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We consider a Rankin-Selberg integral which unfolds to a Shalika period forthe unitary similitude group $\mathrm{GU}_{2,2}$ attached to a quadraticextension $E/\mathbf{Q}$. We show that it represents the degree 5 $L$-functionof $\mathrm{GSp}_4$ twisted by the quadratic character of $E/\mathbf{Q}$ ofcuspidal automorphic representations which contribute to the thetacorrespondence for the pair $(\mathrm{PGSp}_4,\mathrm{PGU}_{2,2})$. Along theway, we obtain a local criterion to determine the multiplicity of local Shalikamodels for unramified representations of $\mathrm{GU}_{2,2}$ at unramifiedinert primes and prove a Casselman-Shalika formula for them.

Shalika models on $\mathrm{GU}_{2,2}$ and the degree 5 $L$-function of $\mathrm{GSp}_4$