Let $\mathcal{A_g}$ be the moduli space of principally polarized abelian varieties over $\mathbb C$, $\mathcal{M_g}$ be the moduli space of smooth projective curves of genus $g$ over $\mathbb C$, and we regard both as Deligne-Mumford stacks or orbifolds. By computations we know their Euler characteristics :

$\chi(\mathcal{A_g})=\chi(\mathcal M_{1})\chi(\mathcal M_{2})...\chi(\mathcal M_{g})=\prod_{k=1}^g \zeta(1-2k)$.

Is there a conceptual explanation of this using Torelli map? Can we update it to an identity of Poincare polynomials?