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Traffic classification, i.e. the identification of the type of applications flowing in a network, is a strategic task for numerous activities (e.g., intrusion detection, routing). This task faces some critical challenges that current deep learning approaches do not address. The design of current approaches do not take into consideration the fact that networking hardware (e.g., routers) often runs with limited computational resources. Further, they do not meet the need for faithful explainability highlighted by regulatory bodies. Finally, these traffic classifiers are evaluated on small datasets which fail to reflect the diversity of applications in real-world settings. Therefore, this paper introduces a new Lightweight, Efficient and eXplainable-by-design convolutional neural network (LEXNet) for Internet traffic classification, which relies on a new residual block (for lightweight and efficiency purposes) and prototype layer (for explainability). Based on a commercial-grade dataset, our evaluation shows that LEXNet succeeds to maintain the same accuracy as the best performing state-of-the-art neural network, while providing the additional features previously mentioned. Moreover, we illustrate the explainability feature of our approach, which stems from the communication of detected application prototypes to the end-user, and we highlight the faithfulness of LEXNet explanations through a comparison with post hoc methods.