Consider the function
$$ G(s,x) : = \sum_{n=1}^\infty \frac{\exp(2\pi i nx)}{n^s} $$ where $x\in[0,1)$ and $s\in\mathbb{C}$. The series is absolutely convergent for $Re(s) > 1$.
If $x\in\mathbb{Q}\cap[0,1)$, then it is not difficult to see that $G(s,x)$ can be written as a finite sum of certain Dirichlet $L$ series which have analytic continuation, functional equation and other nice properties. Now, can we use the fact that $\mathbb{Q}\cap [0,1)$ is dense in $[0,1)$ and show that $G(s,x)$ has analytic continuation and functional equation for all $x\in[0,1)$.
The intuition behind the question is that $G$ is continuous in the $x$ variable and holomorphic in the $s$ variable in the region of convergence. Can we use these two facts and the existence of analytic continuation on a dense subset to conclude the existence of analytic continuation everywhere?