Are L-functions uniquely determined by their values at negative integers?

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 :)

Are L-functions uniquely determined by their values at negative integers?

No. The rescaled Riemann zeta function $$\zeta(2s) = \sum_{m=1}^\infty \frac{1}{m^{2s}} = \sum_{n=1}^\infty \frac{a_n}{n^s},$$ corresponding to the coefficient sequence $$a_n = \begin{cases} 1 & \textrm{if n is a square}, \\ 0 & \textrm{otherwise}, \end{cases}$$ is an example of an $$L$$-function that has a meromorphic continuation to all of $$\mathbb{C}$$ and vanishes at the negative integers.

Note that the formulation of your question seems to mix up the notion of an $$L$$-function with the more general notion of a Dirichlet series.

There are also some interesting Dirichlet series that are not $$L$$-functions but still satisfy the properties you are asking about (meromorphic continuation and zeros at the negative integers). One such function is the so-called Witten zeta function of the group $$SU(3)$$, as I proved in "On the number of $$n$$-dimensional representations of $$\operatorname{SU}(3)$$, the Bernoulli numbers, and the Witten zeta function" (see theorem 1.3 on page 5). The coefficient sequence for that function is \begin{align} a_n &= \#\{ j,k\ge 1 : n = jk(j+k) \} \\ &= \textrm{the number of inequivalent irreducible} \\ & \quad \textrm{ representations of SU(3) of dimension n/2.} \end{align}

• That's a neat counterexample! Also thanks a lot for the reference. I was trying to come up with a notion of "higher rank" analogs of zeta functions, Bernoulli functions, gamma functions etc. where the usual ones correspond to SU(2). It looks like the Witten zeta function in your paper might be related to it. – Henry Jul 14 '19 at 19:41
• @Henry yes, Witten zeta functions have very interesting properties, and there are still many things that are not understood about them. For example, it’s not known whether the phenomenon of trivial zeros extends to the Witten zeta functions of other Lie groups beyond $SU(2)$ (the Riemann zeta case) and $SU(3)$. – Dan Romik Jul 14 '19 at 21:36

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,

Let $$F(s) =\sum_{n=1}^\infty a_n n^{-s}= \sum_{j=1}^J c_j F_j(s)$$ be a linear combination of L-functions $$F_j$$ of degree $$\le d$$, that is $$(s-1)^r F_j(s)$$ entire and its functional equation contains at most $$d$$ gamma factors $$\Gamma(s/2+a)$$.

Then for $$m > d$$ or for $$m=d,|x| < C$$, $$\Gamma(ms)F(s)x^{-s}$$ decays fast enough to apply the residue theorem $$\begin{eqnarray}f_m(x)&=&\frac{1}{2i\pi} \int_{2-i\infty}^{2+i\infty} F(s) \Gamma(ms) x^{-s}ds \\ &=& \sum Res(F(s) \Gamma(ms) x^{-s}) \\ &=& \sum Res_{s=1}(F(s) \Gamma(ms)x^{-s}) + \sum_{k=0}^\infty \frac{(-1)^k}{k!m} F(-k/m) x^{k/m}\end{eqnarray}$$ (for $$m = d$$ it is valid only for $$|x| < C$$ but the exponential decay of $$\Gamma(ms)F(s)$$ on $$\Im(s)=2$$ implies $$f_m$$ is analytic so it is determined by $$x \in (0,C)$$)

And hence $$F(s) = \frac{1}{\Gamma(ms)} \int_0^\infty f_m(x) x^{s-1}dx$$

is fully determined by its principal part at $$1$$ and its values at $$s=-k/m$$.

In other words your claim holds only for L-functions of degree $$d=1$$.