Let \(k\) be an algebraic number field of finite degree over \(\mathbb{Q}\). For a field \(K\) of characteristic zero let \({\mathcal M} {\mathcal M}_ K\) denote the \(\mathbb{Q}\)-linear category of mixed motives in the sense of U. Jannsen or P. Deligne. More generally, one may consider mixed motives (over \(K\)) with multiplication by a number field \(E\). These give an \(E\)-linear category \({\mathcal M} {\mathcal M}_ K (E)\) equivalent to the pseudo-abelian completion of \({\mathcal M} {\mathcal M}_ K \otimes E\). A similar result can be obtained for \({\mathcal M} {\mathcal H}_ K (E)\), where \({\mathcal M} {\mathcal H}_ K\) is the category of real mixed Hodge structures, if \(K = \mathbb{C}\) or \(\mathbb{R}\). In this last case \({\mathcal M} {\mathcal H}_ K\) will mean the category of real mixed Hodge structures equipped with an involution \(F_ \infty\) that respects the weight filtration and whose \(\mathbb{C}\)-linear extension to \(H_ \mathbb{C}\) \((H\) a real mixed Hodge structure) maps \(F^ \bullet H_ \mathbb{C}\) into \(\overline F^ \bullet H_ \mathbb{C}\). Now let \(M\) be a mixed motive in \({\mathcal M} {\mathcal M}_ k (E)\). Then one defines the \(L\)-function \(L(M,s)\) of \(M\) as the product \(L(M,s) = \prod_{\mathfrak p} L_{\mathfrak p} (M,s)\), where the product runs over the finite primes of \(k\) and where \(L_{\mathfrak p} (M,s)\) is the local \(L\)-function of \(M\) at \({\mathfrak p}\). This is an \((E \otimes \mathbb{C})\)-valued function of the form \((\text{det}_ \mathbb{C} (1 - F_{\mathfrak p} N{\mathfrak p}^{-s} | M^{I_{\mathfrak p}}_{\ell, \imath, \sigma})^{-1})_{\sigma \in \operatorname{Hom} (E, \mathbb{C})}\). \(F_{\mathfrak p}\) is the geometric Frobenius at \({\mathfrak p}\). The subscripts \(\ell\) and \(\imath\) mean that one takes the tensor product of the \(\ell\)-adic realization of \(M\) with \(\mathbb{C}\) over \(\mathbb{Q}_ \ell\) via the embedding \(\imath : \mathbb{Q}_ \ell \hookrightarrow \mathbb{C}\). \(I_{\mathfrak p}\) is the inertia subgroup of the absolute Galois group of \(k_{\mathfrak p}\) at \({\mathfrak p}\). If \({\mathfrak p}\) lies over the prime number \(p\), one takes \(\ell \neq p\). It is conjectured that \(L_{\mathfrak p} (M,s)\) does not depend on \(\ell\) or \(\imath\). For any prime \({\mathfrak p}\) at infinity one defines an \((E \otimes \mathbb{C})\)-valued \(L\)-factor \(L_{\mathfrak p} (M,s)\) in terms of gamma-functions determined by the Hodge structure of the Betti realization of \(M\). More specifically, for \(K = \mathbb{C}\) or \(\mathbb{R}\) and \(H\) an object of \({\mathcal M} {\mathcal H}_ K(E)\) one defines a \((E \otimes \mathbb{C})\)-invariant filtration \(\gamma^ \nu H_ \mathbb{C} = F^ \nu H_ \mathbb{C} \cap \overline F^ \nu H_ \mathbb{C}\) and the corresponding graduation determines the powers of the various gamma-functions. Finally one defines the completed \(L\)-function of the motive \(M\), \(\hat L(M,s)\), as the product \(\hat L(M,s) = L(M,s) \prod_{{\mathfrak p} | \infty} L_{\mathfrak p} (M,s)\). If \(w_{\max}\) is the maximal weight of \(M\) this completed \(L\)-function is holomorphic and non-vanishing in the halfplane \(\text{Re} (s) > w_{\max}/2 + 1\). One conjectures that \(\hat L(M,s)\) has a meromorphic continuation to the whole complex plane, and satisfies a functional equation of the form \(\hat L(M,s) = \varepsilon (M,s) \hat L (M^*,1-s)\), where \(M^*\) is the dual motive and \(\varepsilon (M,s) = ae^{bs}\) for some \(a \in (E \otimes \mathbb{C})^*\) and \(b \in E \otimes \mathbb{C}\). Furthermore, it is conjectured that \(\hat L(M,s) = {\hat L_ 1 (M,s) \over \hat L_{02} (M,s)}\), where \(\hat L_ 1 (M,s)\) is entire of genus one and \(\hat L_{02} (M,s)\) is a polynomial in \(s\) with integer zeros. One can give a conjectural description of \(\varepsilon (M,s)\) as a product of local \(\varepsilon\)-factors which are obtained from the restrictions of \(M \otimes k_{\mathfrak p}\) of \(M\) to \(\text{Spec}(k_{\mathfrak p})\).
For the entire collection see [Zbl 0788.00053].