Edit: So with a few more months of math under my belt, I recognize some of the issues with this question. I still hope for an answer, so let me say a few things.
Within the Langlands philosophy, L-funtions associated to various automorphic representations are the invariant which ought to parametrize an extremely complicated and far-reaching interaction between automorphic representations of separate groups. This of course refers to Functoriality.
I get that the conjecture (ie: the definition of the Selberg class) is (vaguely) that the nice L-functions found in number theory and algebraic geometry should come from automorphic obects in some way, but the definition of the automorphic L-function is still mysterious to me.
I can read the definitions, and the $\mathrm{GL}_n$ theory is truly beautiful. But it all seems like a big analogy chase, having seen the useful nature of Euler products and Dirichlet series in the past. In particular, I can't see why one would expect a connection between these L-functions and a correspondence as vast as functoriality suggests.
Is there a more satisfying rational for the definition of automorphic L-functions than "we get Euler products, and by analogy they ought to be important"?
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.
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!
Best Answer
This is not exactly an answer, but should illustrate how tentative the opinion of experts on L-functions is, when it comes to explain what L-function really are. Two quotes from the first volume on number theory by Kazuya Kato, Nobushige Kurokawa and Takeshi Saito:
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