Kurokawa–Ochiai vanishing conjecture for Witten zeta functions

Let $G$ be a compact Hausdorff topological group, and define its Witten zeta function by

$$\zeta_G(s):=\sum_{\rho}\frac{1}{(\operatorname{dim}\rho)^s},$$

where the sum runs over the finite-dimensional irreducible representations of $G$. Kurokawa–Ochiai vanishing conjecture. If $G$ is infinite, then

$$\zeta_G(-2)=0.$$

This conjecture is motivated by examples including $G=\operatorname{SU}(3)$ and $G=\operatorname{SL}_2(\mathbb{Z}_p)$. Substantial progress is known for $p$-adic groups, whereas little was known for Lie groups before the paper; the paper proves a stronger vanishing statement for compact simply connected Lie groups.