Let $n$ be a positive integer, $a\_1,\ldots,a\_{n-1}\in\mathbb{Z}$. Suppose that for every $u\in(\mathbb{Z}/n\mathbb{Z})^\times$,
$$
\tag{$\star$}
\sum\_{i=1}^{n-1} i a\_{(ui\!\!\!\mod n)} = 0.
$$
Then the product of values of the Gamma function
$$
\tag{$\star\star$}
\prod\_{i=1}^{n-1}\Gamma\left(\frac{i}{n}\right)^{a\_i}
$$
is known to be an algebraic number. This is a consequence of the reflection and multiplication formulas for $\Gamma$, and the result is proved in an appendix by Koblitz and Ogus to [1].


I would like to know if the converse of this result is expected to be true. Since the converse is almost certainly open if it is true, I will ask:



> 
> 
> > 
> > Are there any examples where the product $(\star\star)$ is known to be algebraic but $(\star)$ does not hold?
> > 
> > 
> > 
> 
> 
> 


[1] Pierre Deligne, *Valeurs de fonctions $L$ et périodes d'intégrales*, Proceedings of Symposia in Pure Mathematics, Vol. **33** (1989), part 2, pp. 313-346



Converse of a result of Koblitz and Ogus on algebraic products of gamma values
A convenient way to formulate this kind of questions is to use the language of *distributions* introduced by Kubert and Lang. I gave a short account in [this answer](https://mathoverflow.net/a/318995/6506) to a previous MO question.


Your question is equivalent to asking whether the $\Gamma$ distribution is universal. Rohrlich has conjectured that this is the case, see Lang, *Cyclotomic fields I and II*, Chapter 2, Appendix, p. 66.


Lang has made the stronger conjecture that every *polynomial* relation between the values of $\Gamma$ at rational arguments is in the ideal generated by the standard relations, see $\S$3.1.5 in [this survey](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/AWSLecture3.pdf) of Michel Waldschmidt.


It seems that these conjectures are wide open. According to Waldschmidt (see $\S$5.3 in the [same survey](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/AWSLecture5.pdf)), already the case $n=5$ seems open.


