Low-lying zeros of a family of quadratic Hecke $L$-functions via ratios conjecture

TL;DR

This paper uses the ratios conjecture to analyze the distribution of low-lying zeros of quadratic Hecke L-functions over Gaussian fields, confirming previous results under certain conditions.

Contribution

It derives lower order terms of the 1-level density for these zeros, extending understanding beyond main terms and aligning with prior hypotheses.

Findings

Results consistent with previous work under GRH

Lower order terms explicitly derived

Supports conjectural predictions for zero distributions

Abstract

In this paper, we apply the ratio conjecture of $L$-functions to derive the lower order terms of the $1$-level density of the low-lying zeros of a family quadratic Hecke $L$-functions in the Gaussian field. Up to the first lower order term, we show that our result is consistent with that obtained from previous work under the generalized Riemann hypothesis, when the Fourier transforms of the test functions are supported in $(-2, 2)$.