Let $\chi = \prod_{v} \chi_{v}: \mathbb{A}^{\times}/F^{\times} \to \mathbb{C}^{\times}$ be a finite order Hecke character and let $\Phi = \prod_{v} \Phi_{v}$ be a Schwartz function on $\mathbb{A}$. Define local and global zeta functions as $$ \zeta_{v}(s, \chi_v, \Phi_v) = \int_{F_{v}^{\times}} \Phi_{v}(x)\chi_{v}(x) |x|_{v}^{s}d^{\times}x_{v} \\ \zeta(s, \chi, \Phi) = \int_{\mathbb{A}^{\times}} \Phi(x) \chi(x) |x|^{s} d^{\times} x $$ One can show that local zeta functions absolutely converges when $\Re(s) > 0$, and global zeta function absolutely converges when $\Re(s) > 1$. I want to know what makes convergence region to be shifted by one from local to global.
Here's more explicit example - consider the usual Riemann zeta function $$ \zeta(s) = \sum_{n\geq 1} \frac{1}{n^{s}} = \prod_{p} \frac{1}{1-p^{-s}} $$ It is easy to show that $\zeta(s)$ absolutely converges for $\Re(s) > 1$, while the local zeta function $$ \frac{1}{1-p^{-s}} = 1 + \frac{1}{p^{s}} + \frac{1}{p^{2s}} + \cdots $$ converges for $\Re(s) > 0$.