I am conducting research on photons and was wondering if they have relativistic mass. I already know that they they have zero rest mass. Any answers are welcome!
Relativistic mass is obsolete. See Why is there a controversy on whether mass increases with speed? . Therefore this is not a question that modern physicists would consider of interest. Furthermore, the usual motivation given for using the relativistic mass convention is that it lets you use Newton's second law without modification, but Newton's second law isn't going to apply to a photon, no matter how hard you try.
If you did want to assign a photon a relativistic mass, there is no other parameter that could determine the mass besides its energy $E$, and based on units the mass would then have to be of the form $m=kE/c^2$, where $k$ is a unitless constant. Probably $k=1$, since we can take $p=mv$ as the definition of the relativistic mass, and $p=E/c$ for a photon.
It's quite wrong to assert that relativistic mass is obsolete. Gary Oas of Standford university did a study of a sample of the SR and GR textbooks. Of the ones published between 1995 and 2005 there were 60% of them which use relativistic mass.
In this video, Alan H. Guth of MIT describes why the mass of a photon is not zero. http://www.newenglandphysics.org/common_misconceptions/DSC_0003.MOV