Fix a complex number s and a real number x, does there exist an analytic continuation of the Dirichlet series

$L(s,x):=\sum_{k=1}^{\infty}\frac{\sin^2(2\pi k x)}{k^s}$

to the whole complex plane except 1?

If yes, is there some functional equation verified which makes it possible to calculate $L(0,x)$?

If yes, what about the modulus of continuity of $x\mapsto L(0,x)$? ($L(\frac{3}{2},x)$ seems to be a nice case.)

Thanks for any comments Chri