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In this article, we construct a family of integrals which represent theproduct of Rankin-Selberg $L$-functions of $\mathrm{GL}_{l}\times\mathrm{GL}_m$ and of $\mathrm{GL}_{l}\times \mathrm{GL}_n $ when $m+n<l$. When$n=0$, these integrals are those defined by Jacquet--Piatetski-Shapiro--Shalikaup to a shift. In this sense, these new integrals generalizeJacquet--Piatetski-Shapiro--Shalika's Rankin-Selberg convolution integrals. Westudy basic properties of these integrals. In particular, we define local gammafactors using this new family of integrals. As an application, we obtain a newproof of Jacquet's local converse conjecture using these new integrals.

Product of Rankin-Selberg convolutions and a new proof of Jacquet's local converse conjecture