The second moment of $GL(3) \times GL(2)$ $L$-functions at special points

TL;DR

This paper establishes Lindelöf-consistent bounds for the second and sixth moments of certain GL(3) x GL(2) L-functions at special points, advancing understanding of their size and distribution.

Contribution

It provides the first Lindelöf-consistent estimate for the sixth moment of GL(2) L-functions and refines bounds for the second moment of GL(3) x GL(2) L-functions at special points.

Findings

Second moment estimate aligns with Lindelöf Hypothesis.

First sixth moment bound consistent with Lindelöf.

Improves understanding of L-function moments at special points.

Abstract

For a fixed SL(3, Z) Maass form g, we consider the family of L-functions L(g \times u_j, s) where u_j runs over the family of Hecke-Maass cusp forms on SL(2,Z). We obtain an estimate for the second moment of this family of L-functions at the special points \half + it_j consistent with the Lindel\"{o}f Hypothesis. We also obtain a similar upper bound on the sixth moment of the family of Hecke-Maass cusp forms at these special points; this is apparently the first occurrence of a Lindel\"{o}f-consistent estimate for a sixth power moment of a family of GL(2) L-functions.