My question is parallel to J. Borger' question:
Can the failure of the multiplicativity of Euler factors at bad primes be corrected?
As emphasized by Scholbach in his paper on special values of L-functions, the L-functions of motives over Z is multiplicative in triangles.
However, the archimedean L-factor is not. Is there a notion of motives over $Z_\infty$ that corrects this problem, making the archimedean $L$-factor multiplicatives in triangles?
One may define motives for varieties over $Z_\infty$ in the sense of Durov by the usual $A^1$-homotopical machinery (Zariski or Scholze-Bhatt pro-étale topology). Maybe some kinds of "Kaehler motives" may be better adapted. Even the linear structure analogous to Galois representations is not clear in this setting, because Hodge structures don't give multiplicative archimedean L-factors.
