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Corrected typos since question was already on front page. (I apologise in advance if there edits are unwelcome or just bad.)
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edit approved Feb 15, 2020 at 20:07
Emily
edit approved Feb 15, 2020 at 20:07
Emily
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Distribution of signs of autiomorphicautomorphic forms

LetLet's say we have an automorphic form $f$ on $GL(2)$ that is selfdualself-dual. In particular, the associated L-function $L(s, f)$$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?

Distribution of signs of autiomorphic forms

Let say we have an automorphic form $f$ on $GL(2)$ that is selfdual. 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?

Distribution of signs of automorphic forms

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?

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Gory
Gory
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Distribution of signs of autiomorphic forms

Let say we have an automorphic form $f$ on $GL(2)$ that is selfdual. 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?