I'm trying to read the "An elementary introduction to the Langlands program" by Gelbart and got stuck on page 11.
In this section the author states there is a connection between modular forms and "nice" L-functions(entire on some strip, satisfying functional equation, ...). And I'm confused on the explicit form of the connection:
1.is it true that for Riemann's zeta function the corresponding modular form is the theta function?
2.if we look at theorem 1(on pp.12) for Riemann's zeta function, $a_n=1$ for $n=1,...$. Thus $f(z)=\frac{exp(2\pi i z)}{1-exp(2\pi i z)}$, which should not be a modular form in any sense. what's wrong with the argument?