  Let $R_{1,2}$ be scalar Riesz transforms on $\mathbb{R}^2$. We prove that the
$L^p$ norms of $k$-th powers of the operator $R_2+iR_1$ behave exactly as
$|k|^{1-2/p}p$, uniformly in $k\in\mathbb{Z}\backslash\{0\}$, $p\geq2$. This
gives a complete asymptotic answer to a question suggested by Iwaniec and
Martin in 1996. The main novelty are the lower estimates, of which we give
three different proofs. We also conjecture the exact value of
$\|(R_2+iR_1)^k\|_p$. Furthermore, we establish the sharp behaviour of weak
$(1,1)$ constants of $(R_2+iR_1)^k$ and an $L^\infty$ to $BMO$ estimate that is
sharp up to a logarithmic factor.
