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In the United States, the Federal Communications Commission has adopted rules permitting commercial wireless networks to share spectrum with federal incumbents in the 3.5 GHz Citizens Broadband Radio Service band. These rules require commercial systems to vacate the band when sensors detect radars operated by the U.S. military; a key example being the SPN-43 air traffic control radar. Such sensors require highly-accurate detection algorithms to meet their operating requirements. In this paper, using a library of over 14,000 3.5 GHz band spectrograms collected by a recent measurement campaign, we investigate the performance of thirteen methods for SPN-43 radar detection. Namely, we compare classical methods from signal detection theory and machine learning to several deep learning architectures. We demonstrate that machine learning algorithms appreciably outperform classical signal detection methods. Specifically, we find that a three-layer convolutional neural network offers a superior tradeoff between accuracy and computational complexity. Last, we apply this three-layer network to generate descriptive statistics for the full 3.5 GHz spectrogram library. Our findings highlight potential weaknesses of classical methods and strengths of modern machine learning algorithms for radar detection in the 3.5 GHz band.