TITLE: Prime race modulo $12$. When is the first sign change?
QUESTION [5 upvotes]: Define $\pi(x;q,a)$ as the number of primes less than or equal
to $x$ which are congruent to $a$ modulo $q$.
Up to $x=10^{11}$ we have $\pi(x;12,1) \le \pi(x;12,-1)$.
What is the smallest $x$ for which $\pi(x;12,1) > \pi(x;12,-1)$?
Assuming GRH the sign changes infinitely often.

REPLY [11 votes]: The comments above point to the paper Chebyshev's Bias by Rubinstein and Sarnak, and the paper of Martin.  For context, let's consider first primes mod $3$.
Assuming you're familiar with the terminology of Rubinstein and Sarnak, they compute the bias of primes to be $2$ mod $3$ over $1$ mod $3$ to be $0.9990\ldots$  (This is $\delta(P_{3;N;R})$ in their notation).  In 1978, Bays and Hudson computed that $608,981,813,029$ was the smallest integer $x$ such that $\pi_{3,2}(x)<\pi_{3,1}(x)$.  
The paper of Martin shows the bias of primes to be $11$ mod $12$ over $1$ mod $12$ is much stronger: $\delta_{12;11,1}=0.999977$ (in Martin's slightly different notation.)  Since this is a logarithmic scale, the 'extra' $0.000977$ means that the bias is almost two orders of magnitude stronger.
Meanwhile, Ford and Hudson showed in that your $x$ will be less than $10^{84}$.