Working over a $p$-adic field with absolute value $|\cdot|$, let $\chi$ be a character on ${\mathfrak o}^\times$ with conductor $n\ge 1$, meaning that $n$ is the smallest integer such that $\chi$ is trivial on $U_n=1+{\mathfrak p}^n$. I'm trying to calculate the integral, with ${\rm Re}(s)\gg 1$, $J(\chi,|\cdot|^s)=\int_{\mathfrak o^\times}\chi(x)|x-1|^s\ dx$ Note that if we replace $|\cdot|^s$ with a character $\chi'$ with conductor $n$, this would reduce to the typical Jacobi sum, and we'd have the formula $J(\chi,\chi')={G(\chi,\psi)G(\chi',\psi)\over G(\chi\chi',\psi)}$ where $\psi$ is an additive character with conductor $n$ and $G(\chi,\psi)$ is the Gauss sum $G(\chi,\psi)=\int_{\mathfrak o^\times}\chi(x)\psi(x)\ dx$ Of course, this formula wouldn't work for $\chi'$ unramified.
Is anyone aware of how to calculate this sort of Jacobi sum?