On the second moment of twisted higher degree $L$-functions

Assuming the Ramanujan conjecture, the zero density estimate and some subconvexity type bound, we describe a general method of obtaining the log-saving upper bound for the second moment of a standard twisted higher degree $L$-function in the $q$-aspect. Specifically, let $L(s, F)$ be a standard $L$-function of degree $d\geq 3$. Under the above hypotheses, the bound \[ \sideset{}{^*}{\sum }_{{\chi \,({\rm mod}\, q)}}|L({1}/{2}, F\times \chi)|^2\ll_{F,\eta } \frac{q^{{d}/{2}}}{\log^{\eta }q} \] holds for some small $\eta \gt 0$.