Are all Dirichlet coefficients of any element of the Selberg class necessarily algebraic?

Question

The title says it all: do we know at least one element of the Selberg class having at least one transcendental coefficient in its development in a Dirichlet series for $\Re(s)>1$? Or are such coefficients necessarily algebraic? Many thanks in advance.

Answer 1

No, there is nothing special about the coefficients of generic Selberg class $L$-functions. In the theory of automorphic forms, there is spectral decomposition of $L^2(\mathop{SL}(2) \backslash \mathcal{H})$ consisting of Eisenstein series and Maass forms. Both have associated $L$-functions.

It is believed that the generic nonzero coefficient of a generic Maass form is transcendental. Stated slightly differently (to ensure well-definedness up from multiplication by constants), it is believed that the generic Hecke eigenvalue of a generic Maass form is transcendental. In recent years, mathematicians including Andy Booker and Min Lee, both at the University of Bristol, have shown some results towards these conjectures, but nothing concrete as far as I know.

Correspondingly, the $L$-functions $L(s, \mu_j)$ associated to Maass forms $\mu_j(z)$ are all expected to be members of the Selberg class, and expected to have almost entirely transcendental coefficients.