0
votes
2answers
295 views

The origin of the root number $w(C/ℚ)=±1$ (the sign of the functional equation)

The motivation for this question is the same as in my previous question in MO: http://mathoverflow.net/questions/115179/real-root-1-of-the-hasse-weil-l-function-of-c-over I am just curious to know ...
26
votes
1answer
2k views

A puzzling remark of Manin (ICM 1978)

Manin ends his 1978 ICM talk with this remark: I would also like to mention I. M. Gel'fand's suggestion that the $\zeta$-functions of certain special differential operators should have an arithmetic ...
28
votes
10answers
3k views

Why are functional equations important?

People who talk about things like modular forms and zeta functions put a lot of emphasis on the existence and form of functional equations, but I've never seen them used as anything other than a ...