Let me ask some basic question on the zeta function.

As you know well, the Riemann zeta function $\sum_{n=1}^{\infty}\frac{1}{n^s}$ for $Re(s)>1$ has a meromorphic continuation to the whole complex plane and has a simple pole at $s=1$. We denote it by $\zeta(s)$. On the other hand, the zeta function has an Euler product whose local factor at $p$ is $\frac{1}{1-p^{-s}}$ that we call $\zeta_{p}(s)$. In contrast to the global zeta function $\zeta(s)$, this local zeta function is defined in itself on the whole complex plane except where $p^s=1$. Then I am wondering whether there is a relation between the zeta function $\zeta(s)$ and the product of all these local zeta functions, especially on the half-plane $Re(s)<1$. It is obvious these two are not equal, because while the former one equals $-\frac{1}{2}$ at s=0, the latter has an infinite order pole at $s=0$. If these are not equal, then what would could be said about the local factors of $\zeta(s)$?

Could you give me some explanation on this?