It is said, that
The 6° wide UTM zones, numbered 1–60.
But where is the origin? Zone 1 is from longitude [-3..+3], [0..6], [-180..-174]? Where is it said?
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The answer depends on what you mean by "origin"
There are indeed 60 UTM zones numbered 1-60 and it starts at 180 W, so UTM zone 1 is used for longitude in [-180° -174°].
That being said, the origin of the center projection is located in the middle of the extent of each zone. The latitude of the center of projection remains 0 for all zones but the longitude of origin changes for each zone. So, for example, the longitude of the origin of zone 1 is at -177. (-180° + 3°)
Finally, the origin of the metric coordinate system is often shifted. With UTM zones xx of EPSG 326xx (UTM zone xx North WGS84), this shift is 500000 m. The role of the shift is to avoid negative X coordinates. Consequently, the lat/long coordinate of the origin of this UTM zone is located approximately 4°30' West from the center of projection. Sometimes there is also a false northing. You can check for these values in the definition of each UTM projection, e.g. on spatialreference.org