In general, artery-specific calcification analysis comprises the simultaneous calcification segmentation and quantification tasks. It can help provide a thorough assessment for calcification of different coronary arteries, and further allow for an efficient and rapid diagnosis of cardiovascular diseases (CVD). However, as a high-dimensional multi-type estimation problem, artery-specific calcification analysis has not been profoundly investigated due to the intractability of obtaining discriminative feature representations. In this work, we propose a Multi-task learning network with Multi-view Weighted Fusion Attention (MMWFAnet) to solve this challenging problem. The MMWFAnet first employs a Multi-view Weighted Fusion Attention (MWFA) module to extract discriminative feature representations by enhancing the collaboration of multiple views. Specifically, MWFA weights these views to improve multi-view learning for calcification features. Based on the fusion of these multiple views, the proposed approach takes advantage of multi-task learning to obtain accurate segmentation and quantification of artery-specific calcification simultaneously. We perform experimental studies on 676 non-contrast Computed Tomography scans, achieving state-of-the-art performance in terms of multiple evaluation metrics. These compelling results evince that the proposed MMWFAnet is capable of improving the effectivity and efficiency of clinical CVD diagnosis.