I have arrived at an elementary-looking "result" via a sketchy argument. Having unsuccessfully searched for the statement and its "proof" in the literature, I would like to hear if anyone knows whether or not it is true, and if they might provide me with a reference? The statement is as follows: let $f:\mathbb{N}\rightarrow \mathbb{C}$ and
$$F(s)=\sum_{1}^{\infty}\frac{f(n)}{n^s}$$
converge for $\sigma>\sigma_c>0$, then

$$\int_{\sigma-i\infty}^{\sigma+i\infty}|F(s)|^2\frac{ds}{s}$$

converges for $\sigma>\sigma_c>0$, and vice-versa.

EDIT 1: "Vice-versa" meaning that the converse implication also holds.

EDIT 2: The integrand is to be taken as an analytic continuation of the series definition of $F$, assuming a-priori that one exists there.

Sketch of proof: I begin by constructing an auxillary Dirichlet series:
$$D(s)=\sum_{1}^{\infty}\frac{d(n)}{n^s}$$
where
$$d(n)=2\left(f(1)+f(2)+\cdots+\frac{f(n)}{2}\right)\bar{f}(n).$$
Thus

$$\sum_{1}^{n}d(m)=\left|\sum_{1}^{n}f(m)\right|^2$$
is positive. The abscissa of convergence of $D(s)$ is
$$\sigma_d=\limsup_{n\rightarrow\infty}\frac{\log\sum_{1}^{n}d(m)}{\log n}=2\limsup_{n\rightarrow\infty}\frac{\log\left|\sum_{1}^{n}f(m)\right|}{\log n}=2\sigma_c,$$
provided $\sigma_c>0$. I argue that the convergence of the Dirichlet series $D(2s)$ is equivalent to the convergence of the integral above as follows: If $\sigma>\sigma_c$, then
$$D(2\sigma)=\sum_{1}^{\infty}\frac{d(n)}{n^{2\sigma}}=\sum_{1}^{\infty}\left(\frac{1}{\pi i}\int_{\sigma-i\infty}^{\sigma+i\infty}\sum_{1}^{\infty}\frac{f(m)}{m^{s}}\frac{n^{s}ds}{s}\right)\frac{\bar{f}(n)}{n^{2\sigma}},$$
where both $D(2s)$ and $F(s)$ converge absolutely if $\sigma>\sigma_c+1$. Since $F(s)$ has no poles for $\sigma>\sigma_c>0$, so for fixed $n$ the (conditionally convergent) integral is unchanged for $\sigma>\sigma_c$. If one can justify interchanging the order of summation and integration, we get

$$D(2\sigma)=\frac{1}{\pi i}\int_{\sigma-i\infty}^{\sigma+i\infty}|F(s)|^2\frac{ds}{s}.$$