Random L-functions

I wonder if one could prove that RH is true almost surely for some class of L-functions whose Dirichlet coefficients would be defined from a random variable.

As the Selberg class is expected to contain L-functions fulfilling this conjecture, a natural requirement would we to define those random L-functions so that a positive fraction thereof lie in the Selberg class. Another requirement would be that such a class of random L-functions would be stable under product and Rankin-Selberg convolution while containing the constant function s\mapsto 1 as well as the Riemann zeta function so as to endow this class with the structure of what I call an L-rig (see https://mathoverflow.net/questions/372349/are-there-infinitely-many-l-rigs).

We may first, for the sake of simplicity, restrict ourselves to the case of “random Dirichlet L-functions” as it is known that the Rankin-Selberg convolution of two Dirichlet L-functions is itself a Dirichlet L-function. A possible way to define them would be, given a modulus q, to define a probability measure on the set D_{q} of Dirichlet characters modulo q to which we would adjoin a smaller set R_{q} of random arithmetical functions n\mapsto f(n) defined for each prime power less than q, fulfilling f(1)=1 and extended both multiplicatively and periodically.

The resulting set L_{q} would generate an L-rig \mathcal{L}_{q} whose automorphism group \mathcal{G}_{q} could be studied. Namely we require any element \varphi of \mathcal{G}_{q} to fulfill \varphi(1)=1, \varphi(\zeta)=\zeta and for all (F,G)\in\mathcal{L}_{q}^{2}, \varphi(F.G)=\varphi(F).\varphi(G) and \varphi(F\otimes G)=\varphi(F)\otimes\varphi(G), where \otimes is the tensor product induced by the Rankin-Selberg convolution.

Doing so, would a positive proportion of elements of \mathcal{L}_{q} be fixed under a subgroup of \mathcal{G}_{q} of order at most 2?