Repeating behaviour of FRB 121102: periodicity, waiting times and energy distribution.

Since the discovery of repetition it has been clear that the detections of fast radio burst (FRB) 121102 are clustered. Recently, it was argued that it is periodic, raising the question of whether the clustering reflected a not-yet-defined periodicity. We performed an extensive multi-wavelength campaign with Effelsberg, Green Bank telescope and the Arecibo observatory to shadow the Gran Telescope Canaria (optical), NuSTAR (X-ray) and INTEGRAL (gamma-ray). We detected 36 bursts with Effelsberg, one with a pulse width of 39\,ms. We tested the periodicity hypothesis using 165-hr of Effelsberg, and find a periodicity of 161$\pm$5 days. We predict the source to be active from 2020-07-09 to 2020-10-14 and, posteriorly, from 2020-12-17 to 2021-03-24. With the bursts detected, we compare the waiting times between consecutive bursts with a Weibull distribution with shape parameter $k<1$, and a Poissonian distribution. We conclude that the strong clustering was indeed a consequence of a periodic activity and show that if few events of millisecond scale separation are excluded, the sample agrees with a Poissonian distribution. We model the bursts cumulative energy distribution, with energies from ${\sim}10^{38}$-$10^{39}$ erg, and find that it is well described by a power-law with slope of $\gamma=-1.1\pm 0.2$. We exclude a time changing slope to reconcile the discrepancies between the published values and propose that a single power-law might not fit the data over many orders of magnitude. With one burst detected during simultaneous NuSTAR observations, we place a 5-$\sigma$ upper limit of $6-40\times10^{46}$ erg on the 3--79\,keV energy of an X-ray burst counterpart.