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The stability properties of the fundamental linear conjugate-order system are studied. The values of the complex order for which this system is stable and causal are determined. It is shown that all stable, causal systems have orders that lie within unit-radius circles centered at ±1 or on the imaginary axis in the order plane. Plots are given to illustrate these results for several examples. It is shown that as the system bandwidth moves to large or small values relative to unity, the stability region in the order plane becomes more fragmented.