In this paper, a family of six-weight cyclic codes over Fp\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {F}_{p}$$\end{document} whose duals have three zeros is presented, where p\documentclass[12pt]{minimal}
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\begin{document}$$p$$\end{document} is an odd prime. Furthermore, the weight distributions of these cyclic codes are determined.