Given a 0-connective motivic spectrum E is an element of SH(k) over a perfect field k, we determine (h) under bar (0) of the associated motive ME is an element of DM(k) in terms of (pi) under bar (0). (E). Using this, we show that if k has finite 2-etale cohomological dimension, then the functor M W SH(k) - DM(k) is conservative when restricted to the subcategory of compact spectra and induces an injection on Picard groups. We extend the conservativity result to fields of finite virtual 2-etale cohomological dimension by considering what we call real motives.
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