Let $X$ be an arithmetic scheme, that is, a scheme of finite type over the integers. We denote the set of closed points of $X$ by $|X|$. For every $x\in|X|$, write $N(x)$ for the cardinality of the residue field $\kappa(x)$.

The *arithmetic zeta function* of $X$ is defined as
$$\zeta_X(s)=\prod_{x\in|X|}\frac{1}{1-N(x)^{-s}}.$$

This definition (up to a change in variable) can be found in

A. Grothendieck, Formule de Lefschetz et rationalité des fonctions $L$,

Séminaire Bourbaki279 (1964), 41-55.

Grothendieck attributes this definition to Weil, but as far as I know, Weil only defined the *Hasse-Weil zeta function*: if $X$ is a smooth projective variety over $\mathbb{F}_q$ and $N_r=|X(\mathbb{F}_{q^r})|$, then
$$Z_X(t)=\exp\left(\sum_{r=1}^\infty N_r(X)\frac{t^r}{r}\right).$$

Of course, it is easy to show these two functions satisfy $$\zeta_X(s)=Z_X(q^{-s}),$$ but Weil did not address the notion of the zeta function of a scheme, at least not in the original paper:

A. Weil, Numbers of solutions of equations in finite fields,

Bull. Amer. Math. Soc.55 (1949), 497-508.

Where was the zeta function *of an arithmetic scheme* first defined? If anyone knows the actual paper in which this first appears, that would be optimal. (Of course, it's very possible that this definition was well-known but unpublished for some time; that would be an acceptable answer too.)