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

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.