Context : The star-forming nature of long \gamma -ray burst ( GRB ) host galaxies provides invaluable constraints on the progenitors of GRBs and 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 .

Aims : We observed the galaxy hosting GRB 080605 at z = 1.64 using optical/NIR spectroscopy and high-resolution HST/WFC3 imaging in the rest-frame wavelength range between 1150 and 8700 ~ { } \AA .
These data allow us to study a z > 1 GRB host in unprecedented detail and investigate the relation between GRB hosts and field galaxies .

Methods : We avail of VLT/X-shooter optical/NIR spectroscopy to measure the metallicity , electron density , star-formation rate ( SFR ) , and reddening of the host of GRB 080605 .
Specifically , we use different strong-line diagnostics to robustly measure the gas-phase metallicity within the interstellar medium ( ISM ) for the first time based on [ N ii ] at this redshift .

Results : The host of the energetic ( E _ { \gamma,iso } \sim 2 \times 10 ^ { 53 } erg ) GRB 080605 at z \sim 1.64 is a morphologically complex , vigorously star-forming galaxy with an H \alpha -derived SFR of 31 ^ { +12 } _ { -6 } ~ { } M _ { \sun } yr ^ { -1 } .
Its ISM is significantly enriched with metals .
Specifically , [ N ii ] /H \alpha = 0.14 \pm 0.02 which yields an oxygen abundance 12 + \log ( O / H ) 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 Z _ { \sun } .
For its measured stellar mass ( M _ { * } = 8.0 ^ { +1.3 } _ { -1.6 } \times 10 ^ { 9 } M _ { \sun } ) and SFR this value is consistent with the fundamental metallicity relation defined by star-forming field galaxies .
The absence of strong Ly \alpha emission constrains the escape fraction of resonantly-scattered Ly \alpha photons to f _ { esc } \lesssim 0.08 .

Conclusions : Our observations provide a detailed picture of the conditions in the ISM of a highly star-forming galaxy with irregular morphology at z \sim 1.6 .
They include the first robust metallicity measurement based on [ N ii ] for a GRB host at z > 1 and 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 at high redshift .