This question loosely builds on this one.

Take the following infinite product:

$$N(s,\chi_k)=\prod_{n=2}^{\infty} \left(\frac {1}{1-\dfrac{\chi_k(n)}{n^s}} \right)$$

with $\chi_k$ a Dirichlet character with period $k$. The product appears to have quite a few closed forms e.g.:

*For principal characters:*

$\displaystyle N(2,\chi_{1_1})=2$

$\displaystyle N(2,\chi_{2_1})=\frac{4}{\pi} =N(2,\chi_{4_1}),\cdots$

$\displaystyle N(2,\chi_{3_1})=\frac{3\sqrt{3}}{\pi} =N(2,\chi_{9_1}),\cdots$

$\displaystyle N(2,\chi_{6_1})=\frac{2\sqrt{3}}{\pi} =N(2,\chi_{36_1}),\cdots$

*For non-principal characters:*

$\displaystyle N(1,\chi_{3})=\frac{3\sqrt{3}}{2\,\pi}$

$\displaystyle N(1,\chi_{4})=\frac{2\sqrt{2}}{\pi}$

$\displaystyle N(1,\chi_{6})=\frac{\pi}{3}$

Has this function been studied before? If so, I would be grateful for a reference.

Does this product over $n$ converge for non-principal characters in the domain $\Re(s)>\frac12$ (numerical evidence suggests it does), or is this just as problematic to prove as it is for the product over primes?