A two variable Rankin–Selberg integral for $${\textrm{GU}}(2,2)$$ and the degree 5 L-function of $${\textrm{GSp}}_4$$

We give a two-variable Rankin–Selberg integral for generic cusp forms on \(\textrm{PGL}_4\) and \(\textrm{PGU}_{2,2}\) which represents a product of exterior square L-functions. As a residue of our integral, we obtain an integral representation on \(\textrm{PGU}_{2,2}\) of the degree 5 L-function of \({\textrm{GSp}}_4\) twisted by the quadratic character of E/F of cuspidal automorphic representations which contribute to the theta correspondence for the pair \((\textrm{P}{\textrm{GSp}}_4,\textrm{P}{\textrm{GU}}_{2,2})\).