  In this paper, we prove fundamental properties, such as multiplicativity, of
gamma factors defined by Rankin-Selberg integrals of Shimura type for pairs of
generic representations $(\pi, \tau)$ of $\mathrm{U}_{2\ell}(F)$ and
$\mathrm{GL}_n(E)$ 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 $\mathrm{GL}_{2\ell}(F)$ and
$\mathrm{GL}_n(F) \times \mathrm{GL}_n(F)$. As a corollary, we see that the
gamma factors arising from Langlands--Shahidi method and our gamma factors
coincide after a suitable normalization.
