189-726A: L-functions and Modular Forms Assignment 1

Due: Wednesday, September 28

1. Let F be a number field. Show that the Dedekind zeta-function of F and the Artin L-function attached to the induced representation

ρ := Ind_{F}^{Q}1 : G_{Q} ⟶ GL_{d}(C), d = [F : Q]

are equal.

2. Prove the functional equation Γ(s + 1) = sΓ(s), where Γ(s) is the Γfunction, defined for ℜ(s) > 0 by the integral ∫_{0}^{∞} e^{−t}t^{s} \frac{dt}{t}. Show that Γ(s) extends to a meromorphic function on C which is holomorphic except at the non-positive integers, where it has simple poles. What is the residue of Γ(s) at s = −n, for n ≥ 0?

3. Show that the integral

∫_{0}^{∞} ω(t)t^{s/2} \frac{dt}{t}

converges absolutely when ℜ(s) > 1.

4. Let τ(χ) be the Gauss sum attached to a primitive Dirichlet character χ of conductor q. Show that |τ(χ)| = \sqrt{q} and that τ(χ̄) = χ(−1)\overline{τ(χ)}.