  From the generalized Riemann hypothesis for motivic L-functions, we derive an
effective version of the Sato-Tate conjecture for an abelian variety A defined
over a number field k with connected Sato-Tate group. By effective we mean that
we give an upper bound on the error term in the count predicted by the
Sato-Tate measure that only depends on certain invariants of A. We discuss
three applications of this conditional result. First, for an abelian variety
defined over k, we consider a variant of Linnik's problem for abelian varieties
that asks for an upper bound on the least norm of a prime whose normalized
Frobenius trace lies in a given interval. Second, for an elliptic curve defined
over k with complex multiplication, we determine (up to multiplication by a
nonzero constant) the asymptotic number of primes whose Frobenius trace attain
the integral part of the Hasse-Weil bound. Third, for a pair of abelian
varieties defined over k with no common factors up to k-isogeny, we find an
upper bound on the least norm of a prime at which the respective Frobenius
traces have opposite sign.
