TITLE: Beilinson conjectures
QUESTION [21 upvotes]: Continuing an amazingly interesting chain of answers about motivic cohomology, I thought I should learn about the Beilinson conjectures, referred there.
I have found some references, and they seem to present the conjectures from different sides, e.g. there's the statement about vanishing but then there are also connections to motivic polylogarithms. 
What I miss from these articles in a general picture that would allow us to start somewhere natural. So,

how would you describe an introduction into Beilinson conjectures in motivic homotopy?

Sorry for such a loaded question — I really don't know how to make it fit MathOverflow format better. One could theoreticlly post lost of specific questions on the topic, but to ask the right questions in this case you might need to know more than I do. Also, I know there are some technical developments, e.g. the language of derived stacks, and my hope would be that somebody could make a connection to these conjectures using some  clear and suitable language.

REPLY [2 votes]: Beilinson's conjecture, together with Tate's conjecture (on algebraic cycles on smooth projective varieties over $\mathbf F_p$) and with Soulé's conjecture (on the pole orders of zeta functions for schemes of finite type over $Spec \mathbf Z$) can be unified into one statement: subject to certain "standard" conjectures on motivic t-structures, they are (jointly) essentially equivalent to the following: for a geometric motive $M$ over $Spec \mathbf Z$ (i.e., a compact object in the triangulated category $DM(Spec Z)$), there is a natural pairing
$$Hom(1, M) \otimes Hom(M, \hat 1(1)[2]) \to Hom(1, \hat 1(1)[2]).$$
Here, 1 is the monoidal unit (i.e., the motive of $Spec Z$ itself), and $\hat 1$ is the fiber of the map $1 \to H_D$, where $H_D$ is the spectrum representing Deligne cohomology. This object was introduced in joint work of Andreas Holmstrom and myself (Arakelov motivic cohomology I, Journal of Algebraic Geometry (2015), Vol. 24, pp. 719–754).
Conjecturally (See my paper Special L-values of geometric motives in Asian Journal of Mathematics (2017), Vol. 21 (2) pp. 225–264):

The pairing is perfect (of finite-dimensional R-vector spaces, provided that motivic (co)homology is with real coefficients).
The special L-value of $M$ can be read off from the pairing by considering $\mathbf Q$-structures on the above $\mathbf R$-vector spaces.

This conjectural duality is somewhat in the spirit of Artin-Verdier duality, or also of Poincaré duality.