Ok, so this is a rather loose question (but one that has been on my mind for some time) so I don't expect any rigorous (but hopefully serious and thoughtful) answers.

Think of an L-function attached to some motive (I refrain from giving a more detailed description of what this means; if in doubt think of Artin L-functions). Is there in any sense a ''space'' (vector space, scheme,...) of L-functions?

I know that this is not a very precise question, but my thoughts on this are not very precise either (more like a vague dream world), but I know that there are people out there on this forum who know a lot about L-functions. Hopefully some of these can make my dream world a little more real, or shatter it completely.

I should also say that the recent philosophical question modularity gave me the courage (!) to ask the present one.

Edit: I should perhaps stress that I'm not quite sure myself of what I mean but I'm hoping for experts to give their informed opinions of what could be true.

Edit: Ok, to accomodate Gerhard P's comment below in some small way let me give the following motivation.

Suppose given a Galois representation (over some fixed base). I'm particularly thinking of those rep's with finite image. To such a representation you can attach an L-function as the Artin L-function of the fixed field of the kernel of the rep. Suppose now that you deform the representation in "a moduli space" (or in some parameter space) of Galois representations. Is there a sense in which the L-function deforms with it to a family of L-functions in some "space" of L-functions?

This can (probably) be generalized to other motives.

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    $\begingroup$ Nothing that simple, unfortunately. In our current understanding they are discretely occuring in every sense of the word. However, they should conjecturally form a kind of "tensor category" (or at least automorphic representations should). $\endgroup$ – David Hansen Sep 14 '11 at 15:10
  • $\begingroup$ But, say, vector spaces over finite fields are discrete... (hoping, hoping, hoping; I know I'm being naïve) $\endgroup$ – Daniel Larsson Sep 14 '11 at 15:18
  • $\begingroup$ Seriously, what you're saying is that there is no "continuous deformation" of an L-function (of something) that stays an L-function (of something, perhaps, else) under this deformation. Right? $\endgroup$ – Daniel Larsson Sep 14 '11 at 15:35
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    $\begingroup$ @Rob Harron: You can deform any L-function by $L(s,\pi)\mapsto L(s+it,\pi)$ for $\pi$ cuspidal on $GL_n$ and a fixed real $t$, and it's a conjecture of Sarnak that these are the only possible deformations (of standard L-functions of cusp forms anyway; anything reasonable should be a product of such). $\endgroup$ – David Hansen Sep 14 '11 at 18:46
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    $\begingroup$ Rob, I wouldn't call that deforming the data at infinity, because $|\cdot|_\infty^a$ is not a Hecke character (thinking adelically). In order to shift the L-function, you need to use $\prod_{p\le\infty}|\cdot|_p^a$, in which case you are shifting everything by the same amount. I wasn't thinking about it, but it is fair to say that any automorphic representation $\pi$ (on $GL_2$ for simplicity) lies in the family of automorphic representations $\pi \otimes|\cdot|^s$ (using the Converse Theorem to get back to $GL_2$), but I think this just amounts to changing the central character. (cont) $\endgroup$ – B R Sep 14 '11 at 19:18

Just to summarize some of the comments:

Strong Multiplicity One says that if the local factors of two (cuspidal) automorphic L-functions agree at all but finitely many places, then they agree at all places ("all but finitely many places" can conjecturally be replaced with "all places in a set of Dirichlet density greater than $1-1/2n^2$"). So, in this sense, the space of automorphic L-functions is rigid (you can't make changes to a small number of places and get something automorphic). This can fail for residual representations but I don't have good examples to share.

In another sense, you could deform an L-function $L(s,\pi)$ by considering $L(s,\pi\otimes|\cdot|^z)=L(s+z,\pi)$, where $\pi\otimes|\cdot|^z$ is $\pi$ with its central character twisted by $|\cdot|^z$ (some care is necessary if $z$ is not imaginary). Also, Eisenstein series exist in meromorphically parametrized families, so their L-functions also exist in families. For example, the L-function for the classical non-holomorphic Eisenstein series $E_z$ is $\xi(s+z)\xi(s+1-z)$.

Regarding vector spaces, note that the sum of two automorphic L-functions will (generally) not have an Euler product, so won't be automorphic in the usual sense.

Finally, just to explicitly tie the above in with the original question, Artin L-functions are conjecturally cuspidal automorphic for $GL_n$, and motivic $L$-functions are conjecturally automorphic for some group $G$ which can them be conjecturally transfered to $GL_n$.


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