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- On Dirichlet series and critical strips 3 answers
What form does the Riemann hypothesis have for a global L-function?
I'm thinking of the Hasse-Weil zeta function associated with a certain elliptic curve, which has CM. I believe that if the Riemann hypothesis is valid then the Riemann hypothesis for such a Hasse-Weil zeta function is valid. Thus, I think that this problem is as difficult as Riemann hypothesis.
I want to complement my statement. Suppose that the elliptic curve E has CM.
It is known that the L-series of E is the Hecke L-series. Thus, we can show that L(E, s) is a meromorphic function over the entire complex plane. Considering Artin's conjecture, it seems that L(E, s) has no poles.
Since L(E, s) is a meromorphic function, 1/L(E, s) is also a meromorphic function. The function 1/L(E, s) can't be zero because L(E, s) has no poles.
The Hasse-Weil zeta function is given as Z(s)Z(s-1)/ L(E, s), where Z(s) is the Riemann zeta function. Therefore, if the Hasse-Weil zeta function has zeros then the all zeros are the zeros of Z(s)Z(s-1).