In this work, we study complex-valued data detection performance in massive multiple-input multiple-output (MIMO) systems. We focus on the problem of recovering an $n$-dimensional signal whose entries are drawn from an arbitrary constellation $\mathcal{K} \subset \mathbb{C}$ from $m$ noisy linear measurements, with an independent and identically distributed (i.i.d.) complex Gaussian channel. Since the optimal maximum likelihood (ML) detector is computationally prohibitive for large dimensions, many convex relaxation heuristic methods have been proposed to solve the detection problem. In this paper, we consider a regularized version of this convex relaxation that we call the regularized convex relaxation (RCR) detector and sharply derive asymptotic expressions for its mean square error and symbol error probability. Monte-Carlo simulations are provided to validate the derived analytical results.