Unit of impedance

simulate this circuit – Schematic created using CircuitLab

For the circuit above (for steady state I guess, if it is important) impedance of the capacitor is calculated using $$Z_c=\frac{1}{j\cdot\omega\cdot C}$$ So,$$Z_c =-j2000\;\Omega$$My question is why unit of $$\Z_c\$$ is not$$\\;\frac{\Omega}{rad}\$$ because $$\\omega\$$ has unit of $$\\frac{rad}{second}\$$ and C has unit of Farad which is equivalent $$\\frac{second}{\Omega}\$$ and I assumed $$\j\$$ is dimensionless.

I will be glad if you help me. Also I am not sure whether or not my question is reasonable.

The units of $$\\omega\$$ are 1/s. That's also true of the Laplace transform's complex frequency variable s.