• L-functions
• SelbergClass
• Degree4

We give Selberg data for some degree 4 L-functions of the Selberg class.

Suppose that we have an L-function associated with a Siegel modular form of genus two, weight k, and level 1, specifically a spinor L-function. Then its Selberg data is

Next, consider the Rankin-Selberg convolution of the L-functions of a weight k and a weight \ell newform, that is of a (2,q_1,\{(k-1)/2,(k+1)/2\},1) and a (2,q_2,\{(\ell-1)/2,(\ell+1)/2\},1), where (q_1,q_2)=1. The result is an element of the Selberg class with data

Now suppose we have the symmetric cube of the L-function of a weight k squarefree level q newform. Its functional equation is of type

Note that the eigenvalue data associated with the symmetric cube of a weight 2 modular form matches the data associated with the spinor L-function of a genus 2 weight 3 Siegel modular form, to get \biggl\{\frac52, \frac32, \frac 32, \frac 12 \biggr\}.

Now suppose we have a Hilbert modular form for a real quadratic field. Its L-function will have functional equation of type

Now suppose we have an Artin L-function associated with a four-dimensional Galois representation. Its functional equation will be of type

where each \mu_i is either 0 or 1 and all possibilities can occur.

Finally, suppose we have the Hasse-Weil L-function of a genus two curve. It is expected that such an L-function will have a functional equation of type

where q will be related to the discriminant.