Let B be a fixed matrix over non-archimedean field F. Let X be a random invertible matrix over $O_F$(Entries in $O_F$ and The determinant is unit), $O_F$ is the integer ring of F. What is the Expectation of the p-adic absolute value of determinant of B+X.
Explicitly, what I am asking is
$$f(B,s)=\int_{\mathrm{GL}_n(O_F)}|\det(B+X)|^s\mathrm{d}X$$
Where $B=\mathrm{diag}(\pi^{a_1},\cdots,\pi^{a_n})$ Here by $\mathrm{diag}$ I mean diagonal matrix. And $a_1,\cdots a_n$ are integers. $\pi$ is the uniformizer of $O_F$.
Is there any easy expression of $f(B,s)$ in terms of $a_1,\cdots,a_n$? or any method to compute this integral?