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Minhyong Kim introduced arithmetic Chern-Simons invariants for totallyimaginary number fields as arithmetic analogues of the Chern-Simons invariantsfor 3-manifolds. In this paper, we extend Kim's definition for any numberfield, by using the modified \'etale cohomology groups and fundamental groupswhich take real places into account. We then show explicit formulas of mod 2arithmetic Dijkgraaf-Witten invariants for real quadratic fields $\mathbb{Q}(\sqrt{p_1 p_2 \cdots p_r})$, where $p_i$ is a prime number congruent to 1 mod4, in terms of the Legendre symbols of $p_i$'s. We also show topologicalanalogues of our formulas for 3-manifolds.

On mod 2 arithmetic Dijkgraaf-Witten invariants for certain real quadratic number fields