Say an increasing arithmetic sequence $ (s_n)_{n\ge 0} $ is 'sensible' if every element thereof is the degree of some function belonging to the Selberg class. Let $ a_s $ its reason and $ f_s : =1/a_s $. For example the sequence defined by $ s_n=n $ is sensible.
Is the Selberg class stable under the transformation $ a_{n}(F)\mapsto a_{n}(F)e^{2i\pi.n.f_s}$ where $ a_{n} (F)$ is the $ n $ -th coefficient in the Dirichlet series defining $ F $ for $\Re(s)>1 $?
I believe the answer is no. Elements of the Selberg class have Euler products, and twisting the coefficients by additive characters destroys this multiplicativity.