To preface, I am a student of automorphic representation theory, and I know full well the definition of the L-function attached to an automorphic representation.

I am intending to give a talk on the question in the title to a group of graduate students and young researchers. While the history of and ubiquity of L-functions is an aspect of what I want to explain, there is a nagging question in the back of my mind that I do not know how to approach:

Why is it that a Dirichlet series with analytic continuation and functional equation is such a potent idea? What is a unifying idea behind these constructions?

For many (all?) instances of L (or zeta)-functions in number theory, representation theory, (Artin, Hasse-Weil, Dirichlet, etc.) and perhaps many other fields I know less about (Selberg's zeta function) , the hope is that these are all instances of automorphic L-functions and are related in deep ways to an automorphic representation.

But on the automorphic side of things, I don't understand what the L-function actually is. The converse theorems gives me a partial answer in that the L-function is some local-global object (The definition as an Euler product) which encodes, along with its twists, automorphy of the representation.

This then leads me to other questions for another time, and still seems more about why the L-function is useful, as opposed to what the L-function is.

My question, then, is

What is the (conjectural) underlying idea of what an L-function is, either in the automorphic case or more generally? Is there a sense of why such a construction gives a powerful way of connecting different areas of mathematics?

I have read Bump's Book Automorphic Forms and Representations, a few articles such as Iwaniec's and Sarnak's enjoyable article Perspectives on the Analytic theory of L-functions, as well as many of the brilliant responses to related questions here on MO.

As this is my first question, I apologize if my question is not clear, or is duplicate to a question I have not yet found. Thank you for your help!

Edit: In terms of an answer, let me say this: I was hoping that there is a known way, perhaps in terms of the relevant group, to see why the L-function construction should be so fundamental to so many theories.

I mean, we take an interesting sequence and construct either a Dirichlet series or an Euler product, and the fact that this construction conjecturally relates automorphic lifts through functoriality, contains valuable data in its special values (eg: Birch andSwinnerton-Dyer), etc. is a jarring idea.

I hope not to appear glib by this comment; while much theory in the given instance is involved with what this sequence is, I still think my surprise has merit.

If there isn't a known answer in this sense, as was indicated by @Myshkin's answer, then I will be happy with intuition or heuristic understanding that is in this direction. Please let me know if this is still too broad or unclear. Thank you!