Let $E/Q$ be an elliptic curve. The Sato--Tate conjecture, now a theorem, tells us that the angles $\theta p=\cos -1(ap2p)$ are equidistributed in $[0,\pi ]$ with respect to the measure $2\pi \sin 2\theta d\theta$ if $E$ is non-CM (resp.~$12\pi d\theta +12\delta \pi /2$ if $E$ is CM). In the non-CM case, Akiyama and Tanigawa conjecture that the discrepancy [ D_N = \sup_{x\in [0,\pi]} \left| \frac{1}{\pi(N)} \sum_{p\leqN} 1_{[0,x]}(\theta_p) - \int_0^x \frac{2}{\pi}\sin^2\theta, d\theta\right| ] asymptotically decays like $N-12+ϵ$, as is suggested by computational evidence and certain reasonable heuristics on the Kolmogorov--Smirnov statistic. This conjecture implies the Riemann hypothesis for all $L$-functions associated with $E$. It is natural to assume that the converse (generalized Riemann hypothesis implies discrepancy estimate'') holds, as is suggested by analogy with Artin $L$-functions. We construct, for compact real tori, fake Satake parameters'' yielding $L$-functions which satisfy the generalized Riemann hypothesis, but for which the discrepancy decays like $N-ϵ$ for any fixed $ϵ>0$. This provides evidence that for CM abelian varieties, the converse to ``Akiyama--Tanigawa conjecture implies generalized Riemann hypothesis'' does not follow in a straightforward way from the standard analytic methods. We also show that there are Galois representations $\rho :Gal(Q―/Q)\to GL2(Zl)$, ramified at an arbitrarily thin (but still infinite) set of primes, whose Satake parameters can be made to converge at any specified rate to any fixed measure $\mu$ on $[0,\pi ]$ for which $\cos \ast \mu$ is absolutely continuous with bounded derivative.