Why is the Orbifold Euler characteristic of $M_{1, 1}$ equal to $\zeta(-1) $

Question

The orbifold euler characteristic of $M_{1, 1}$= $PSL(2, \mathbb{Z}) $/$\mathbb{H}$is equal to -1/12. This is due to the fact that $M_{1,1}$ has one two-cell with $\mathbb{Z_{2}}$ automorphism, two one-cells with $\mathbb{Z_{2}}$ automorphism and two points with automorphisms$\mathbb{Z_{4}}$ and$\mathbb{Z_{6}}$. Is there any alternative perspective to the analytic contiuation of the Riemann Zeta Function that this might indicate?