A topic related to hypergeometric motives is that of Jacobi sum motives. These are indeed simpler, and in fact the tame prime information for hypergeometric motives can be determined from Jacobi motives, possibly twisted by Kummer and Tate characters.
The classical Jacobi sums were indicated by Weil to come from Grössencharacters [Wei52], and this functionality is also included, with it indeed being the preferred method to compute Euler factors and the L-series, once the reciprocity correspondence has been established and the Grössencharacter identified.
Let nj∈(Z) and xj∈(Q)/(Z) with θ=∑j nj< xj > an element of the free group on (Q)/(Z) with ∑njxj∈(Z).
Letting m be the least common multiple of the denominators of the xj, the field of definition Kθ is a subfield of (Q)(ζm), corresponding by class field theory to quotienting out by ((Z)/m(Z))star by elements which leave θ fixed when scaling by them. When scaling by -1 fixes θ this field Kθ is totally real, and otherwise it is a CM field.
For primes p with gcd(p, m)=1, we consider Gauss sums corresponding for prime ideals p in (Q)(ζm), defined by Ga/mψ(p)= - ∑_(x∈(F)ppstar) ((x/p))am ψ((Tr)_((F)p)^((F)pp) x), where ψ is a nontrivial additive character on (F)p and the power residue symbol takes values in the roots of unity of (Q)(ζm) with ((x/p))am ≡ x(q - 1)a/m ((mod) p). The associated Jacobi sum evaluation for θ at p is then given by ∏j Gxj(p)nj with the result being independent of the choice of additive character ψ. This defines the Jacobi sum for good primes p up to a choice of ζm into (C).
If one is just interested in Euler factors over (Q) and not Kθ, then a p-adic method using the Gross-Koblitz formula can also be used. There are known bounds on the conductor of the resulting L-function, the first being that of Weil [Wei52].
A Jacobi motive can also be Kummer twisted by tρ for some rational ρ and nonzero rational t. This can increase the field of definition so that m includes the denominator of ρ. This corresponds to multiplying the various Jacobi sum evaluations by suitable roots of unity.
Often one wants to Tate twist the Jacobi sum to gets its effective weight, and for this reason the full unit is sometimes called a Jacket motive (for Jacobi, Kummer, and Tate).