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.)
Meanwhile, Ford and Hudson showed in that your $x$ will be less than $10^{84}$.