A multivariate integral representation on ${\mathrm{G}\mathrm{L}}_2\times {\mathrm{G}\mathrm{S}\mathrm{p}}_4$ inspired by the pullback formula

We give a two variable Rankin-Selberg integral inspired by consideration of Garrett’s pullback formula. For a globally generic cusp form on ${\mathrm{G}\mathrm{L}}_2\times {\mathrm{G}\mathrm{S}\mathrm{p}}_4$, the integral represents the product of the $\mathrm{S}\mathrm{t}\mathrm{d}\times \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}$ and $\mathbf{1}\times \mathrm{S}\mathrm{t}\mathrm{d}$ $L$-functions. We prove a result concerning an Archimedean principal series representation in order to verify a case of Jiang’s first-term identity relating certain non-Siegel Eisenstein series on symplectic groups. Using it, we obtain a new proof of a known result concerning possible poles of these $L$-functions.