special values of L-functions terminology

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Apr 17, 2013 at 17:12	comment	added	GH from MO		Dear David and David, my comment has nothing to do with the original question. I just responded to the term "loathe that normalization" (which is now "dislike that normalization"). Of course I respect the advantages of the algebraic/motivic normalization.
Apr 17, 2013 at 16:41	comment	added	David Farmer		I wrote the question in the "arithmetic" normalization, with $s$ going to $w+1-s$ in the functional equation. I know that "most" L-functions are not motivic, but my understanding is that the concept of special/critical values only makes sense for motivic L-functions. You wouldn't say, for example, that every integer is a critical point for the L-function of a Maass form. There are times when you want to normalize the L-function so that $s$ goes to $1-s$, but this is not one of those times.
Apr 17, 2013 at 16:27	history	edited	David Loeffler	CC BY-SA 3.0	slightly less combative :-)
Apr 17, 2013 at 16:06	comment	added	David Loeffler		I see what you mean, but the normalization is inconvenient when the $L$-function really is motivic and you are interested in special values, which was the context of the question.
Apr 17, 2013 at 15:40	comment	added	GH from MO		Most $L$-functions are not motivic (as far as we know), so there is no $w$ or algebraicity as a guide for normalization. Instead, there is the functional equation (in any normalization), so it makes a lot of sense to normalize it as Riemann did. Needless to say many things we know about motivic $L$-functions follow from putting them into a larger family of (non-motivic) $L$-functions.
Apr 17, 2013 at 15:24	history	answered	David Loeffler	CC BY-SA 3.0