Moments of L'(1/2) in the Family of Quadratic Twists

TL;DR

This paper establishes asymptotic formulas for moments of derivatives of GL(2) L-functions in quadratic twist families with fixed root number -1, under GRH, with implications for elliptic curves.

Contribution

It provides new asymptotic results for moments of L-function derivatives in quadratic twist families, including secondary terms and mixed moments, under GRH.

Findings

Asymptotic formula for the second moment with a secondary term

Asymptotic for the moment of two distinct modular forms

First moment with controlled weight and level dependence

Abstract

We prove the asymptotic formulae for several moments of derivatives of GL(2) L-functions over quadratic twists. The family of L-functions we consider has root number fixed to -1 and odd orthogonal symmetry. Assuming GRH we prove the asymptotic formulae for (1) the second moment with one secondary term, (2) the moment of two distinct modular forms f and g and (3) the first moment with controlled weight and level dependence. We also include some immediate corollaries to elliptic curves via the modularity theorem and the work of Gross and Zagier.