A type of state-dependent Gaussian Z-interference channels is studied, in which transmitters 1 and 2 wish to send two messages to receivers 1 and 2, and only receiver 1 is interfered by transmitter 2's signal. Both receivers are corrupted by the same but differently scaled state sequence. The state information is assumed to be known noncausally at both transmitters. The channel is partitioned into very strong, strong, and weak interference regimes based on the strength of the interference. Respectively for the very strong and strong regimes, the capacity region and points on the capacity region boundary are characterized under certain channel parameters by designing joint dirty paper coding between two transmitters to cancel the state at both receivers. For the weak interference regime, the sum capacity is characterized by independent dirty paper coding at two transmitters. Comparison between the state-dependent regular and Z-interference channels indicates that although with one interference-free link, Z-interference channel does not necessarily perform better, because the dirty paper coded interference can be useful to help to fully cancel the state via joint dirty paper coding between the transmitters.