Zadoff-Chu (ZC) sequences have been used as synchronization sequences in modern wireless communication systems, replacing the conventional pseudorandom noise sequences due to their perfect autocorrelation properties. We first study the problem of ambiguity between a timing offset and a frequency offset, which arises when a ZC sequence is used as a synchronization signal. We then show how a frequency offset can impair the timing property of a ZC sequence, causing irreducible timing errors. An analytical framework, particularly the timing spectrum, is developed, which fully characterizes a ZC sequence's timing properties and its fundamental limitations as a time synchronization sequence in the presence of a frequency offset between the transmitter and the receiver. This analytical framework provides a powerful analytical tool for timing signal design and performance analysis of ZC sequences.