A note on a Hecke type integral for $\mathrm{S}\mathrm{p}(2n)\times \mathrm{G}\mathrm{L}(1)$

Let $\pi$ be an irreducible cuspidal automorphic generic representation of ${\mathrm{S}\mathrm{p}}_{2n}(\mathbb{A})$ and let $\chi :F^{\times }\mathrm{\backslash }{\mathbb{A}}^{\times }\to {\mathbb{C}}^{\times }$ be a unitary idele class character. In this note, we present a Rankin-Selberg integral of Hecke type for the twisted standard partial $L$-function $L^S(s,\pi \times \chi )$ of degree $2n+1$.