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

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.

15 pages. Revisions have been made with accordance to the referee's report and it is to appear in Compos. Math

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• One level density of low-lying zeros of quadratic and quartic Hecke $L$-functions
• Bounds for moments of cubic and quartic Dirichlet $L$-functions
• One-level density and non-vanishing for cubic $L$-functions over the Eisenstein field
• Weighted low-lying zeros of L-functions attached to Siegel modular forms
• Lower order terms of the one level density of a family of quadratic Hecke $L$-functions
• Moments and One level density of certain unitary families of Hecke {$L$}-functions
• Moments and One level density of sextic Hecke $L$-functions of $\mathbb{Q}(\omega )$