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    $\begingroup$ More generally, you may look at the values of the gamma function at rational arguments, and they are not motivic, i.e. they are not periods, but any product $\Gamma(s_1) \cdots \Gamma(s_k)$ for $s_1,\ldots,s_k\in \mathbb{Q}$ satisfying $\sum_i s_i\in \mathbb{Z}$ is motivic. So you want to say that individual gamma-values are motivic of fractional weight. For instance $\Gamma(1/2)=\sqrt{\pi}$ wants to be the square root of the Tate motive. $\endgroup$ Commented Dec 13, 2021 at 10:20
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    $\begingroup$ Do these live in MMHS, or some over-category of that? $\endgroup$ Commented Dec 13, 2021 at 10:22
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    $\begingroup$ @Balazs I presume the $\Gamma$ correspond to twisted de Rham cohomology classeson $\mathbb{A}^{1}$ (evaluated along rapid decay cycles etc) for some potential $t^{n}$ (maybe after étale pullback) so yes this should be encodable in EMHS $\endgroup$ Commented Dec 13, 2021 at 11:29
  • $\begingroup$ @Balasz I don't know exact definitions, but what EBz wrote makes sense. By the way, the Hodge decomposition for $\Gamma(s)$ should be $H^{s,1-s}=\mathbb{C}$, so the weight is in fact always integer, but the Hodge filtration has jumps at fractional indices, similar to how the original post suggests. $\endgroup$ Commented Dec 13, 2021 at 12:45