Let LE(s, χ) be the Hasse-Weil L-function of an elliptic curve E defined over Q and twisted by a Dirichlet character χ of order k and of conductor fχ. Keating and Snaith introduced the way to study L-functions through random matrix theory of certain topological groups. Conrey, Keating, Rubinstein, and Snaith and David, Fearnley, and Kisilevsky developed their ideas in statistics of families of critical values of LE(1, χ) twisted by Dirichlet characters of conductors ≤ X and proposed conjectures regarding the number of vanishings in their families and the ratio conjectures of moments and vanishings which are strongly supported by numerical experiments. 
In this thesis, we review and develop their works and propose the ratio conjectures of moments and vanishings in the family of LE(1, χ) twisted by Dirichlet characters of conductors fχ ≤ X and order of some odd primes, especially 3, 5, and 7 inspired by the connections of L-function theory and random matrix theory. Moreover, we support our result on the ratio conjectures of moments and vanishings of the families for some certain elliptic curves by numerical experiments.