On page 501 in the following ICM1983 conference proceeding
https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1983.1/ICM1983.1.ocr.pdf
you will find a conjecture on the L-function $L(X,s)$ of an arithmetic scheme $X/\mathbb{Z}$. We define the L-function $L(X,s)$ as follows:
$$C1.\text{ }L(X,s):= \prod_{x\in X^{cl}}\frac{1}{1-N(x)^{-s}}$$
where $x\in X$ is a closed point and $N(x)$ is the number of elements in the residue field of $x$. The function $L(X,s)$ generalize the Riemann zeta function and is conjectured to satisfy the following formula (Conjecture 2.2.iii):
$$C3.\text{ } ord_{s=j}(L(X,s))= \chi(X,j)$$
where $\chi(X,j)$ is the $j$'th Euler characteristic of $X$.
Example: If $E^*$ is an elliptic curve over a number field with $E/S$ its corresponding Neron model, it follows the conjecture
$$C_{ell}.\text{ } ord_{s=1}(L(E,s))=\chi(E,1)$$
is a K-theoretic version of the BSD conjecture for $E^*$.
On page 501 in the paper on the above link the author mentions the following conjecture (this is a translation):
Conjecture 1: If $X$ is of finite type over $\mathbb{Z}$ it follows the function $L(X,s)$ has an analytic continuation to the complex plane.
Question: What is known on Conjecture 1?
Example. If $K$ is a number field with ring of integers $\mathcal{O}_K$ and $S:=Spec(\mathcal{O}_K)$ it follows $L(S,s)$ is the Dedekind zeta function of $K$. Is Conjecture 1 known for $L(S,s)$? I ask for references?
Note: The "L-function" $L(X,s)$ defined in C1 is also referred to as the "zeta function" of $X$, written $\zeta_X(s)$.
Another reference is a paper of Serre in
"Arithmetical Algebraic Geometry": Proceedings of a Conference Held at Purdue University December 5-7, 1963 O F G Schilling
Serre uses "zeta function" and "L-function" for the function $L(X,s)$ from C1.
Example: Assume $\pi: X \rightarrow S$ where $S:=Spec(\mathcal{O}_K)$ and $K$ is a number field, and $X_t:=\pi^{-1}(t)$ with $t\in S^{cl}$ a closed point. Assume $X_t$ is a smooth projective scheme of finite type over $\kappa(t)$ for any $t$. It follows there is a product formula
$$L(X,s)=\prod_{t\in S^{cl}} L(X_t,s)$$
and $L(X_t,s)$ is a "rational function" in the following sense: If $Z(X_t,T)$ is the Weil zeta function of $X_t$ it follows
$$L(X_t,s)= Z(X_t,q^{-s})$$
where $q:=\#\kappa(t)$ is the number of elements in $\kappa(t)$. Hence $L(X,s)$ is an "infinite product of Weil zeta functions" which are known to be rational functions by the work of Dwork/Grothendieck/Artin...