To every local admissible representation $\pi_v$ of $\mathrm{GL}_n(k_v)$, there is a local $L$-function $L(s,\pi_v)$. For a global admissible representation $\pi=\otimes_v \pi_v$ of $\mathrm{GL}_n(\mathbb{A}_k)$, the corresponding global $L$-function is defined as $L(s,\pi)=\prod_v L(s,\pi_v)$. However, unless $\pi$ is automorphic, this global $L$-function will not have the usual analytic properties (cf. [converse theorems][1]). So it is very special for a global admissible representation to be automorphic. Automorphicity of $\pi=\otimes_v \pi_v$ connects the various local factors $\pi_v$. In particular, if you change finitely many local factors $\pi_v$ in a given automorphic representation $\pi=\otimes_v \pi_v$ , the resulting global admissible representation will no longer be automorphic (cf. [multiplicity one theorems][2]).

BTW I recommend Goldfeld-Hundley's two-volume textbook on automorphic forms (in the same series as Bump's textbook). Also, there is this [related earlier post of mine][3] that you might find useful.


  [1]: https://en.wikipedia.org/wiki/Converse_theorem
  [2]: https://en.wikipedia.org/wiki/Multiplicity-one_theorem
  [3]: https://mathoverflow.net/questions/349122/are-the-l-functions-of-a-normalized-newform-and-the-corresponding-cuspidal-repre/349124#349124