The globally growing data centers energy consumption and their carbon emissions pose significant environmental challenges. In this context, discerning server fabrication and operational energy contribution to data center carbon footprint is key to identifying effective mitigation strategies. This study partitions server carbon footprints into a) fabrication (embodied carbon), b) static operational power, and c) dynamic operational power, and proposes a novel 2D representation for analyzing data centers carbon impacts. This representation highlights the contributions of these three a-c factors, for any server load and any carbon intensity of electricity. To showcase our methodology and representation, we conducted experimental power measurements on four diverse servers under various load conditions, and combined them with Life Cycle Assessment (LCA) methods for their embodied carbon. Our results show that operational energy generally dominates the total footprint. Indeed, high static power consumption, due to poor energy proportionality in current hardware, is a major carbon emission factor, especially at low loads. We conclude that optimization efforts should follow this sequence: 1) improve server utilization, 2) prioritize low-carbon electricity, 3) maximize server lifetime. Hence, fabrication impact is primarily relevant only when servers are powered by low-carbon electricity. Our representation shows that reducing static power waste through future hardware with better energy proportionality is a priority to design and operate sustainable data centers.