Moments and One level density of quadratic Hecke $L$-functions of $\mathbb{Q}(\omega)$

TL;DR

This paper explicitly evaluates quadratic Hecke Gauss sums over , studies moments and non-vanishing of quadratic Hecke L-values, and analyzes low-lying zeros to understand their distribution.

Contribution

It provides explicit evaluations of quadratic Hecke Gauss sums and establishes new results on moments, non-vanishing, and low-lying zeros of quadratic Hecke L-functions over .

Findings

Explicit evaluation of quadratic Hecke Gauss sums.

Quantitative non-vanishing results for L-values.

One level density result for low-lying zeros.

Abstract

In this paper, we evaluate explicitly certain quadratic Hecke Gauss sums of $\mathbb{Q}(\omega), \omega=\exp \left( \frac {2\pi i}{3}\right)$. As applications, we study the moments of central values of quadratic Hecke $L$-functions of $\mathbb{Q}(\omega)$, and establish quantitative non-vanishing result for the $L$-values. We also establish an one level density result for the low-lying zeros of quadratic Hecke $L$-functions of $\mathbb{Q}(\omega)$.