A Multi-variable Rankin-Selberg Integral for a Product of $GL_2$-twisted Spinor $L$-functions

We consider a new integral representation for $L(s_1, Π\times τ_1) L(s_2, Π\times τ_2),$ where $Π$ is a globally generic cuspidal representation of $GSp_4,$ and $τ_1$ and $τ_2$ are two cuspidal representations of $GL_2$ having the same central character. As and application, we find a new period condition for two such $L$ functions to have a pole simultaneously. This points to an intriguing connection between a Fourier coefficient of a residual representation on $GSO(12)$ and a theta function on $\widetilde{Sp}(16).$ A similar integral on $GSO(18)$ fails to unfold completely, but in a way that provides further evidence of a connection.

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