Let $a$ be an integer which is neither a square nor $-1$. Artin's conjecture states that there are infinitely many primes $p$ for which $a$ is a primitive root modulo $p$. My question is whether there is anything on the literature for

(1) the (conjectural) size of the **smallest such prime $p$**.

(2) Conditional or unconditional upper or lower bounds.

Regarding (2), one can chase down the implied constants in Hooley's paper (Hooley, Christopher (1967). "On Artin's conjecture." J. Reine Angew. Math. 225, 209-220) to show that his asymptotic must kick in after $x\geq x_0=|a|^{C\log \log 3|a|}$ for some absolute $C>0$. Thus, under GRH the least prime is at most $$|a|^{C\log \log 3|a|}.$$

in generalis hard! $\endgroup$