central/critical/special values of L-functions terminology

Question

I have a question about the terminology for special values of L-functions. Is the following a correct description of standard usage:

Suppose L(s) is an L-function which satisfies a functional equation relating $s$ to $w+1-s$, where $w$ is the (motivic) weight. ADDED LATER: I am assuming the L-function is motivic, otherwise (please correct me if I am wrong) there is nothing special about the value at any integer.

1) If $m$ is an integer then $L(m)$ is a special value of the L-function.

2) If $m$ is an integer and neither $m$ nor $w+1-m$ is a pole of a $\Gamma$-factor of the L-function, then $m$ is a critical point and $L(m)$ is a critical value of the L-function.

3) $L(\frac{w+1}{2})$ is the central value of the L-function.

4) If $\frac{w+1}{2}$ is not an integer, then the central value is not a special value.

I am pretty sure 2) is correct, unless Deligne's notion of critical point is not the only one. I am also pretty sure 3) is correct, since the central point of the functional equation is pretty unambiguous. It is 1) and 4) that I am hoping the experts can clarify.

Answer 1

I think everything you write is correct, and moreover very clear. For example, the value $\zeta(1/2)$ is not a special value of the Riemann Zeta function: $w=0$ in this case. On the contrary, if $E$ is an elliptic curve, $w=1$ and $L(E,(w+1)/2)=L(E,1)$ is a special and the central value of $E$, and it is the value of interest for the Birch-Swinnerton Dyer. There is a dichotomy between the $L$-function of motives with even, and with odd, motivic weights, and a Tate twist won't change in which world your $L$-motive is, because it adds an even integer to the weight.

The Bloch-Kato's conjecture aims at understanding the order $n$ (and then the value of the $n$-th derivative) of an $L$-function $L(M,s)$ at any point $s \in \mathbb Z$, in terms of algebraic informations on the motive $M$ giving rise to that $L$-function. Proving that the order $n$ of $L(M,s)$ at some $s \in \mathbb Z$ is at least what the Bloch-Kato conjecture predicts is completely elementary (provided we have the functional equation for $L(M,s)$ at every $s \in \mathbb Z$ except when $s$ is the central value. When $w$ is even, this means that the lower bound of Bloch-Kato for the order of $L(M,s)$ at any integer $s$ is known. It remains to prove the upper-bound, and then to compute precisely the value of the suitable derivative, which of course is not an easy task. When $w$ is odd, both proving the BK lower bound and proving the BK upper bound on the order of $L(M,s)$ at the central point $(w+1)/2$ are open. That makes this point the center of all attentions. Example as above: $L(E,1)$ for $E$ an elliptic curve, or an abelian variety.

One can complete this lexicon with: 5) The near central points are defined, in the case $w$ even only, as follows: the point $w/2$ and $w/2+1$.

Answer 2

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.