The "maximal" field associated to the Selberg class Hello,
for any given function $F$ of the Selberg class $\mathcal{S}$, let $A_{F}$ be the set of coefficients $a_{n}$ of the Dirichlet series defining $F(s)$ for $\Re(s)>1$, and let $A=\bigcup_{F\in\mathcal{S}}A_{F}$. Is it true that $\mathbb{Q}(A)=\mathbb{C}$? Same question for $F$ running through $\mathbb{P_{\mathcal{S}}}$, where $\mathbb{P_{\mathcal{S}}}$ is the set of all primitive functions of $\mathcal{S}$.
Thank you in advance.
 A: This is a very serious question, probably intractable. 
The "arithmetic theory of automorphic forms", interpreted first as pertaining to holomorphic Siegel-Hilbert or other discrete-series-at-archimedean-places automorphic forms, does connect immediately to algebraic-geometric rationality notions...
The next larger regime is "cohomological" repns... at infinity, only, since finite primes seem to have no obvious constraint.
But... all Maass cuspforms for SL(2,Z)? I think no one has any expectation that associated data would be algebraic, etc. Not that there is "proof" to the contrary, but that there is no reason to think otherwise.
My own opinion is that we do not know the proper question about this, at this time. 
A: Take any Dirichlet series $L(s)=\sum_{n \geq 1} \frac{a_n}{n^s}$ in the Selberg class. If $L(s)$ has no pole at $s=1$, then for any $\theta \in \mathbf{R}$, the additive twist $L_{\theta}(s)=L(s+i\theta) = \sum_{n \geq 1} \frac{a_n n^{-i\theta}}{n^s}$ still belongs to the Selberg class. It follows that your set $A$ contains all complex numbers of absolute value $1$ and that $\mathbf{Q}(A)=\mathbf{C}$.
So you probably want to ask your question in a more precise form, by considering Dirichlet series only up to additive twist.
As pointed out by Emerton, it is believed that all functions in the Selberg class come (in a sense which might not be formulated very precisely at the moment) from arithmetic automorphic forms, see for example the abstract of this preprint of J. B. Conrey and A. Ghosh. Since the set of all arithmetic automorphic forms is (provably) countable, the set of coefficients of the Dirichlet series which are (conjecturally) associated to them is countable, so cannot generate the field $\mathbf{C}$, whose transcendance degree over $\mathbf{Q}$ is the cardinality of the continuum.
But, as pointed out by Paul Garrett, the question in terms of elements the Selberg class is likely to be completely intractable (like many open questions on the Selberg class).
