  We study the two-dimensional joint distribution of the first hitting time of
a constant level by a continuous-state branching process with immigration and
their primitive stopped at this time. We show an explicit expression of its
Laplace transform. Using this formula, we study the polarity of zero and
provide a necessary and sufficient criterion for transience or recurrence. We
follow the approach of Shiga, T. (1990) [A recurrence criterion for Markov
processes of Ornstein-Uhlenbeck type. Probability Theory and Related Fields,
85(4), 425-447], by finding some $\lambda$-invariant functions for the
generator.
