2 formulated a completely new question

Arepartial$\zeta$-functionsOntherelationofspecialvaluesof motivic ?Lfunctionsandpartialzetas

My first question is now if the partial zetas $\zeta_\sigma(s)$ are motivic?

(EDIT: this doesn't seem likely, see the comments)

but I don't know if this answers my question.

BEGIN (EDIT)

Even if the partical zetas are not motivic it is pretty clear that they are not very far from being motivic. It seems reasonable to ask if known procedures in order to compute special values of motivic L functions carry over in a modified version to partial zetas.

Let us for example consider the case of the Dedekind zeta function $\zeta_K(s)$ by the analytic class number formula we know that the leading coefficient $\zeta_K^*(0)$ at $s=0$ is given by $$\zeta_K^*(0) = \frac {h_K R_K} {\omega_K}$$ where $h_K$ is the class number, $R_K$ the regulator and $\omega_K$ the number of roots of unity.

At present, my "favourite method" to prove the above formula is by means of calculating the volume of the Arakelov-Picard group of degree zero divisors $Pic^{(0)}(K)$. This is for example explained beautifully in the article of van der Geer and Schoof (http://arxiv.org/abs/math/9802121).

One question is now the following: Is it known if the volume of certain subspaces of the Arakelov-Picard group Pic(K) or certain quotients of the latter can be related to the leading coefficients at $s=0$ of partial zeta functions?

If not the Arakelov-Picard group, are their attempts to my question using volumina of different spaces?

(Unfortuanetly, I do not even know if the Arakelov-Picard group can be used to compute the leading coefficients of the Artin L functions $L(\chi,s)$ from above, but at least we have conjectures (Beilinson, ...) at hand which describe the leading coefficient...)

END (EDIT)

(I know that at least some parts of the many Stark conjectures appear under the roof of the equivariant Tamagawa number conjecture together with Beilinson's, but I don't know if this contains an answer to my question.)

1

Are partial $\zeta$-functions motivic?

Let $K$ be a number field, $L$ a finite abelian extension and $\chi \in \widehat{Gal(L/K)}$ a (non-trivial) character. If we multiply out the associated Artin L-function $L(\chi,s)$ we can write this as a finite sum of "partial" $\zeta$-functions $\zeta_\sigma (s)=\zeta_{\chi,\sigma}(s)$, namely we have $$L(\chi,s) = \sum_{\sigma \in Gal(L/K)} \chi(\sigma) \zeta_\sigma (s)$$ My question is now if the partial zetas $\zeta_\sigma(s)$ are motivic?

By finite Fourier inversion we can express every partial zeta as linear combination of motivic L-functions: $$\zeta_\sigma(s) = \frac 1 {[L:K]}\sum_{\chi \in \widehat{Gal(L/K)}}\overline\chi (\sigma) L(\chi,s)$$ but I don't know if this answers my question.

The motivation for my question is the following:

(Roughly speaking) the partial zetas $\zeta_{\sigma}$ appear prominently in the work of Stark on explicit class field theory, their first coefficients of the Taylor expansion at $s=0$ are expected to be logarithms of units in $L$, thereby producing explicit abelian extensions of $K$.

On the other hand Beilinson's program provides conjectural information about special values of motivic L-functions and I would like to understand if Stark's conjectures fit into this program or if they provide a refinement.

(I know that at least some parts of the many Stark conjectures appear under the roof of the equivariant Tamagawa number conjecture together with Beilinson's, but I don't know if this contains an answer to my question.)