Millimeter-wave (mmWave) communication systems rely on large-scale antenna arrays to combat large path-loss at mmWave band. Due to hardware characteristics and deployment environments, mmWave large-scale antenna systems are vulnerable to antenna element blockages and failures, which necessitate diagnostic techniques to locate faulty antenna elements for calibration purposes. Current diagnostic techniques require full or partial knowledge of channel state information (CSI), which can be challenging to acquire in the presence of antenna failures. In this letter, we propose a blind diagnostic technique to identify faulty antenna elements in mmWave large-scale antenna systems, which does not require any CSI knowledge. By jointly exploiting the sparsity of mmWave channel and failure pattern, we first formulate the diagnosis problem as a joint sparse recovery problem. Then, the atomic norm is introduced to induce the sparsity of mmWave channel over continuous Fourier dictionary. An efficient algorithm based on alternating direction method of multipliers (ADMM) is proposed to solve the formulated problem. Finally, the performance of the proposed technique is evaluated through numerical simulations.