Weighted first moments of some special quadratic Dirichlet $L$-functions

TL;DR

This paper derives asymptotic formulas for the weighted first moments of central values of certain quadratic Dirichlet L-functions and proves their non-vanishing at s=1/2, focusing on primes splitting in a quadratic field.

Contribution

It provides new asymptotic formulas for moments of quadratic Dirichlet L-functions with conductors involving split primes and establishes their non-vanishing at the critical point.

Findings

Asymptotic formulas for weighted first moments of quadratic Dirichlet L-functions.

Non-vanishing results at s=1/2 for these L-functions.

Focus on conductors with primes splitting in a quadratic field.

Abstract

In this paper, we obtain asymptotic formulas for weighted first moments of central values of families of primitive quadratic Dirichlet $L$-functions whose conductors comprise only primes that split in a given quadratic number field. We then deduce a non-vanishing result of these $L$-functions at the point $s=1/2$.