While I was playing with the product representation for the function $$\frac{\sin x}{x}$$ and the Nicolas equivalence to the Riemann hypothesis, I wondered if it is possible to find unconditionally some upper and lower bound for the sequence $$\log|\sin N_n|\tag{1}$$ for integers $n\geq 1$, where $N_n=\prod_{k=1}^n p_k$ is the primorial of order $n$ (this MathWorld), thus here $p_k$ denotes the $k$th prime number.
Question. Is it known or can you compute any (not obvious) lower and upper bounds of our sequence $$\text{lower bound}<\log|\sin N_n|<\text{upper bound}\tag{2}$$ for the segment, say us $1\leq n\leq 100$? Many thanks.
I've calculated some terms, and I believe that the question has mathematical meaning. If it is in the literature please answer this as a reference request.