Mean-zero Sato–Tate law for conjugation-stable sets of analytic L-functions

Let $\mathcal{S}$ be a finite set of entire analytic $L$-functions that is stable under conjugation, so that $\overline{\mathcal{S}}=\mathcal{S}$. Let $(a_f)_{f\in\mathcal{S}}$ be complex numbers satisfying $a_{\overline{f}}=\overline{a_f}$. Mean-zero Sato–Tate conjecture. The sequence

$$\left(\sum_{f\in\mathcal{S}}a_f\lambda_f(p)\right)_p$$

is equidistributed in an interval of $\mathbf{R}$ according to a Sato–Tate law with mean value $0$. A prime number theorem gives the mean-zero conclusion when the $L$-function is entire and nonvanishing on $\operatorname{Re}(s)=1$; the stated extension to finite conjugation-stable sets remains open.