Let $G=SO(2,0)(\mathbb{R})$, a quasi-split group with signature (2,0). Let $e$ be an element in $O(2,0)(\mathbb{R}) \backslash SO(2,0)(\mathbb{R})$.

Let $\pi$ be an irreducible generic admissible representation of $G$. Let $\pi^e$ be $e$-conjugate of $\pi$ defined by $\pi^e(g):= \pi(e^{-1}g e)$.

Let $\sigma$ and $\sigma^e$ be the Weil group representation corresponding to $\pi$ and $\pi^e$ through local Langlands correspondence for $G=SO(2,0)(\mathbb{R})$.

Then I am wondering whether the Artin gamma function of $\sigma$ and $\sigma^e$ are equal.

Thanks in advance!