The comments above point to the paper [Chebyshev's Bias][1] by Rubinstein and Sarnak, and the paper of [Martin][2].  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][3] 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][4] showed in that your $x$ will be less than $10^{84}$.

[1]: https://projecteuclid.org/download/pdf_1/euclid.em/1048515870
[2]: https://arxiv.org/pdf/math/0010086.pdf
[3]: http://www.ams.org/journals/mcom/1978-32-142/S0025-5718-1978-0476616-X/S0025-5718-1978-0476616-X.pdf
[4]: https://www.impan.pl/download/pdf/aa100-4-1