A copolymer including ethylene units and units of one or more C3-10 alpha-olefins. The copolymer has a number average molecular weight of less than 5,000 g/mol, as measured by GPC. The ethylene content of the copolymer is less than 80 mol %. At least 70% of molecules of the copolymer have an unsaturated group, and at least 70% of said unsaturated groups are located in a terminal vinylidene group or a tri-substituted isomer of a terminal vinylidene group. The copolymer has a crossover temperature of −20° C. or lower and/or a certain ethylene run length. Also disclosed are a method for making the copolymer and polyolefins plasticized with 1-40 wt % of the copolymer.