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The atmospheric turbulence (AT) causes distortion of phase fronts of orbital angular momentum (OAM) beams, which hinders the recognition of OAM modes. Convolutional neural network (CNN), a deep learning (DL) technique, can be utilized to realize the effective recognition of OAM modes. In this article, we propose a properly designed six-layer CNN model that can achieve high recognition accuracy (RA) of OAM modes at a reasonable computing complexity. We used intensity images of Laguerre-Gaussian (LG) beams to train our CNN model and explored the relationship between the RA for different single OAM modes and many factors including the number of training epochs, AT intensity, transmission distance, and the number of single OAM modes. Our CNN model can obtain an RA of 97.1% under moderate turbulence and 80% under strong turbulence, which are better than some CNN models proposed previously. Compared with these previous CNN models, our CNN model can also reduce the time consumption for at most 70%. Our research could contribute in achieving higher data capacity in OAM-based free space optical (FSO) communication systems.