One level density of low-lying zeros of families of $L$-functions

TL;DR

This paper establishes one level density results for low-lying zeros in specific families of $L$-functions, including those from holomorphic Hecke eigenforms and Dirichlet $L$-functions of quadratic, cubic, and quartic characters.

Contribution

It provides new density results for low-lying zeros in families of $L$-functions associated with Hecke eigenforms and Dirichlet characters, expanding understanding of their zero distributions.

Findings

Proved one level density results for holomorphic Hecke eigenforms twisted with quadratic characters.

Established density results for cubic and quartic Dirichlet $L$-functions.

Enhanced understanding of the distribution of low-lying zeros in these families.

Abstract

In this paper, we prove some one level density results for low-lying zeros of families of $L$-functions. More specifically, the families under consideration are that of $L$-functions of holomorphic Hecke eigenforms of level 1 and weight $k$ twisted with quadratic Dirichlet characters and that of cubic and quartic Dirichlet $L$-functions.