I am interested on integrals of "Tate style", say of the form $$\int_{F^\times} \Phi(a) \kappa(a) |a|^s d^\times a$$
Here, $F$ is a local field, $|\cdot|$ the associated absolute value, $\Phi$ a Schwatz-Bruhat function and $\kappa$ a finite function (i.e. continuous and such that its translates span a finite-dimensional vector space). Is it clear that this function is a rational function in $q^{-s}$? Do we know what it looks like?
Such integrals often appear, and it seems natural to consider subintegrals on $a$'s of fixed valuation. But unlike in Tate's thesis for instance, here $\kappa$ is not really a character, and $\phi$ is not totally invariant, so that I end up with $$q^{-ns}\int_{O^\times} \Phi(\omega^n a) \kappa(\omega^n a) |a|^s d^\times a$$ but I am not able to evaluate further or sum over $n$.