Explanation for $\chi(\operatorname{SL}_2(\mathbb{Z})) = -1/12$ with zeta function $\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$One can compute the (group cohomological) Euler characteristic of $\SL_2(\mathbb{Z})$ via
$$ \chi(\SL_2(\mathbb{Z})) = \chi(\mathbb{Z}/2) \cdot \chi(\PSL_2(\mathbb{Z})) = \frac{1}{2}\cdot \left(\frac{1}{2} + \frac{1}{3} - 1\right) = -\frac{1}{12} = \zeta(-1) $$
(since $ \PSL_2(\mathbb{Z}) \cong \mathbb{Z}/2 * \mathbb{Z}/3 $).
Alternatively it follows as $\SL_2(\mathbb{Z})$ has $F_2$ as an index-$12$ subgroup.
I was wondering whether the connection with $\zeta$ is coincidental.
The only (far fetched) connection I could come up with is that the (representation theoretic) zeta function of $\SL_2(\mathbb{C})$ is the usual $\zeta$, as its irreducible representations have dimensions $1, 2, 3, \dotsc$. (Here $\zeta_G(s) = \sum_{V\in \operatorname{Irr}(G)} \dim(V)^{-s}$.)
 A: (Expanding my comment into an answer)
It is not a coincidence. Relating the Euler characteristic of certain arithmetic groups to the Zeta function is a theorem due to Harder [1] from 1971. It is expanded on in Brown's "Cohomology of Groups", Chapter IX.8.
Taken from Gruenberg's AMS review of Brown's book is the following overview of the idea:

Let $G$ be an algebraic subgroup of $\operatorname{GL}_n$ defined over $\mathbb{Q}$ and $\Gamma$ an arithmetic subgroup of $G(\mathbb{Q})$. Then $\Gamma$ is a discrete subgroup of the Lie group $G(\mathbb{R})$. If $K$ is a maximal compact subgroup of $G(\mathbb{R})$, then $X = G(\mathbb{R})/K$ is diffeomorphic to a euclidean space of dimension say $d$. 
[...] 
Number theory enters through the work of Harder (1971). The Gauss-Bonnet measure on $X$ lifts to a unique invariant measure $\mu$ on $G(\mathbb{R})$. Harder proved the deep theorem that $\chi(\Gamma) = \mu(G(\mathbb{R})/\Gamma)$. This leads to an explicit fromula [sic.] for $\chi(G(\mathbb{Z}))$ in terms of values of the zeta-function. For example, $\chi(\operatorname{SL}_2(\mathbb{Z})) = \zeta(-1)$ and since $\zeta(-1) = -1/12$, this gives a third way of arriving at the Euler characteristic of $\operatorname{SL}_2(\mathbb{Z})$.

Edit: One can find more values using this method, of course. Some are given in Brown's book (p. 255-256). For example, we have
$$
\chi(\operatorname{SL}_n(\mathbb{Z})) = \prod_{k=2}^n \zeta(1-k)
$$
and
$$
\chi(\operatorname{Sp}_{2n}(\mathbb{Z})) = \prod_{k=1}^n \zeta(1-2k).
$$
Thus for example, we find $\chi(\operatorname{SL}_n(\mathbb{Z})) = 0$ for $n \geq 3$ and $\chi(\operatorname{Sp}_{4}(\mathbb{Z})) = \zeta(-1)\zeta(-3) = -\frac{1}{1440}$.
[1] Harder, G., A Gauss-Bonnet formula for discrete arithmetically defined groups, Ann. Sci. Éc. Norm. Supér. (4) 4, 409-455 (1971). ZBL0232.20088.
