The second moment of the $G{L}_3$ standard $L$-function on the critical line

Abstract

We obtain a strong bound on the second moment of the $G{L}_3$ standard $L$-function on the critical line. The method builds on the recent work of Aggarwal, Leung, and Munshi which treated shorter intervals. We deduce some corollaries including an improvement on the error term in the Rankin-Selberg problem, and on certain subconvexity bounds for $G{L}_3\times G{L}_2$ and $G{L}_3$ $L$-functions. As a byproduct of the method of proof, we also obtain an estimate for an average of shifted convolution sums of $G{L}_3$ Fourier coefficients.