  In this paper, we will study the connections between the mirror symmetry of
Calabi-Yau threefolds and Deligne's conjecture on the special values of the
$L$-functions of critical motives. Using the theory of mirror symmetry, we will
develop a method to compute the Deligne's period for a Calabi-Yau threefold in
the mirror family of a one-parameter mirror pair. We will give two examples to
show how this method works, and we will express the Deligne's period in terms
of the classical periods of the threeform. Using this method, we will compute
the Deligne's period of a Calabi-Yau threefold studied in a recent paper by
Candelas, de la Ossa, Elmi and van Straten. Based on their numerical results,
we will explicitly show that this Calabi-Yau threefold satisfies Deligne's
conjecture. A second purpose of this paper is to introduce the Deligne's
conjecture to the physics community, and provide further evidence that there
might exist interesting connections between physics and number theory.
