\frac{\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};\xi_p)}{\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}})}\right)\left(\prod_{p\not \in S,p<\infty}\frac{\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};Z)}{\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}})}\right)