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slightly less combative :-)

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edited Apr 17, 2013 at 16:27

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How are you normalizing your $L$-function? I ask because in some fields, it's conventional to normalize everything by shifting so the functional equation always takes $s$ to $1-s$. If the original motivic weight is odd, this means that what I would call critical values may sometimes get moved to half-integers rather than integers. (This is why I personally loathedislike that normalization.)

If your $L$-function is the $L$-function of a motive and you don't do any strange shiftings (so your usage is consistent with the motives literature), then (2) and (3) are unambiguously correct, (4) is a consequence of (1), and (1) accords with how I use the term but I've never seen it written down as a formal definition.

E.g. if the $L$-function is $L(f, s) = \sum_{n \ge 1} a_n(f) n^{-s}$ for $f = \sum_{n \ge 1} a_n q^n$ a modular cusp form of weight $k$, then the functional equation sends $s$ to $k - s$, the special values are at $s \in \mathbb{Z}$, the critical values are at $\{1, \dots, k-1\}$, and if $k$ is odd then the central value at $s = k/2$ is not a critical or special value.

How are you normalizing your $L$-function? I ask because in some fields, it's conventional to normalize everything by shifting so the functional equation always takes $s$ to $1-s$. If the original motivic weight is odd, this means that what I would call critical values may sometimes get moved to half-integers rather than integers. (This is why I personally loathe that normalization.)

If your $L$-function is the $L$-function of a motive and you don't do any strange shiftings (so your usage is consistent with the motives literature), then (2) and (3) are unambiguously correct, (4) is a consequence of (1), and (1) accords with how I use the term but I've never seen it written down as a formal definition.

E.g. if the $L$-function is $L(f, s) = \sum_{n \ge 1} a_n(f) n^{-s}$ for $f = \sum_{n \ge 1} a_n q^n$ a modular cusp form of weight $k$, then the functional equation sends $s$ to $k - s$, the special values are at $s \in \mathbb{Z}$, the critical values are at $\{1, \dots, k-1\}$, and if $k$ is odd then the central value at $s = k/2$ is not a critical or special value.

How are you normalizing your $L$-function? I ask because in some fields, it's conventional to normalize everything by shifting so the functional equation always takes $s$ to $1-s$. If the original motivic weight is odd, this means that what I would call critical values may sometimes get moved to half-integers rather than integers. (This is why I personally dislike that normalization.)

If your $L$-function is the $L$-function of a motive and you don't do any strange shiftings (so your usage is consistent with the motives literature), then (2) and (3) are unambiguously correct, (4) is a consequence of (1), and (1) accords with how I use the term but I've never seen it written down as a formal definition.

E.g. if the $L$-function is $L(f, s) = \sum_{n \ge 1} a_n(f) n^{-s}$ for $f = \sum_{n \ge 1} a_n q^n$ a modular cusp form of weight $k$, then the functional equation sends $s$ to $k - s$, the special values are at $s \in \mathbb{Z}$, the critical values are at $\{1, \dots, k-1\}$, and if $k$ is odd then the central value at $s = k/2$ is not a critical or special value.

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created Apr 17, 2013 at 15:24

37k
3
89
194

How are you normalizing your $L$-function? I ask because in some fields, it's conventional to normalize everything by shifting so the functional equation always takes $s$ to $1-s$. If the original motivic weight is odd, this means that what I would call critical values may sometimes get moved to half-integers rather than integers. (This is why I personally loathe that normalization.)

If your $L$-function is the $L$-function of a motive and you don't do any strange shiftings (so your usage is consistent with the motives literature), then (2) and (3) are unambiguously correct, (4) is a consequence of (1), and (1) accords with how I use the term but I've never seen it written down as a formal definition.

E.g. if the $L$-function is $L(f, s) = \sum_{n \ge 1} a_n(f) n^{-s}$ for $f = \sum_{n \ge 1} a_n q^n$ a modular cusp form of weight $k$, then the functional equation sends $s$ to $k - s$, the special values are at $s \in \mathbb{Z}$, the critical values are at $\{1, \dots, k-1\}$, and if $k$ is odd then the central value at $s = k/2$ is not a critical or special value.