Hi friends,

I have some questions concerning the critical values of motives, in the sense of Deligne. I will only look at motives of the form $h^i(X)$ where $X$ is a smooth projective algebraic variety over $\mathbb{Q}$. If I understand correctly, the notion of critical value depends only on the Hodge numbers.

Introduce the notations:

$$ \Gamma_\mathbb{C}(s):=2(2\pi)^{-s}\Gamma(s), \quad \Gamma_\mathbb{R}(s):=\pi^{-s/2}\Gamma(\frac{s}{2}) $$ where $\Gamma$ is the usual Gamma function. Then one define the $L$-factor at infinity as follows.

Let us first consider the case where $i$ is odd. Then:

$$ L_\infty(h^i(X), s)=\prod_{p < q} \Gamma_{\mathbb{C}}(s-p)^{h^{p, q}} $$ where $p+q=k$ and $h^{p, q}$ is the corresponding Hodge number.

**Example**: If $i=3$ and $h^{3, 0}=0$ (for instance $X$ is an hypersurface in $\mathbb{P}^4$), then

$L_\infty(h^3(X), s)=\Gamma_{\mathbb{C}}(s-1)^{h^{2, 1}}$

When $i$ is even, the definition is more involved, as one has also to consider the action of complex conjugation on $H^{p, p}$, which decomposes this space as $H^{p^+}\oplus H^{p^{-}}$. Let

$$ h^{p^{\pm}}:=\dim_{\mathbb{C}} H^{p^{\pm}}. $$

Then $$ L_\infty(h^i(X), s)=\prod_{p < q} \Gamma_{\mathbb{C}}(s-p)^{h^{p, q}}\cdot \Gamma_{\mathbb{R}}(s-\frac{i}{2})^{h^{i/2+}} \Gamma_{\mathbb{R}}(s-\frac{i}{2}+1)^{h^{i/2-}} $$

After having introduced this: an integer $n$ is said to be critical for $M=h^i(X)$ if $L_\infty(M, s)$ has no pole at $s=n$.

In his paper Deligne also asks that $L_\infty(\hat{M}, 1-s)$ has no pole at $n$. Is that necessary or it is a consequence of the former provided one has a functional equation?

Anyway, I would like to know if for my example the critical integers are all integers $n \geq 2$.

**Question 2** What about a $K3$ surface? Can one determine the critical integers for the transcedental lattice? In that case there is a $\Gamma(s)$ coming from $h^{2,0}=1$. What about the real Gamma factor?

Thanks for your help!