A $p$-adic "Jacobi sum" with an unramified character 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?
 A: I think I've figured out how to do it. It's essentially the same argument as in the "classical" case, just interpreted slightly differently. Here's the sketch, which may lose some change-of-measure constants.
$$J(\chi,|\cdot|^s)=\int_{\mathfrak o^\times}\chi(x)|x-1|^s\ dx$$
Extend $\chi$ by zero to all of $k$ and apply Fourier inversion to get
$$=\int_{\mathfrak o^\times}\int_k |x-1|^s\psi(-ax)\int_{\mathfrak o^\times}\chi(y)\psi(ay)\ dy\ da\ dx$$
where $\psi$ has conductor $0$. The integral over $y$ vanishes unless ${\rm ord}(a)=-n$, so the integral becomes
$$=\int_{\mathfrak o^\times}\int_{\mathfrak o^\times}|x-1|^s\psi(-\varpi^{-n}ux)\int_{\mathfrak o^\times}\chi(y)\psi(\varpi^{-n}uy)\ dy\ du\ dx$$
$$=\int_{\mathfrak o^\times}\int_{\mathfrak o^\times}|x-1|^s\psi(-\varpi^{-n}ux)\bar\chi(u)\int_{\mathfrak o^\times}\chi(y)\psi(\varpi^{-n}y)\ dy\ du\ dx$$
$$=G(\chi,\psi_{\varpi^{-n}})\int_{k}\int_{\mathfrak o^\times}{\bf 1}_{\mathfrak o^\times}(x)|x-1|^s\psi(-\varpi^{-n}ux)\bar\chi(u)\ du\ dx$$
$$=G(\chi,\psi_{\varpi^{-n}})\int_{k}\int_{\mathfrak o^\times}{\bf 1}_{\mathfrak o^\times}(1+x)|x|^s\psi(-\varpi^{-n}ux)\psi(-\varpi^{-n}u)\bar\chi(u)\ du\ dx$$
$$=q^{-ns-1}G(\chi,\psi_{\varpi^{-n}})\int_{k}\int_{\mathfrak o^\times}{\bf 1}_{\mathfrak o^\times}(1-\varpi^{n}u^{-1}x)|x|^s\psi(x)\bar\psi(\varpi^{-n}u)\bar\chi(u)\ du\ dx$$
Now, as a function of $x$, the characteristic function ${\bf 1}_{\mathfrak o^\times}(1-\varpi^{n}u^{-1}x)$ differs from ${\bf 1}_{\mathfrak o}(\varpi^{n}x)$ only on a set of measure zero, so we have 
$$J(\chi,|\cdot|^s)=q^{-ns-1}\big|G(\chi,\psi_{\varpi^{-n}})\big|^2\int_{\varpi^{-n}\mathfrak o}|x|^s\psi(x)\ dx$$
This final integral can be calculated using the well-known formula for $\int_{|x|=p^i}\psi(x)\ dx$, giving (if I did everything correctly),
$$J(\chi,|\cdot|^s)=q^{-ns-1}\big|G(\chi,\psi_{\varpi^{-n}})\big|^2{2-q^s\over 1-q^{-s}}$$
