In some old notes, I found the following conjecture:
Let $\mathbb T$ denote the unit circle and let $\chi:{\mathbb N} \to {\mathbb T}$ be fully multiplicative. Then the L-series $$ L(\chi,s)=\sum_{n=1}^\infty\frac{\chi(n)}{n^s}, $$ defined for ${\rm Re}(s)>1$, cannot be extended to a holomorphic function on ${ {\rm Re}(s)\ge 0}$.
Who formulated this conjecture? What related results are known?
