# L-rigs, symmetry of roots, Ramanujan and Riemann conjectures

I define L-rigs in Are there infinitely many L-rigs?

An interesting property of such classes of L-functions is that if any two elements thereof fulfill RH, then so does their product. On the other hand, the Ramanujan conjecture (which is assumed as elements of an L-rig are required to belong in the Selberg class) is stable under the Rankin-Selberg convolution.

A reformulation of RH is that any non-trivial zero $$s$$ of an L-function $$F$$, where here by L-function I mean any element of the maximal L-rig, is invariant under the involution $$s\mapsto 1-\bar{s}$$. Analogously, a reformulation of RC is that any Satake parameter at an unramified prime is invariant under the involution $$\alpha\mapsto\bar{\alpha}^{-1}$$.

Another common point is that the Satake parameters at $$p$$ are the reciprocal roots of the local factor $$F_{p}$$ of $$F$$, while non trivial zeros of $$F$$ are roots of its complete L-function. Call then the non trivial zeros of $$F$$ "$$\times$$-essential roots" of $$F$$ and its Satake parameters at unramified primes "$$\otimes$$-essential roots" of $$F$$.

Now, viewing the permutation groups $$G_{\times}(F)$$ of the $$\times$$-essential roots of $$F$$ and $$G_{\otimes}(F)$$ of the $$\otimes$$-essential roots of $$F$$ preserving it and compatible with the structure of L-rig as Galois groups, are those two Galois groups isomorphic? In other words, in an L-rig, is the Ramanujan conjecture equivalent to the Riemann hypothesis?