The metal-enriched host of an energetic gamma-ray burst at z ~ 1.6

(Abridged) Long gamma-ray burst (GRB) host galaxies might open a short-cut to the characteristics of typical star-forming galaxies throughout the history of the Universe. Due to the absence of near-infrared (NIR) spectroscopy, however, detailed investigations, specifically a determination of the gas-phase metallicity of gamma-ray burst hosts, was largely limited to redshifts z < 1 to date. Here, we report observations of the galaxy hosting GRB 080605 at z = 1.64 using optical/NIR spectroscopy and high-resolution HST/WFC3 imaging. We avail of VLT/X-shooter spectroscopy to measure the metallicity, electron density, star-formation rate (SFR), and reddening of the host. Specifically, we use different strong-line diagnostics based on [N II] to robustly measure the gas-phase metallicity for the first time for a GRB host at this redshift. The host of the energetic (E_iso ~ 2 x 10^53 erg) GRB 080605 is a morphologically complex, vigorously star-forming galaxy with an H\alpha-derived SFR of 31 M_sun/yr. Its ISM is significantly enriched with metals with an oxygen abundance between 8.3 and 8.6 depending on the adopted strong-line calibrator. This corresponds to values in the range of 0.4-0.8 times the solar value. For its measured stellar mass of 8 x 10^9 M_sun and SFR this value is consistent with the fundamental metallicity relation defined by star-forming field galaxies. Our observations directly illustrate that GRB hosts are not necessarily metal-poor, both on absolute scales as well as relative to their stellar mass and SFR. GRB hosts could thus be fair tracers of the population of ordinary star-forming galaxies as a whole at high redshift.