If $f$ is a modular form and we let $a_n$ be the Fourier coefficients, then the $L$-Series associated to $f$ is $$ L(s,f)=\sum_n\frac{a_n}{n^s} $$ Usually, we only define this for cusp forms that is when $a_0=0$, but I am curious if, $f$ is not a cusp form does it make sense to ask what the $L$-series is? Can you just throw away $a_0$ in that case, or do we not care about $L$-series in this case?
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asked Apr 8, 2022 at 16:08
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$\begingroup$ Try it and see. There is no reason I know that you can not do what you suggest. $\endgroup$– SomosCommented Apr 8, 2022 at 16:50
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$\begingroup$ The $L$-series is most interesting for eigenforms, where it has an Euler product. This also makes sense for Eisenstein series, but, in a sense, we don't get a degree $2$ $L$-series. For example, if $E_{2k}\in S_{2k}(1)$ is the normalised Eisenstein series of weight $2k$, then $L(E_{2k}, s) = \zeta(s)\zeta(s+1-2k)$. $\endgroup$– Mathmo123Commented Apr 9, 2022 at 17:03
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1$\begingroup$ @Mathmo123 you still get a degree $2$ $L$-function, but it's no longer primitive: it factorises as the product of two degree $1$ $L$-functions. $\endgroup$– Peter HumphriesCommented Apr 9, 2022 at 17:33
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$\begingroup$ @PeterHumphries thanks - that's why I said "in a sense", but I'd forgotten what the term for a "reducible" $L$-function was! $\endgroup$– Mathmo123Commented Apr 10, 2022 at 8:33
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If $f = \sum_{n \geq 0} a(n) q^n$ is any modular form, one can construct the Dirichlet series $$ L(s, f) := \sum_{n \geq 1} \frac{a(n)}{n^s} $$ (note the sum is only over $n \geq 1$, omitting the constant term of the Fourier expansion). This Dirichlet series will have meromorphic continuation and a functional equation, coming essentially from the modularity of $f$.
If $f$ is a cuspform, then the Dirichlet series has analytic continuation. If $f$ is a Hecke eigenform, then the Dirichlet series has an Euler product (and is called an $L$-function).
The theory is a bit more general and extends to less well behaved objects. For example, if $f$ is a half-integral weight modular form (where there are no Euler products), then the Dirichlet series still exists and has meromorphic continuation. In practice this can be useful, as many quadratic forms can be related to general theta functions, which are generically half integral weight objects on an appropriate congruence subgroup. Thus estimating growth of these quadratic forms is closely related to the behavior of Dirichlet series of modular forms.
answered Apr 11, 2022 at 10:53
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$\begingroup$ This doesn't answer the question; the question is about removing the condition that the modular form have vanishing zeroeth coefficient, i.e. a cusp form instead of an Eisenstein series. The Dirichlet series also has a meromorphic continuation if $f$ is noncuspidal, though it may have poles, and if the modular form is a Hecke eigenform (which can still be the case for Eisenstein series!) then it also has an Euler product. $\endgroup$ Commented Apr 11, 2022 at 14:23
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$\begingroup$ @PeterHumphries Is that different than what I noted? It might be that I'm missing something very obvious --- I certainly do that often enough. $\endgroup$– davidlowryduda ♦Commented Apr 11, 2022 at 14:28
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$\begingroup$ The discussion about half-integral weight modular forms is irrelevant; these have $L$-series but not $L$-functions (since, as you mention, they don't have Euler products), and these $L$-series don't fit into the general picture of $L$-functions (in the Selberg class sense, or instead in terms of Langlands' theory of $L$-functions of automorphic representations of reductive algebraic groups). $\endgroup$ Commented Apr 11, 2022 at 14:44
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$\begingroup$ In particular, you gain nothing extra by studying analytic properties of $L$-series of half-integral weight modular forms that you couldn't already gain by using standard techniques in the analytic theory of integral weight automorphic forms; it's a red herring. $\endgroup$ Commented Apr 11, 2022 at 14:45
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$\begingroup$ Aha. My point there is that the analytic properties of the $L$-series come entirely from the modularity of the form (as opposed to arithmetic properties of the form). The OP asks only about $L$-series after all, not about $L$-functions. As an aside, I'll also note that I think there are things to be gained from studying half integral weight forms. As a concrete example, the error term in the Gauss sphere problem concerns spectral data of weight $3/2$ forms. It may be that we are both saying that the governing aspects here is entirely the modularity. $\endgroup$– davidlowryduda ♦Commented Apr 11, 2022 at 14:54