The second moment of $GL(3)\times GL(2)$ $L$-functions 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 Lindelöf 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 Lindelöf-consistent estimate for a sixth power moment of a family of GL(2) L-functions.

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• The Second Moment of ${\mathrm{GL}}_3\times {\mathrm{GL}}_2$ $L$-functions at Special Points
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• Rankin-Selberg periods for spherical principal series