On the orthogonal symmetry of $L$-functions of a family of Hecke Gr\"{o}ssencharacters

TL;DR

This paper investigates the symmetry type of a family of L-functions associated with Hecke Grossencharacters, providing asymptotic formulas, moment bounds, and evidence for orthogonal symmetry through analytic number theory and random matrix theory methods.

Contribution

It offers new asymptotic formulas for L-values, bounds on moments, and evidence supporting orthogonal symmetry of the family, using novel analytic techniques.

Findings

Asymptotic formula with power savings for average L-values

Upper bound for the second moment close to conjectured size

Computed the one level density with limited Fourier support

Abstract

The family of symmetric powers of an $L$-function associated with an elliptic curve with complex multiplication has received much attention from algebraic, automorphic and p-adic points of view. Here we examine one explicit such family from the perspectives of classical analytic number theory and random matrix theory, especially focusing on evidence for the symmetry type of the family. In particular, we investigate the values at the central point and give evidence that this family can be modeled by ensembles of orthogonal matrices. We prove an asymptotic formula with power savings for the average of these L-values, which reproduces, by a completely different method, an asymptotic formula proven by Greenberg and Villegas--Zagier. We give an upper bound for the second moment which is conjecturally too large by just one logarithm. We also give an explicit conjecture for the second moment of this family, with power savings. Finally, we compute the one level density for this family with a test function whose Fourier transform has limited support. It is known by the work of Villegas -- Zagier that the subset of these L-functions from our family which have even functional equations never vanish; we show to what extent this result is reflected by our analytic results.