Growth and nonvanishing of restricted Siegel modular forms arising as Saito-Kurokawa lifts

We study the analytic behavior of the restriction of a Siegel modular form to $\mathbb{H}\times \mathbb{H}$ in the case that the Siegel form is a Saito-Kurokawa lift. A formula of Ichino links this behavior to a family of $G{L}_3\times G{L}_2$ $L$-functions.