})^{k(k+1)/2} \frac{(1-p^{-1})^{-k}\sum_{m=0}^{\lfloor k/2\rfloor} {k\choose 2m}p^{-m}+p^{-1}}{1+p^{-1}}$.

(5)

$A_1=\prod_p(1-1/(p(p+1)))$.

Comments

(6)

Here $k$ is the power of $L(1/2,\chi_d)$ itself, so $k=1$ is the first moment. In the table of arithmetic factors $a_k$ the same letter is half the exponent, and $k=1$ is the second moment.

(7)

This table stores the arithmetic factor $A_k$ alone; the leading constant in (1) pairs it with the symplectic random-matrix factor $f_{USp}(k)$.

(8)

The rational powers in (3) are taken on positive real bases. Since $p>1$, both $1-p^{-1/2}$ and $1+p^{-1/2}$ are positive.

Programs

(P1)

PARI/GP

\\ positive integer k only
default(realprecision, 140)
k = 3
e = k*(k + 1)/2
s = sum(m = 0, floor(k/2), binomial(k, 2*m)/p^m)
prodeulerrat((1 - 1/p)^e * ((s/(1 - 1/p)^k + 1/p)/(1 + 1/p)), 1, 2)

References

[1]

J. Brian Conrey, David W. Farmer, Jon P. Keating, Michael O. Rubinstein and Nina C. Snaith, Integral moments of L-functions, Proceedings of the London Mathematical Society 91 (2005), 33-104. (arXiv)

Links

[2]

Wikipedia: Dirichlet L-function

[3]

PARI/GP reference: prodeulerrat

[4]

OEIS: A065463

Similar tables

Arithmetic factors $a_k$ in the moments of the Riemann zeta function — stores the analogous arithmetic factor for the unitary zeta-moment family

Data properties

Entries are of type: real number

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

Table is complete: no (it holds $k=1/2,1,3/2,\ldots,6$ and every integer $7\leq k\leq12$)

How they were obtained:

The generator computes the small-prime local factors directly, computes the middle prime range from the exact power-series coefficients of the logarithm of the local factor in (3), and computes the remaining prime tail from prime-zeta values.

The written 100 digits were computed with 140-digit settings and recomputed with 160-digit settings; the written digits agreed. The checks include (5) and the integer rows against PARI's prodeulerrat applied to (4) [3].