In this thesis, we use numerical methods in conjunction with trace or explicit formula to obtain various number theoretical results. The main results are: the derivation of an explicit version of the trace formula that will enable us to compute the low-lying eigenvalues of the spectrum of all congruence subgroups ┌o(N,X) for non-squarefree level, N, and even Dirichlet character, X; we prove new cases of the Artin Conjecture for S5-Artin Representations; we prove an upper bound for ranks of high-ranked elliptic curves. We also use the numerical method of computing an optimal test-function for explicit formulae to investigate the relationship between the rank and zero-repulsion of L-functions corresponding to elliptic curves