Rankin-Selberg convolution for the Duke-Imamoglu-Ikeda lift

For two Hecke eigenforms $h_1$ and $h_2$ in the Kohnen plus space of half-integral weight, let $I_n(h_1)$ and $I_n(h_2)$ be the Duke-Imamoglu-Ikeda lift of $h_1$ and $h_2$, respectively, which are Siegel cusp forms with respect to $S{p}_n(\text{ZZ})$. Moreover, let $E_{n\mathrm{/}2+1\mathrm{/}2}$ be the Cohen Eisenstein series of weight $n\mathrm{/}2+1\mathrm{/}2$. We then express the Rankin-Selberg convolution $R(s,I_n(h_1),I_n(h_2))$ of $I_n(h_1)$ and $I_n(h_2)$ in terms of a certain Dirichlet series $D(s,h_1,h_2,E_{n\mathrm{/}2+1\mathrm{/}2})$, which is similar to the triple convolution product of $h_1,h_2$ and $E_{n\mathrm{/}2+1\mathrm{/}2}$. We apply our formula to mass equidistribution for the Duke-Imamoglu-Ikeda lift assuming the holomorphy of $D(s,h_1,h_1,E_{n\mathrm{/}2+1\mathrm{/}2})$.