  We report a high-resolution spectroscopic investigation of the long-range
states of the $^{23}$Na$^{87}$Rb molecule near its
Na($3^2S_{1/2}$)+Rb($5^2P_{3/2}$) asymptote. This study was performed with
weakly bound ultracold molecules produced via magneto-association with an
inter-species Feshbach resonance. We observed several regular vibrational
series, which are assigned to the 5 attractive long-range states correlated
with this asymptote. The vibrational levels of two of these states have sharp
but complex structures due to hyperfine and Zeeman interactions. For the other
states, we observed significant linewidth broadenings due to strong
predissociation caused by spin-orbit couplings with states correlated to the
lower Na($3^2S_{1/2}$)+Rb($5^2P_{1/2}$) asymptote. The long-range $C_6$ van der
Waals coefficients extracted from our spectrum are in good agreement with
theoretical values.
