Moments of derivatives of modular $L$-functions

Abstract

Let $f$ be an Hecke eigenform for the group $\Gamma_{0}(q)$ and $\chi_{d}$ be a primitive quadratic character of conductor $|d|$. In this article, we prove an asymptotic for the second moment of the derivative of $L(s, f \otimes \chi_{8d})$ at the central point $1/2$, which was previously known under GRH by Petrow \cite{petrow}.