No, you are not right. The local $L$-function $L(s,\chi_v)=(1-\chi_v(\omega)q^{-s})^{-1}$ has infinitely many poles, namely the solutions of the equation $q^s=\chi_v(\omega)$. All these poles are simple of course. Also, it has no zeros. The local $L$-function can be more complicated, but it is usually a product of the above kind of factors. Similarly, a normalized gamma function like $\pi^{-s/2}\Gamma(s/2)$, or a product of shifted versions of it, which occur as local $L$-functions at archimedean places, has infinitely many poles but no zeros.
The local $L$-function has nice properties and tell a lot about the underlying local object (e.g. an admissible representation of $GL_n(\mathbb{Q_p})$). The $L$-function of a global object (e.g. a Dirichlet character) is the product of the local $L$-functions associated with the various local factors of the global object (e.g. various local characters coming from the same Dirichlet character). It has even nicer properties, and properties that are even harder to access. More precisely, the product formula is only valid in a half-plane (usually $\Re s>1$), and elsewhere it is defined by analytic continuation. This also explains why the poles of the local factors are not poles of the global $L$-function.
You have to learn the theory to get familiar with these things. I recommend the following books (already the first one answers your questions):
Davenport: Multiplicative number theory
Neukirch: Algebraic number theory
Weil: Basic number theory
Bump: Automorphic forms and representations
Goldfeld-Hundley: Automorphic representations and $L$-Functions for the general linear group