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This is a follow-up to that rather old question of mine : Would a proof of Ramanujan Conjecture together with other known results about automorphic L-functions imply the Grand Riemann Hypothesis?

I would like to know if there are known cases of such automorphic L-functions for which it is proven that any permutation of its Satake parameters defined for a given finite place is an isometry of the complex plane.

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