Your restriction that all $a_n$ are integers is too restrictive to define $L$-functions in general (even most Dirichlet $L$-functions are not like that), and you left out the Euler product. Anyway, examples with the restrictions you imposed can be found among the zeta-functions of number fields.

For a number field $F$, its zeta-function $\zeta_F(s)$ has order of vanishing $r_2(F)$ (the number of conjugate pairs of complex embeddings of $F$) at negative odd integers and $r_1(F) + r_2(F)$ (here $r_1(F)$ is the number of real embeddings of $F$) at negative even integers. Always $r_1(F) + r_2(F)$ is positive, and $r_2(F) > 0$ exactly when $F$ is not totally real, so $\zeta_F(s)$ vanishes at all negative integers when $F$ is not totally real. For example, the zeta-function of $\mathbf Q(i)$, or more generally any cyclotomic field other than $\mathbf Q$ itself, vanishes at all negative integers. The zeta-function of a number field is not identically 0 as a function since it tends to 1 (its constant term) as ${\rm Re}(s) \rightarrow \infty$ or since $a_p = [F:\mathbf Q]$ when $p$ is a prime splitting completely in $F$ (there are infinitely many such $p$).

The zeta-function of $\mathbf Q(i)$ equals the product $\zeta(s)L(s,\chi_4)$ where $\chi_4$ is the nontrivial character mod $4$, where $\zeta(s)$ vanishes at negative even integers but not negative odd integers and $L(s,\chi_4)$ vanishes at negative odd integers but not negative even integers. So you might feel that something like $\zeta_{\mathbf Q(i)}(s)$ is not a good example since it is a product where each piece ($\zeta(s)$ or $L(s,\chi_4)$) is not an example of the kind you seek. You could use instead $L$-functions of elliptic curves (over $\mathbf Q$, say) all of which satisfy the conditions you impose and most (the ones for non-CM elliptic curves) are not expected to break up into two parts which vanish only on negative even or negative odd integers. Or use the Artin $L$-function of the $2$-dimensional irreducible representation of ${\rm Gal}(F/\mathbf Q) \cong S_3$ where $F$ is the splitting field over $\mathbf Q$ of $x^3-2$: it satisfies all of your conditions.

The answer by David Loeffler here gives a formula for the order of vanishing at negative integers of Hecke $L$-functions over a number field $F$. From the formula you see that the order of vanishing is positive at all negative integers if $F$ is not totally real. As he writes, "the orders of vanishing for negative integers $s$ are completely determined by the Gamma-factors in the functional equation" so when you have a functional equation that involves $\Gamma(s)$, which has poles at all negative integers, this forces there to be zeros of the $L$-function at negative integers since the product of the (dual) $L$-function and $\Gamma$-factor is nonzero at positive integers to the right of the critical strip and then you use the functional equation to relate that to values at negative integers,