Fractional Josephson vortices carry a magnetic flux Φ, which is a fraction of the magnetic flux quantum Φ₀≈2.07×10^-15 Wb. We consider a fractional vortex which spontaneously appears at a phase discontinuity. Its properties are very different from the properties of the usual integer fluxon. In particular, a fractional vortex is pinned and may have one of two possible polarities – just like a usual spin 1/2 particle. The fractional vortex may also oscillate around its equilibrium position with an eigenfrequency which is expected to be within the Josephson plasma gap. Using microwave spectroscopy, we investigate the dependence of the eigenfrequency of a fractional Josephson vortex on its magnetic flux Φ and on the bias current. The experimental results are in good agreement with theoretical predictions. Positive result of this experiment is a cornerstone for further investigation of more complex fractional vortex systems such as fractional vortex molecules and tunable bandgap materials.