The Langlands parameter of a simple supercuspidal representation: Symplectic groups

Let $\pi$ be a simple supercuspidal representation of the symplectic group $Sp_{2l}(F)$, over a $p$-adic field $F$. In this work, we explicitly compute the Rankin-Selberg $\gamma$-factor of rank-$1$ twists of $\pi$. We then completely determine the Langlands parameter of $\pi$, if $p \neq 2$. In the case that $F = \mathbb{Q}_2$, we give a conjectural description of the functorial lift of $\pi$, with which, using a recent work of Bushnell and Henniart, one can obtain its Langlands parameter.