Usually the Riemann zeta function $\zeta(s)$ gets multiplied by a "gamma factor" to give a function $\xi(s)$ satisfying a functional equation $\xi(s)=\xi(1-s)$. If I changed this gamma factor by a non-zero constant complex number, the functional equation would still hold.
More generally, for zeta functions of number fields, one gets a gamma factor for each real place and a gamma factor for each complex place. In a parallel universe, we could have defined the gamma factor for each real place to be (say) $\pi^{1/2}$ times what we usually use, and the gamma factor at the complex places to be (say) $1/2$ of what we usually use (and I think I was taught to use $2.(2\pi)^s$ so we could just knock off that first 2) and all functional equations would still be exactly the same (because the extra factors would be the same on both sides).
More generally, for Dirichlet $L$-functions and Hecke $L$-functions: I now need a gamma factor for the sign function on the non-zero reals, and again we could use a different choice.
Now some general yoga of gamma factors tells us that really there are only 3 choices to be made (because Hodge structures basically always decompose into the types covered above)—and I've mentioned them all already.
Upshot: did we, at some point, make 3 arbitrary choices of constants, and the entire theory of $L$-functions of motives wouldn't care what choices we made, so we could have made other choices? Note for example that conjectures on special values of $L$-functions don't take the gamma factors into account (well, the ones I know don't; they predict values of the incopleted $L$-function without the gamma factors). Note also that when defining these things for Hecke characters a la Tate, again pretty arbitrary choices are made at the infinite places—there is no one canonical function that is its own Fourier transform, because we can change things by constants again.
Am I totally wrong here or are there really 3 arbitrary choices that we have made, and we could have made others?
Best Answer
In Deninger's first paper on zeta-regularized determinants, his construction of the Archimedian L-factor at the real place for Q (i.e., the gamma factor for Riemann zeta) is off by a factor of sqrt(2) in comparison to Riemann's gamma factor.