Vanishing of Witten zeta function at negative integers

We introduce a new analytic method for studying Witten zeta function of a root system $\mathrm{\Phi }$, based on a refined manipulation of an integral representation involving the Hurwitz zeta function. As an application, we prove high-order vanishing at negative even integers. This technique also describes non-trivially, the arithmetic nature of the leading term, in which the highest root of $\mathrm{\Phi }$ makes a surprising appearance.