Let's say we have an automorphic form $f$ on $GL(2)$ that is self-dual. In particular, the associated L-function $L(s,f)$ satisfies a functional equation with sign $\varepsilon_F = \pm 1$.

Is it known that the proportion of such automorphic forms with given sign (say $-1$) is exactly $1/2$?

I know many results about distributions of signs for coefficients and eigenvalues of automorphic forms, however when I think of this question I wonder whether it is well-known or difficult?