On the Second Moment of Twisted Higher Degree $L$-functions

Abstract

Assuming the Ramanujan conjecture, the zero density estimate and some subconvexity type bound, we describe a general method to obtain the log-saving upper bound for the second moment of standard twisted higher degree $L$-function in the $q$-aspect. Specifically, let $L(s, F)$ be a standard $L$-function of degree $d\geq3$. Under these foundational hypotheses. the bound \[ \sideset{}{^*}{\sum}_{\chi \pmod q}\Big|L\big(\frac{1}{2}, F\times \chi \big)\Big |^2\ll_{F,\eta} \frac{q^{\frac{d}{2}}}{\log^{\eta}q} \] holds for some small $\eta>0$