A multi-variable Rankin–Selberg integral for a product of $$GL_2$$-twisted Spinor L-functions

Abstract

We consider a new integral representation for $$L(s_1, \Pi \times \tau _1) L(s_2, \Pi \times \tau _2),$$ where $$\Pi $$ is a globally generic cuspidal representation of $$GSp_4,$$ and $$\tau _1$$ and $$\tau _2$$ 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 Sp(16). A similar integral on GSO(18) fails to unfold completely, but in a way that provides further evidence of a connection.