Is there a known elementary function bound in terms of $a,b,n$ for the $n$-th prime equal to $b$ modulo $a$ (coprime to $b$)?

Bounds on Linnik's constant answer this for the first prime in each progression. Is there a known analogue for an $n$-th prime in a progression? And I found some references on an error term for the prime number theorem for arithmetic progressions. But I don't see how to turn these into a construction for arbitrary $a,b,n$.

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