On gamma factors of Rankin--Selberg integrals for ${\mathrm{U}}_{2\mathrm{\ell }}\times {\mathrm{R}\mathrm{e}\mathrm{s}}_{E\mathrm{/}F}{\mathrm{G}\mathrm{L}}_n$

In this paper, we prove the fundamental properties of gamma factors defined by Rankin-Selberg integrals of Shimura type for pairs of generic representations $(\pi ,\tau )$ of ${\mathrm{U}}_{2\mathrm{\ell }}(F)$ and ${\mathrm{G}\mathrm{L}}_n(E)$ for a local field $F$ of characteristic zero and a quadratic extension $E$ of $F$. We also prove similar results for pairs of generic representations $(\pi ,{\tau }_1\times {\tau }_2)$ of ${\mathrm{G}\mathrm{L}}_{2\mathrm{\ell }}(F)$ and ${\mathrm{G}\mathrm{L}}_n(F)\times {\mathrm{G}\mathrm{L}}_n(F)$. As a corollary, we prove that the gamma factors arising from Langlands–Shahidi method and our gamma factors coincide.