Let $f$ be a function which can be represented by a Dirichlet serie (for $\Re s$ big enough) $$f(s)=\sum_{n=1}^{\infty}{\frac{a(n)}{n^s}}$$ and which can be meromorphically extend to $\mathbb{C}$ by the functionnal equation $$\frac{f(s)\Gamma(s)}{(2\pi)^s}=\frac{f(k-s)\Gamma(k-s)}{(2\pi)^{k-s}}$$ Can I conclude that it exists $a(0)$ such that $$F(\tau)=a(0)+\sum_{n=1}^{\infty}{a(n)e^{2i\pi n \tau}}$$ is a modular form of weight $k$ on the full modular group ?
Thanks a lot !
The answer is more or less yes. You need $F$ to be holomorphic on the upper half plane, and for the coefficients to have polynomial growth: $a(n)=O(n^r)$ for some $r>0$. There is also a condition on the poles of completed $L$-function, which determines $a_0$. This is consequence of Hecke's converse theorem.
See for example, corollary 3.2 in these notes: http://www.math.ethz.ch/education/bachelor/seminars/ws0607/modular-forms/Heckes_converse_theorem.pdf