Given co-prime $a,b$, Dirichlet's theorem states that there are infinitely many primes in the arithmetic progression $M = \{ a + bn : n \in \mathbb N\}$. Linnik's theorem asserts that the first such prime is bounded by $cb^L$ for positive constants $c,L$.

Let $M_t = M \cap \{ n > t : n\in \mathbb N\}$ for some positive threshold $t$. Is there an analogue of Linnik's theorem giving bounds for the first prime in $M_t$, preferably polynomial in $b$ and $t$?