With regard to the following equation: $$\prod_{n=1}^{\infty}(1-\frac{z}{n^a})e^{\frac{z}{n^a}+\frac{z^2}{2n^{2a}}}=a,$$
I am trying to answer the following questions: for $a=\frac{\pi}{7}$, has the equation any solutions? If it has, how many?
I have no idea as to how to approach this problem. I have done some exercises in which the infinite product was equal to zero and, in those cases, it was easy to give an answer. But the fact that the product is equal to $a$ complicates it.
Any hints to find a solution will be very helpful.
Thanks in advance.