Let $F$ be a number field. For an irreducible cuspidal automorphic representation $\pi$ of $\operatorname{GL}_n(\mathbb{A}_F)$, we say that $\pi$ is symplectic (or orthogonal) if $L(s,\pi,\wedge^{2})$ (or $L(s,\pi,Sym^2)$) has a pole at $s=1$.

I am wondering that if $\pi=\otimes \pi_v$ is symplectic (or orthogonal), then $\pi_v$‘s are also symplectic (or orthogonal) for all places $v$?

(Here, $\pi_v$ is symplectic (or orthogonal) means that its corresponding Weil-Delign group representation by local Langlands correspondence is such type.)

Any comments are welcome!