Weighted one-level density of low-lying zeros of Dirichlet $L$-functions

TL;DR

This paper computes the weighted one-level density of low-lying zeros of Dirichlet L-functions, confirming conjectures about their distribution when weighted by special values.

Contribution

It introduces a novel weighted density computation for Dirichlet L-functions and verifies key conjectures in the field.

Findings

Confirmed Fazzari's conjecture for the weighted density

Verified the first author's conjecture on the same density

Provided explicit formulas for the weighted one-level density

Abstract

In this paper, we compute the one-level density of low-lying zeros of Dirichlet $L$-functions in a family weighted by special values of Dirichlet $L$-functions at a fixed $s \in [1/2, 1)$. We verify both Fazzari's conjecture and the first author's conjecture on the weighted one-level density for our family of $L$-functions.