This article reports the absolute and convective instability of a cyclindrical liquid jet in a co-flowing inviscid gas stream. An efficient mesh-searching method is developed to determine the absolute instability and the critical Weber number which separates the region of absolute from that of convective instability. If is found that both gas velocity and density may increase or decrease the critical Weber number and may promote or suppress the absolute instability; depending on flow conditions, and that it is the absolute rather than the relative velocity that governs the absolute instability. For Weber numbers larger than the critical values, the instability of liquid jets becomes convective. The absolute gas and liquid velocity at low Weber numbers and the relative velocity between the liquid and gas at high Weber numbers control the spatial growth rate, and may enhance significantly jet breakup process at high gas velocities. Surface tension has both stabilizing (for short waves) and destabilizing (for long waves) effects on the convective growth rate whenever there exists velocity discontinuity across the liquid-gas interface. Otherwise, it always promotes the jet instability. The gas-to-liquid density ratio shows a completely opposite effect for equal and unequal gas and liquid velocities, while liquid viscosity always has a damping effect.