I ran into this question when trying to compute the Atkin-Lehner pseudo-eigenvalue of newforms. Let $k \geq 2$, let $\omega$ be a Dirichlet character modulo $N$ and let $f \in S_k(N,\omega)$ be a newform. We use $\pi_f$ to denote the corresponding representation of $\text{GL}_2(\mathbb{A}_\mathbb{Q})$, and we have $\pi_f = \prod_v \pi_{f,v}$. Then there should be a function $\epsilon(\pi, s)$ such that
$L(\pi, s) = \epsilon(\pi,s)L(\pi^{\vee}, 1-s)$. (1)
Further, fix an additive character $\psi$ of $\mathbb{A}$, then we have the decomposition
$\epsilon(\pi,s) = \prod_v \epsilon(\pi_v,s,\psi_v)$. (2)
Now I am aware of two set of conventions of the local epsilon factors, one from Gelbart and the other from Bushnell-Henniart, but I wasn't able to use them to compute $\epsilon$, and their definitions are different. I'll explain what I tried:
1.I started from GL(1). If $\chi$ is an unramified character of $(\mathbb{Q}_p)^{\times}$, then Gelbart defined $\epsilon(\chi,s) = 1$, where as [BH] defines $\epsilon(\chi,s) = p^{1/2-s}\chi(p)^{-1}$. When $p \nmid N$, the local representation $\pi_{f,p}$ is unramified principal series and the first definition would yield $\epsilon(\pi_p) = 1$ and second one would yield $\epsilon_p = \omega(p)$. This makes me confused about the [BH] definition, since there will be infinitely many $p$ with $\epsilon_p \neq 1$, so I don't know how to make sense of the infinite product over all primes. Also it seems to me that [BH] did not treat the case $v = \infty$.
- I was not able to find the formula for the local constants in Gelbart when the local represenation is special or supercuspidal.
Overall, I'm asking for a definition of these epsilon constants that would make (1) and (2) hold. I've already spent several hours online looking for references without success, so I'd much appreciate experts' help and guidance!