Lower order terms for the moments of symplectic and orthogonal families of $L$-functions

TL;DR

This paper derives formulas for the lower order terms in the asymptotic expansions of moments at the central point for quadratic Dirichlet L-functions and quadratic twists of elliptic curve L-functions, involving determinants of binomial coefficients.

Contribution

It provides explicit formulas for lower order terms in the moments of certain families of L-functions, advancing understanding of their asymptotic behavior.

Findings

Formulas for asymptotic expansion terms of moments of quadratic Dirichlet L-functions.

Formulas for moments of quadratic twists of elliptic curve L-functions.

Study of determinants of binomial coefficient matrices related to these moments.

Abstract

We derive formulas for the terms in the conjectured asymptotic expansions of the moments, at the central point, of quadratic Dirichlet $L$-functions, $L(1/2,\chi_d)$, and also of the $L$-functions associated to quadratic twists of an elliptic curve over $\Q$. In so doing, we are led to study determinants of binomial coefficients of the form $\det (\binom{2k-i-\lambda_{k-i+1}}{2k-2j})$.