Abstract: 〖JP2〗The elementary excitations in a quasi-one-dimensional synthetic spin-orbit-coupled Bose-Einstein condensate are investigated in this paper. With the mean-field approximation and the Bogoliubov approach, the excitation spectrum of the atomic condensate in the zero-momentum and plane-wave phases is calculated, respectively, which depend on the Raman coupling strength. It is shown that the two branches of the excitation spectrum in zero-momentum phase both exhibit symmetry structure. In contrast, the excitation spectrum〖JP〗 exhibits roton minimum structure in the plane-wave phase for small Raman coupling strength, which provides the onset of the phase transition to the stripe phase. It is also shown that the sound speed of the low-frequency excitations decreases sharply and vanishes near the phase transition between the plane-wave and zero-momentum phases. This work gives comprehensive analysis of novel properties of elementary excitations in a synthetic spin-orbit-coupled atomic condensate, which may provide theoretical support for experimental studies on this new kind of many-body system.