For $x\in\mathbb{R}$, consider the following special case of the Lerch zeta function,
$$ L(s,x) = \sum_{n=1}^\infty \frac{e^{2\pi i nx}}{n^s}. $$
The series is absolutely convergent for $Re(s)>1$. If $x\not\in\mathbb{Z}$, then it continues to an entire function. If $x\in\mathbb{Z}$, it reduces to the Riemann zeta function and therefore has a pole at $s=1$.
Consider the following integral,
$$ \int_0^1 L(s,x)L(s,-x)dx. $$
If $Re(s) > 1$, then the above integral equals $\zeta(2s)$. Now the integrand is an analytic function of $s$ for any value of $x\in(0,1)$. Therefore shouldn't the integral also be an analytic function of $s$, but it clearly isn't. What am I missing here?