On the Rankin-Selberg \(L\)-factors for \(\mathrm{SO}_5 \times \mathrm{GL}_2\) (Q6588717)

Let \(\pi\) and \(\tau\) be irreducible smooth generic representations of \(\mathrm{SO}_5\) and \( \mathrm{GL}_2\), respectively over a non-Archimedean local field. The paper shows that the \(L\)- and \(\varepsilon\)-factors attached to \(\pi\times\tau\) defined by the Rankin-Selberg integrals and the associated Weil-Deligne representation coincide. The proof is obtained by elaboration of the relation between the Rankin-Selberg integrals for \(\mathrm{SO}_5 \times \mathrm{GL}_2\) and Novodvorsky's local integrals for \(\mathrm{GSp}(4) \times \mathrm{GL}_2\).