Zeros of polynomials in derivatives of automorphic L‐functions

Let be the set of all cuspidal automorphic representations of , and let be a polynomial in the derivatives of ‐functions associated with representations . We establish an asymptotic formula for the number of nontrivial zeros of with . We explicitly determine the main term of this formula in terms of the dimensions, the arithmetic conductors, and the orders of differentiation of the component ‐functions. Furthermore, we show that, under certain conditions, almost all nontrivial zeros of lie near the critical line .