# Order of non-trivial zeros of an L-function and topological dimension

Let $F$ be a primitive element of the Selberg class of degree $d_{F}>0$, and let's consider the group $G$ of complex isometries of finite order that preserve the critical strip $S_{c}=\{s\in\mathbb{C},0<\Re(s)<1\}$. For any strict subgroup $H$ of $G$, let's denote $\mathcal{D}_{H}$ the set $\{s\in S_{c},\forall h\in H, h(s)=s\}$ and $\mathcal{\hat{D}}_{H}$ the set $\{s\in S_{c},s\in\mathcal{D}_{K}\Rightarrow K=H\}$. Finally, let's denote $dim_{H}$ the topological dimension of any connected component of $\mathcal{\hat{D}}_{H}$. Let's define for any $s\in S_{c}$ the number $v_{F}(s)$ as $v_{F}(s)=0$ if $F(s)\ne 0$, and $v_{F}(s)$ as the order of the zero $s$ if $F(s)=0$.

I formulate the following dimension conjecture:
for every strict subgroup $H$ of $G$, $s\in\mathcal{\hat{D}}_{H}\Rightarrow v_{F}(s)\leqslant\frac{1}{dim_{H}}$.

This conjecture implies both the modified RH for $F$ (all non trivial zeroes of $F$ are real or lie on the critical line) and the Grand Simplicity Hypothesis.

My question is: is there any piece of evidence in the literature that the dimension conjecture as stated above should be true? Any reference about related subjects?