Consider a complex normal surface singularity and its three plurigenera, the m-th L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}–plurigenus of Watanabe, the m-th plurigenus of Knöller and the m-th log-plurigenus of Morales. For any of these invariants we construct a double graded Z[U]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}[U]$$\\end{document}–module, whose Euler characteristic is the chosen plurigenus. The three outputs are compared with the analytic lattice cohomology of the germ, whose Euler characteristic is the classical geometric genus.