I got inspired by quadratic rings and zeta functions. I know for instance that the norm $a^2 + 17 b^2$ for the ring $\Bbb Z[\sqrt{-17}]$ is multiplicative yet the ring $\Bbb Z[\sqrt{-17}]$ is not a UFD !!
The norm of that ring is $a^2 + 17 b^2$
I was thinking about this related zeta analogue , the analytic continuation of the Euler product
$$\zeta_{17}(s) = \prod_p (1 + \frac{1}{p^s - 1})$$
where $p$ is an irreducible element of the form $a + b \sqrt{-17}$.
This function $\zeta_{17}$ has a simple pole at $s=1$ and it has no zero's or poles for $s = 1 + t i$ for nonzero real $t$. (correct me if I am wrong but I think I have proved it)
Now I conjecture that all the zero's of $\zeta_{17}$ are on the real line or have $Re(s) < 0$ or $Re(s) = 1/2$.
Is this true ?
Is this an open problem ?
If open, does this belong to a known generalization of the zeta function ? Keep in mind this is not the function $f(s) = \sum \frac{1}{(a^2 + 17 b^2)^s}$ nor $g(s) = \sum \frac{1}{(a + \sqrt{-17} b)^s}$ and the related ring is not a UFD.
Again , this is not the same function as Closed form for $A=\lim_{k \to \infty} \sum_{0<a^2 + 17 b^2 < k} \frac{1}{a^2 + 17 b^2} - \sum_{k<a^2 + 17 b^2 < k^2} \frac{1}{a^2 + 17 b^2}$?
( although that might interest you. And in fact one can wonder about the regularization of this function at $s=1$ but that is not the question here )
Finally I have to say that I am not clear about how the analytic continuation should actually be done , how far it goes etc.
Using taylor and monodromy theorem works but it is probably not so efficient. We should get half-planes rather than circles I assume. I think taking the log of the euler product will help ?
