Is
$$f(x)=\dfrac{2\pi}{n^2}\sum_{m=1}^{x}\sum_{k=1}^{n}\dfrac{\Im(e^{\dfrac{\pi i m}{\Im(\rho_k)}})k}{\Im(\rho_k)}$$
related to the Bessel family of functions?
Plot for $n=300$
Or is it related to the $sin$ integral as here?
(NB, when $k$ replaces $\Im(\rho_k)$, output is similar, but different scale.)