The Ranking-Selberg integral on $\mathbf{G}\mathbf{S}\mathbf{p}(\mathbf{2})$ for square free levels

We explicitly compute the Rankin-Selberg type integral introduced by Piatetski-Shapiro over adeles for vector-valued Siegel cusp forms of square-free levels ${\mathrm{\Gamma }}_0(N)$. On the way, for particular test functions in the Bessel models of irreducible admissible representations, exact evaluations of the local zeta-integrals are given.