The second moment of derivatives of quadratic twists of modular $L$-functions

TL;DR

This paper establishes an asymptotic formula for the second moment of the first derivative of quadratic twists of modular L-functions, enhancing previous results by including three main terms.

Contribution

It introduces a new asymptotic formula with three main terms for the second moment, improving upon earlier work that only had one main term.

Findings

Derived an asymptotic formula with three main terms

Improved accuracy over previous results by Kumar et al.

Utilized large sieve inequality and shifted L-functions in the proof

Abstract

We prove an asymptotic formula for the second moment of the first derivative of quadratic twists of modular $L$-functions with three leading order main terms. It improves the previous result of Kumar et al. with the first main term. The proof is based on the large sieve type inequality established by Li, with a key input that we convert the problem into computing an asymptotic formula for the completed twisted modular $L$-functions with large shifts.