Multiple Dirichlet series predictions for moments of $L$-functions: unitary, symplectic and orthogonal examples

TL;DR

This paper develops heuristics using multiple Dirichlet series to predict asymptotic moments of various families of $L$-functions, aligning with existing recipes and highlighting specific modifications needed for certain cases.

Contribution

It introduces a novel heuristic approach with multiple Dirichlet series for predicting moments of $L$-functions, extending and refining existing conjectural frameworks.

Findings

Predictions agree with the recipe for most families.

A modification is needed for quadratic twists of elliptic curve $L$-functions.

Residues from Dirichlet series analysis correspond to recipe terms.

Abstract

We devise heuristics using multiple Dirichlet series to predict asymptotic formulas for shifted moments of (1) the family of Dirichlet $L$-functions of all even primitive characters of conductor $\leq Q$, with $Q$ a parameter tending to infinity, (2) the family of quadratic Dirichlet $L$-functions, (3) the family of quadratic twists of an $L$-function associated to a fixed Hecke eigencuspform for the full modular group, and (4) the family of quadratic twists of an $L$-function of a fixed arbitrary elliptic curve over $\mathbb{Q}$ that has a non-square conductor. For each of these families, the resulting predictions agree with the predictions of the recipe developed by Conrey, Farmer, Keating, Rubinstein, and Snaith, except for (4), where the recipe requires a slight modification due to a correlation between the Dirichlet coefficients and the root number of the corresponding $L$-functions. We find a one-to-one correspondence between the residues from the multiple Dirichlet series analysis and the terms from the recipe prediction.