Euler characteristics of quotient stack/orbifold

Question

Let $X$ be a scheme. Consider the quotient stack $[X/G]$ for $G$ a finite group. This is a Deligne--Mumford stack.

Is there a definition of a version of euler characteristic of DM stacks so that $$\chi([X/G]) = \chi(X)/|G|$$

I know something called orbifold Euler characteristic exists. I think this doesn't work. Indeed taking $X=\mathrm{spec}(k)$ and $G$ the cyclic group of order $p$ then the orbifold Euler characteristic of $[X/G]$ is, according to my reading of the wikipedia page, p.

Answer 1

I have learned about three Euler characteristics one can associate to an orbifold/DM stack. Comments/corrections welcome.

For an example consider $\mathbb{P}^1$ quotient by the action of the subgroup of $p^\mathrm{th}$ roots of unity. Let $X$ be an orbifold.

1. Note $X$ admits a locally closed stratification $$X = \bigcup_H Y_H$$ where $Y_H$ is the locally closed subset of points in $X$ with automorphism group $H$. Then Satake defined the Euler-Satake characteristic $$\chi(X) = \sum_H \frac{\chi_c(Y_H)}{|H|}.$$ In this equation $\chi_c$ is Euler characteristic with compact support. In our example the answer is $2/p$. More generally one has for any complex scheme $X$ and finite group $G$, $$\chi(X/G)= \frac{\chi(X)}{|G|}.$$ I think its pretty clear why this is an interesting definition.
2. The stringy orbifold Euler Characteristic is a gadget that physicists came up with when studying string theory. This is the Wikipedia definition so I will leave you to look that up. I will comment only that in our example the Euler characteristic is $2p$.
3. The coarse moduli space Euler characteristic is the Euler characteristic of the coarse moduli space of the orbifold $X$. This is the usual Euler characteristic of a topological space. In our example this is $2$