Pullbacks of Saito-Kurokawa Lifts and a Central Value Formula for Degree 6 $L$-Series

Abstract

We prove an explicit central value formula for a family of complex $L$-series of degree 6 for ${\text{GL}}_2\times {\text{GL}}_3$ which arise as factors of certain Garret-Rankin triple product $L$-series associated with modular forms. Our result generalizes a previous formula of Ichino involving Saito-Kurokawa lifts, and as an application we prove Deligne's conjecture stating the algebraicity of the central values of the considered $L$-series up to the relevant periods.