The second moment of twisted modular $L$-functions and Dirichlet $L$-functions at the central point

Abstract

We prove an asymptotic formula with a power-saving error term for the second moment $$ \sum_{q \in \mathcal{Q}} \; \; \sideset{}{^\flat}\sum_{χ\bmod q} \left| L\big( \tfrac{1}{2} , χ\big) L\big(\tfrac{1}{2},f\otimes χ)\right|^2 $$ of the twisted $\mathrm{GL}(2)$ {$L$}-function and the Dirichlet {$L$}-function at the central point under the assumption of Selberg's eigenvalue conjecture. Here $f$ is a fixed Hecke holomorphic cusp form for $\mathrm{SL}(2,\mathbb{Z})$ and the sum over $\chi$ runs over all primitive even Dirichlet characters modulo $q$, and $$\mathcal{Q}=\left\{q=q_1{q}_2\asymp Q:q_1\le Q_1,q_2\le Q_2,Q_2\asymp Q^{{\delta }_1},(q,6)=1,(q_1,q_2)=1\right\}$$ with $0<{\delta }_1<0.0004$.