We measure the dynamic differential scalar polarizabilities at 10.6 $mu$m for two candidate clock transitions in $^{176}mathrm{Lu}^+$. The fractional black body radiation (BBR) shifts at 300 K for the $^1S_0 leftrightarrow {^3D_1}$ and $^1S_0 leftrightarrow {^3D_2}$ transitions are evaluated to be $-1.36,(9) times 10^{-18}$ and $2.70 ,(21) times10^{-17}$, respectively. The former is the lowest of any established optical atomic clock.
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