Are L-functions uniquely determined by their values at negative integers? In another words, is there a sequence of integers $a_1, a_2, a_3, \cdots$ such that

• the corresponding L-function $$L_{\{a_n\}}(s):=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$$ converges well for $\text{Re}(s) > M$ for some $M\in \mathbb{R}$

• $L_{\{a_n\}}(s)$ has analytic continuation to a meromorphic function on the whole complex plane

• $L_{\{a_n\}}(n)=0$ for all negative integers $n$

• not all of $a_n$ are zero?

Are L-functions uniquely determined by their values at negative integers? In another words, is there a sequence of integers $a_1, a_2, a_3, \cdots$ such that

• the corresponding L-function $$L_{\{a_n\}}(s):=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$$ converges well for $\text{Re}(s) > M$ for some $M\in \mathbb{R}$

• $L_{\{a_n\}}(s)$ has analytic continuation to a meromorphic function on the whole complex plane

• $L_{\{a_n\}}(n)=0$ for all negative integers $n$

• not all of $a_n$ are zero?

Added : It was suggested in the answers that I should have used the term "Dirichlet series of integer sequences" instead of "L-function" as it lacks Euler product. I apologize for the confusion :)