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We study conjugacy classes of germs of non-flat diffeomorphisms of the realline fixing the origin. Based on the work of Takens and Yoccoz, we establishresults that are sharp in terms of differentiability classes and order oftangency to the identity. The core of all of this lies in the invariance ofresidues under low-regular conjugacies. This may be seen as an extension of thefact (also proved in this article) that the value of the Schwarzian derivativeat the origin for germs of $C^3$ parabolic diffeomorphisms is invariant under$C^2$ parabolic conjugacy, though it may vary arbitrarily under parabolic $C^1$conjugacy.

On residues and conjugacies for germs of 1-D parabolic diffeomorphisms in finite regularity