A binary irrational with bits defined by primes

Question

Define a number $q$ in binary notation whose $n$-th bit is $1$ for $n$ prime, and $0$ for $n$ composite. So its 2nd, 3rd, 5th, 7th, 11th, etc. bits are $1$, with all other bits $0$. Here is $q$ out to its $101$-st bit: $$.01101010001010001010001000001 010000010001010001000001000 001010000010001010000010001 000001000000010001$$ What is known about $q$? ($\approx 0.414683_{10}$). Has it been investigated? Does it have a name? It is irrational. But is it transcendental?