It is difficult to say which of the six primary and numerous secondary characteristics of a state are most important: they are all so interrelated that trying to determine the most important is like trying to determine if the chicken or the egg came first. However, in my opinion, the most important is agriculture. Taken by itself it is not the be-all, end-all, but its occurrence is generally what sparks the emergence of the other characteristics. And, it is that it is fairly easy for archaeologists to find tangible evidence of intensive agriculture (for example, changed plant/animal distribution, changes in the features of plants/animals as seen in fossils, sickle sheen on blades). This is in contrast to some other state characteristics, such as state authority and state religion, because the information we have on these latter characteristics is often gathered through ethnohistory, which is generally nonconcrete information that requires extrapolation.
Arguably, nearly all of the primary characteristics are reliant on the preestablishment of intensive agriculture. Intensive agriculture occurs only in nonnomadic groups, and as the agriculture becomes more productive, it necessitates the establishment of storage buildings and some sort of administrators to distribute the harvest (so, more buildings and more people; thus, urbanism). When such large surpluses occur and there is a large enough amount of people devoted to maintaining the agricultural side of things, we also see the emergence of skilled workers who can dedicate their time to specialized crafts without fear of starving. These large surpluses also allow the state to start forming a complex economy–an excess of harvest goods allows it to exchange food and specialized crafts in exchange for items not readily available in its locality, thus developing often-long-distance trade networks.
With the aforementioned establishment of distribution administrators also comes state authority (although not necessarily right away). Those with the power to control the distribution of food eventually find that they have all the power and therefore can make (and enforce) decisions. Here is where we typically also start to see the development of social stratification, with the state authority forming the upper class, then skilled laborers, then farmers. And of course, agriculture and the primary characteristics that emerge because of it influence secondary state characteristics such as epidemic disease (due to the high concentration of people around the productive agriculture zones) and state art, writing, and metallurgy (at a certain point, not everyone has to dedicate all of their time to agriculture; this is an aspect of specialization).
In sum, while all primary characteristics are extremely coinfluencial, agriculture often has to be the first to occur. It is difficult to imagine, for instance, a highly urbanized city that emerged before intensive agriculture was established or a complex economy that came into being before a community had a relatively easy, steady, reliable source of food.