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Multivariate Time Series (MTS) classification has gained importance over the past decade with the increase in the number of temporal datasets in multiple domains. The current state-of-the-art MTS classifier is a heavyweight deep learning approach, which outperforms the second-best MTS classifier only on large datasets. Moreover, this deep learning approach cannot provide faithful explanations as it relies on post hoc model-agnostic explainability methods, which could prevent its use in numerous applications. In this paper, we present XCM, an eXplainable Convolutional neural network for MTS classification. XCM is a new compact convolutional neural network which extracts information relative to the observed variables and time directly from the input data. Thus, XCM architecture enables a good generalization ability on both large and small datasets, while allowing the full exploitation of a faithful post hoc model-specific explainability method (Gradient-weighted Class Activation Mapping) by precisely identifying the observed variables and timestamps of the input data that are important for predictions. We first show that XCM outperforms the state-of-the-art MTS classifiers on both the large and small public UEA datasets. Then, we illustrate how XCM reconciles performance and explainability on a synthetic dataset and show that XCM enables a more precise identification of the regions of the input data that are important for predictions compared to the current deep learning MTS classifier also providing faithful explainability. Finally, we present how XCM can outperform the current most accurate state-of-the-art algorithm on a real-world application while enhancing explainability by providing faithful and more informative explanations.