Let $F$ be a number field and $v$ a finite place of $F$.

Let $\chi_v$ be a unramified unitary character of $F_v$.

Then we define local L-function $L_v(s,\chi_v):=\frac{1}{1-\chi_v(\omega)q^{-s}}$ where $\omega$ is a uniformizer of $F_v$ for $s>1$.

Then I know that it has a unique meromorphic continuation to $\mathbb{C}$ and has simple a pole at s=$0$ and $1$ if $\chi_v=1$ and no poles if $\chi_v \ne 1$. Am I right?

Then I am confusing beacause I knew that meromorphic contimution is unique and $\frac{1}{1-\chi_v(\omega)q^{-s}}$ has meromorphic continuation to $\mathbb{C}$ by itself.(Furthermore, $\frac{1}{1-\chi_v(\omega)q^{-s}}$ has no pole at $s=1$ and may have a simple pole at $s=0$ depending $\chi_v=1$ or not.

But this meromorphic continution does not fit into the above argument. Because in the above meromorphic continuation definition of local $L$-function, $L_v(s,\chi_v)$ may have a pole at $s=1$.

Why people defined local L-function in this way despite the above natural explcit definition? Is it for the convergence of global L-funtion?

I am also wondering why the global $L$-function $L(s,\chi)$ for non trivial character would have at most simple pole although many local $L$-functions have simple pole there.

Is there a point I am misunderstanding?

Since I am jest a beginner in this area, any tiny comments will be appreciated.