Twisted first moment of quadratic and quadratic twist $L$-functions

TL;DR

This paper evaluates the twisted first moment of central values involving quadratic Dirichlet and modular L-functions, providing insights into their average behavior and interactions.

Contribution

It introduces a new method to compute the twisted first moment of these L-functions, advancing understanding of their joint distribution.

Findings

Explicit formula for the twisted first moment

Asymptotic estimates for the average values

Insights into the distribution of quadratic twists

Abstract

We evaluate the twisted first moment of central values of the product of a quadratic Dirichlet $L$-function and a quadratic twist of a modular $L$-function.