Zeros Of L-functions

Question 1

Show that any zero of an L-function is in the critical strip.
Show that if χmodq is primitive and even then: L(-2n,χ)=0
Show that if χmodq is primitive and odd then: L(-(2n+1),χ)=0

Answer 1

L(s,χ) is not zero for Re(s)>1 by its ruler product. Using the functional equation and the fact that Γ has no zeros gives the rest.
Use the completed L function Λ which is entire.

Question 2

Compute the following uniform bounds for L functions, where σ=Re(s)>1.

|L(s,χ)|==2 and χ a character modq:
|Λ(s,χ)|=<C•|s|•(log|s|+logq)

Answer 2

Compare to the ζ function on the real line.
Can assume σ>1/2 and use the functional equation. Then use a simple bound on L calculated at start of course and stirlng’s formula.

Question 3

Prove that the number of zeros of Λ(s,χ) with s=σ+it and |t|=<T (which is larger than 2) is:
O(T(logq+logT))

Answer 3

Note that if ρ=β+iγ is a root with
|γ|<
Σlog(10T/|ρ-2|) where we sum over all zeros of Λ with |ρ-2|<10T.
We then need to use Jensons inequality and then our lemma on bounds for the Λ function to get the bound.