The Akiyama–Tanigawa conjecture sets a bound on the rate of convergence of the Stake parameters of an elliptic curve to the Sato–Tate measure. Their conjecture implies the Riemann Hypothesis for all L-functions associated with the elliptic curve. I construct a range of examples, using Diophantine approximation, which show that GRH does not imply the Akiyama–Tanigawa conjecture for CM abelian varieties.