Does the L-series of a modular form that is not a cusp form make sense?

Question

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?

Answer 1

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.