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 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 $d\ge 3$. Under these foundational hypotheses. the bound $$\text{sideset}{}^{\ast }{\sum }_{\chi (\mathrm{m}\mathrm{o}\mathrm{d}q)}\mid L(\frac{1}{2},F\times \chi ){\mid }^2{\ll }_{F,\eta }\frac{q^{\frac{d}{2}}}{{\log }^{\eta }q}$$ holds for some small $\eta >0$