The $1$-Level Density for Zeros of Hecke $L$-Functions of Imaginary Quadratic Number Fields of Class Number $1$

TL;DR

This paper investigates the distribution of zeros of Hecke L-functions associated with imaginary quadratic fields of class number one, providing both conditional and unconditional results that align with predictions from random matrix theory.

Contribution

It offers a conditional asymptotic formula for the 1-level density of zeros using the Ratios Conjecture and proves an unconditional result matching RC predictions for limited test function support.

Findings

Conditional 1-level density asymptotics derived from Ratios Conjecture.

Unconditional 1-level density results consistent with RC predictions for Fourier support in (-1,1).

Results support the Katz-Sarnak Density Conjecture for this family.

Abstract

Let $\mathbb{K} = \mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic number field of class number $1$ and $\mathcal{O}_{\mathbb{K}}$ its ring of integers. We study a family of Hecke $L$-functions associated to angular characters on the non-zero ideals of $\mathcal{O}_{\mathbb{K}}$. Using the powerful Ratios Conjecture (RC) due to Conrey, Farmer, and Zirnbauer, we compute a conditional asymptotic for the average $1$-level density of the zeros of this family, including terms of lower order than the main term in the Katz-Sarnak Density Conjecture coming from random matrix theory. We also prove an unconditional result about the $1$-level density, which agrees with the RC prediction when our test functions have Fourier transforms with support in $(-1,1)$.