An interesting thing about the GRH is that at oft neglected piece of evidence against it is how the Siegal zero seems to behave like a real object, i.e., having consistent properties.

I'm not sure what you mean here. If it didn't have consistent properties we could show it doesn't exist. Everything looks consistent up until the point you show it isn't real. Do you mean that it has properties that don't look that implausible? That seems like a different argument.