Jacquet's local converse conjecture for general linear groups

Let $F$ be a non-Archimedean local field of characteristic $0$. Let $\mathrm{GL}_l(F)$ denote the general linear group, and let $\pi_1$ and $\pi_2$ be irreducible generic representations of $\mathrm{GL}_l(F)$ with the same central character. For an irreducible generic representation $\tau$ of $\mathrm{GL}_n(F)$ and a nontrivial additive character $\psi$ of $F$, write $\gamma(s,\pi_i\times\tau,\psi)$ for the local twisted Rankin–Selberg gamma factor.

Jacquet's local converse conjecture. If

$$\gamma(s,\pi_1\times\tau,\psi)=\gamma(s,\pi_2\times\tau,\psi)$$

as functions of the complex variable $s$ for every irreducible generic representation $\tau$ of $\mathrm{GL}_n(F)$ with $1\leq n\leq\left[\frac{l}{2}\right]$, then $\pi_1\cong\pi_2$.

The conjecture is a converse theorem characterizing generic representations of general linear groups through their twisted Rankin–Selberg gamma factors. It was proved independently by Chai and by Jacquet and Liu using different analytic methods.