Functional Equation For L-functions

Question 1

When is a character even/odd?

If χ is primitive give the functional equation satisfied by χ in each case. What is Λ(s,χ)?

Answer 1

χ(-1)=1 means χ is even
χ(-1)=-1 means χ is odd

Writing a=0 if even and a=1 if odd, we put:
Λ(s,χ)=
(q/π)^((s+a)/2)•Γ((a+s)/2)•L(s,χ)

Then we get for χ even:

Λ(s,χ)=τ(x)/(sqr(q))•Λ(1-s,χ*)

And for odd χ:

Λ(s,χ)=-i•τ(x)/(sqr(q))•Λ(1-s,χ*)

Question 2

For a primitive character what is its associated theta function?
What is the automorphism of the theta functions?

Answer 2

Even χ:

Θχ(x)=Σχ(n)•exp(-π/q•(xn^2))

Odd χ:

Θχ(x)=Σnχ(n)•exp(-π/q•(xn^2))

where we sum over all n in Z.

They satisfy the automorphy relations:

Θχ(x)=’χ’(1)•Θχ(1/x)/x, χ even
Θχ(x)=-i•’χ’(1)•Θχ(1/x)/(x^3), χ odd

Question 3

What is the poisson summation formula?

Answer 3

Define the Fourier transform of an integrable function f on R by:

‘f’(ξ)=(INTEGRATE /R) f(x)•e(-ξx)dx

Then as long as f is differentiable and decays sufficiently fast and smoothly, we get:

Σf(n) = Σ’f’(n)

Question 4

How do we prove the automorphy of the theta function?

Answer 4

Split the sum of the theta function into residue classes mod q, and then use poisson summation formula on exponential part.
Take out as many constants as possible from the integral we have, and use a substitution to get an integral we can evaluate, and this should leave us with the result.

Question 5

Calculate the following:

Log(ζ(s))
ζ’/ζ(s)
log(L(s,χ))
L’/L(s,χ)

Answer 5

Log(ζ(s))=ΣΣ1/n•p^(ns) by euler product. Sum over n and p.
Then by differentiating:
ζ’/ζ(s)=ΣΣlog(p)/p^(ns)

Similarly:
Log(L(s,χ))=ΣΣχ(p^n)/n•p^(ns)
and
L’/L(s,χ)=ΣΣχ(p^n)•log(p)/p^(ns)

Note, the second is always more useful as it has no branch cut

Question 6

How do we prove the functional equation of the completed L function Λ?

Answer 6

Well we look at case even χ.
We have:
Λ(s,χ)=Γ(s/2)•(q/π)^s/2•L(s,χ)

Show that Γ(s/2)=
2•(INT) exp(-x^2)•x^s•dx/x
And then alter by multiplying
 (q/π)^s/2 to this. 
Then Λ has a form of a sum of some integrals. Swap the integral and sum by absolute convergence, and we get a simple representation of the completed L-function:

Λ(s,χ)=(INT)Θχ(x)•x^s•dx/x

Then split the integral into two sections and use the automorphy of theta. Notice that this gives an analytic continuation of Λ to all of C.
Finally, use the fact that the modulus of any character is 1 to get functional equation.