I would like to know if, under Ramanujan conjecture, the following three distributions are known or conjectured to match:


1. the distribution of spacings between Satake parameters of an L-function $F$ at all unramified primes on the unit circle
2. the distribution of the spacings between non trivial zeros of $F$ under the analogue of RH for $F$ on a circle of the Riemann sphere whose stereographic projection is the critical line
3. the spacings of eigenvalues of some random matrix along the lines of Katz-Sarnak philosophy.



Spacings of Satake parameters under Ramanujan conjecture
If the first item refers to the set of Satake parameters for $X \le p <2 X$ for large $X$, then that distribution will not match the others. This is because the Satake parameters for different primes do not "see" each other: they will not have (quadratic) repulsion as is expected or known in the other two cases.


