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). 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).

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 that you might find useful.

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). 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 cuspidal automorphic representation $\pi=\otimes_v \pi_v$, the resulting global admissible representation will no longer be cuspidal automorphic (cf. multiplicity one theorems).

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 that you might find useful.

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). 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).

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). 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).

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 that you might find useful.