On the second integral moment of $L$-functions

Assume that the generalized Ramanujan conjecture holds on the automorphic $L$-function $L(s,\pi )$ on ${\text{GL}}_d$ over $\mathbb{Q}$ with $d\ge 3$, we can obtain a small log-saving non-trivial bound on the second integral moment of $L(1\mathrm{/}2+it,\pi )$. Specifically the bound $${\int }_T^{2T}\mid L(\frac{1}{2}+it,\pi ){\mid }^2\text{dd}t{\ll }_{\pi }\frac{T^{\frac{d}{2}}}{{\log }^{{\eta }_d}T}$$ holds for a small constant ${\eta }_d>0$.