Let $f:\mathbb{N}\to\mathbb{C}$ and assume that the L-function $$L(f;s)=\sum_{n=1}^\infty \frac{f(n)}{n^s}$$ converges absolutely on some right half-plane $Re(s)>k$.
Is it true that $L(f;\cdot)$ is identically zero if and only if so is $f$? If not, is there any classification of such arithmetic functions?
My guess is that there exists a non-zero function $f$ with zero L-function, but I could not find any example.
Suppose that $L(f,s)$ is identically zero on $\Re(s)>k$, and that $f\neq 0$. Then there is some minimal $N$ for which $f(N)\neq 0$. Hence $$ f(N)=-\sum_{n>N}f(n)\Big(\frac{N}{n}\Big)^\sigma $$ with the right-hand side converging absolutely for $\sigma>k$.
Each term on the right tends to zero as $\sigma\to+\infty$, hence by the dominated convergence theorem the sum tends to zero as well. Therefore $f(N)=0$, contrary to the choice of $N$.