On the second moment of twisted $L$-functions

Abstract: Under various suitable assumptions, we describe a general method to obtain the log-saving upper bound for the second moment of standard twisted $L$-function in the $q$-aspect. Specifically, let $L(s, F)$ be a standard $L$-function of degree $d\geq3$, the bound [ \sideset{}{^*}{\sum}_{{\chi (\mod 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 $\eta>0$.