Let $F$ be a nonarchimedean local field, $\chi$ a ramified character of $F^{\ast}$, $\psi$ a nontrivial character of $F$, and $dx$ a Haar measure on $F$ with respect to which the Fourier transform is defined. The local Tate epsilon factor $\epsilon(\chi,\psi,dx)$ is defined via the local functional equation $$\epsilon(\chi,\psi,dx) \int\limits_{F^{\ast}} f(x)\chi(x) d^{\ast}x = \int\limits_{F^{\ast}} \hat{f}(x) \chi^{-1}(x)|x| d^{\ast}x$$ for all suitable functions $f \in C_c(F)$. Taking $f$ to be a characteristic function of a suitable open compact subgroup of $F$, one can realize $\epsilon(\chi,\psi,dx)$ as a "principal value integral"

$$\epsilon(\chi,\psi,dx) = \int\limits_{F} \chi^{-1}(x) \overline{\psi(x)} d^{\ast}x \tag{1}$$ in the sense that the right hand side becomes constant and equal to the left hand side when evaluated over $\mathfrak p_F^{-k}$ for sufficiently large $k$.

Let $f$ and $d$ be the conductors of $\chi$ and $\psi$. Tate's article in Corvallis II describes $\epsilon(\chi,\psi,dx)$ more specifically as the right hand side of (1) where the integral is only taken over the annulus of $x \in F^{\ast}$ with $\operatorname{ord}(x) = -d-f$.

Here $\operatorname{ord}(c) = d +f$.

My question is, what can be said about the summands

$$\int\limits_{\pi^n \mathcal O^{\ast}} \chi^{-1}(x) \psi(x)dx$$

Are they all equal to zero except for $n = -d-f$? In Tate's thesis, he shows they are zero for $n > -d-f$. I was wondering whether the ones for $n < -d-f$ are individually zero, or whether they just cancel each other out in the long term.