My question concerns the least prime (denoted $p(a, q)$) in the arithmetic progression $a \pmod q$ where $a$ and $q$ are coprime. Quite a time ago Linnik demonstrated that 
$$p(a, q) \ll q^L$$
for some absolute constant $L$. [Wiki page](http://en.wikipedia.org/wiki/Linnik%2527s_theorem) for this theorem lists a number of papers that estimate $L$ with the most recent result by Xylouris who proved that $L \leq 5.2$.


It is also known that the Generalized Riemann Hypothesis implies
$$p(a, q) \ll (q\log q)^2 \text{,}$$
while in 1978, Heath-Brown conjectured even tighter bound:
$$p(a, q) \ll q(\log q)^2 \text{.}$$
I'm wondering whether this last bound, if true (it is still an open problem), implies something non-trivial about $L$-functions?



On the least prime in arithmetic progressions
I haven't looked at such types of questions, however my first thought is no: I don't see (or haven't seen yet) how the existence of one prime (or a small number of them) would force the Dirichlet L-functions (I guess these are which you meant) to look in a certain way.


As an example, take the explicit formula which counts the number of primes in arithmetic progressions by using zeros of some L-functions. 


Now, if you know that the "left side" in the particular equation (counting the primes) is 1 instead of 0, this does not seem to force anything noteworthy for the zeros which appear in the sum of the right side....Some small change in the imaginary parts of a couple of zeros (with very large imaginary part) might be enough to change the total value by 1, which then implies my answer for the special case if you only use the explicit formula. However, most arguments in analytic number theory (where this theorem on the least prime number comes from) tend to give similar behavior.


I hope, you understand the point I am trying to make besides my unclear presentation.


