The peak of the luminosity function is at 555 nm (green); the eye’s visual system is more sensitive to light of this wavelength than any other. For monochromatic light of this wavelength, the irradiance needed to make one lux is minimum, at 1.464 mW/m2.
That is, one obtains 683.002 lux per W/m2 (or lumens per watt) at this wavelength. Other wavelengths of visible light produce fewer lumens per watt.
Solar Radiation is a shorter wave length and different color spectrum, so it has a different factor of Lux to W/m2:
Bright sunlight is approximately 136000 lux = 1075 W/m2 = 126.7
The factor changes slightly depending on time of day, conditions such as cloud cover, moisture in the air etc. and most sources agree on the 126.7 as a reasonable factor to use.
The conversion in the green visible light spectrum spectrum at 555nm is 1W/m2 = 683 lux, and is used of human eyes.
Some of our weather stations allow to calibrate the gain to whatever light spectrum you wish to base solar radiation on.