I have a question about the local epsilon factor for $\operatorname{GL}_1$. If $k$ is a $p$-adic field with normalized absolute value, $\chi$ is an unramified character of $k^{\ast}$, $\psi$ is a nontrivial additive character of $k$, and $dx$ is the self dual measure on $k$ with respect to $\psi$, then
$$\epsilon(s,\chi,\psi) = \epsilon(0,\chi,\psi)q^{-ns}$$ where $a$ is the level of $\psi$ (that is, the integer $a$ such that $\psi$ is trivial on $\mathfrak p^{-a}$, but not on $\mathfrak p^{-a-1}$). This is equation (3.2.6.1) in Tate's article in the Corvallis proceedings. In particular, $\epsilon(s,\chi,\psi)$ is constant whenever $\chi$ is unramified, and $\psi$ is trivial on $\mathcal O_k$ but not on $\mathfrak p^{-1}$.
Now let $k$ be a number field, $\chi = \otimes \chi_v$ be a character of $\mathbb A_k^{\ast}/k^{\ast}$, and $\psi = \otimes_v \psi_v$ a nontrivial character of $\mathbb A_k/k$. The global epsilon factor is defined by
$$\epsilon(s,\chi) = \prod\limits_v \epsilon(s,\chi_v,\psi_v)$$ I understand that this should be a finite product. It follows from the continuity of $\chi$ and $\psi$ that almost all $\chi_v$ are unramified, and almost all $\psi_v$ are trivial on $\mathcal O_{k_v}$. But is it the case that almost all the $\psi_v$ are moreover nontrivial on $\mathfrak p_v^{-1}$?
I guess one way to see this would be to construct a "standard" nontrivial character on $\mathbb A_k/k$ by means of canonical characters of the local fields $k_v$, as Tate does in his thesis. Then use the fact that every other character is equal to $\psi(ax)$ for $a \in k$. But can this be seen directly by an argument with topology?