Second moment of degree three $L$-functions

Abstract

Let $F$ be a Hecke-Maa\ss\ cusp form for $\mathrm{SL}(3,\mathbb{Z})$. We obtain the first non-trivial upper bound of the second moment of $L(F,s)$ in $t$-aspect: $$\int_{T}^{2T}|L(F,1/2+it)|^2 dt\ll_{F,\varepsilon} T^{3/2-3/32+\varepsilon}.$$ Immediate corollaries include improvements over the existing results on the subconvexity bound for self-dual $\mathrm{GL}(3)$ $L$-functions in the $t$-aspect and for self-dual $\mathrm{GL}(3)\times \mathrm{GL}(2)$ $L$-functions in the $\mathrm{GL}(2)$ spectral aspect, the error term in the Rankin-Selberg problem, and the zero density estimate for $\mathrm{GL}(3)$ $L$-functions.