Special Values of Dirichlet \(L\)-Functions

Inspired by a question of Austin Lei, we review the basic theory of special values of Dirichlet \(L\)-functions.

Let’s start by stating the result we are interested in proving today.

Definition 1. A Dirichlet character, denoted \(\chi\pmod N\), is a multiplicative character \(\chi\colon(\mathbb Z/N\mathbb Z)^\times\to\mathbb C^\times\). A Dirichlet character \(\chi\pmod N\) is called primtiive if and only if \(\chi\) does not factor through \((\mathbb Z/d\mathbb Z)^\times\) for each divisor \(d\) of \(N\).

Example 2. There is a unique nontrivial Dirichlet character \(\chi\colon(\mathbb Z/3\mathbb Z)^\times\to\mathbb C^\times\) given by \(\chi(1)=1\) and \(\chi(-1)=-1\).

Definition 3. Given a Dirichlet character \(\chi\pmod N\), we define the Dirichlet \(L\)-function as \[L(s,\chi)=\sum_{n=1}^\infty\frac{\chi(n)}{n^s},\] where by convention \(\chi(n)=0\) whenever \(\gcd(n,N)>1\). This series converges absolutely and uniformly converges on compacts in the region \(\operatorname{Re}s>1\) and therefore defines a holomorphic function there.

Theorem 4. Fix an odd primitive Dirichlet character \(\chi\pmod N\). Then for any odd positive integer \(n\), we have \[\frac{L(n,\chi)}{\pi^{n}}\in\overline{\mathbb Q}.\]

Remark 5. When \(\chi\) is even, one instead has \(L(2m,s)\cdot\pi^{-2m}\in\overline{\mathbb Q}^\times\); for example, this variant applies to the Riemann \(\zeta\)-function by taking \(\chi\) to be trivial. The proof of this contains no new ideas, so we will pay it no attention.

Remark 6. Our method of proof will also give us a way of computing \(L(n,\chi)/\pi^{n}\) in examples.

Our proof is based on Riemann’s initial proof of the functional equation and analytic continuation of \(\zeta\). The idea is to write \(L(s,\chi)\) as a certain Mellin transform.

Lemma 7. Fix a Dirichlet character \(\chi\pmod N\). For \(\operatorname{Re}s>1\), \[\Gamma(s)L(s,\chi)=\int_{\mathbb R^+}\Bigg(\frac1{1-e^{-Nt}}\sum_{n=0}^{N-1}\chi(n)e^{-nt}\Bigg)t^s\frac{dt}t.\]

Proof. This is a direct expansion. Recall that \(\Gamma(s)=\int_{\mathbb R^+}t^se^{-t}\,dt/t\), so a change of coordinates shows that \[\Gamma(s)n^{-s}=\int_{\mathbb R^+}t^se^{-nt}\,\frac{dt}t.\] Summing over all \(n\ge1\), we see that \[\Gamma(s)L(s,\chi)=\int_{\mathbb R^+}\Bigg(\sum_{n=0}^\infty\chi(n)e^{-nt}\Bigg)t^s\,\frac{dt}t.\] We now sum over \(n\) according to residue class\(\pmod N\), which reveals \[\Gamma(s)L(s,\chi)=\int_{\mathbb R^+}\Bigg(\sum_{n=0}^{N-1}\frac{\chi(n)e^{-nt}}{1-e^{-Nt}}\Bigg)t^s\,\frac{dt}t.\] Rearranging completes the proof. \(\blacksquare\)

One can now analytically continue Mellin transforms in a rather formal way.

Proposition 8. For any Schwartz function \(f\colon\mathbb R_{\ge0}\to\mathbb R\), define \[D(f,s)=\frac1{\Gamma(s)}\int_{\mathbb R^+}f(t)t^s\,\frac{dt}t.\] In this situation, \(D(f,s)\) admits a holomorphic continuation to \(\mathbb C\), and \(D(f,-n)=(-1)^nf^{(n)}(0)\) for all nonnegative integers \(n\).

Proof. When \(\operatorname{Re}s>1\), the rapid decay of \(f\) ensures that the integral converges absolutely and uniformly on compacts. Thus, the main issue for continuation is that the integral may be large around \(0\). To control this possible singularity, let \(\varphi\colon\mathbb R_{\ge0}\to[0,1]\) be a smooth bump function for which \(\varphi|_{[0,1]}=1\) and \(\varphi|_{[2,\infty)}=0\). Then we set \(f_1=\varphi f\) and \(f_2=(1-\varphi)f\) so that \(f=f_1+f_2\). We now handle \(f_1\) and \(f_2\) separately.

• It turns out that we may totally ignore \(f_2\). Indeed, \[D(f_2,s)=\frac1{\Gamma(s)}\int_{1}^\infty f_2(t)t^s\,\frac{dt}t.\] The rapid decay of the integrand now immediately grants the analytic continuation of \(D(f_2,s)\) to all \(\mathbb C\). Additionally, due to the poles of \(\Gamma(s)\), we see that \(D(f_2,-n)=0\) for all nonnegative \(n\).
• We handle \(f_1\) by inductively integrating by parts. Indeed, integrating by parts once reveals \[D(f_1,s)=\frac1{\Gamma(s)}\cdot f_1(t)\cdot\frac{t^s}s\bigg|_{t=0}^{t=\infty}-\frac1{s\Gamma(s)}\int_{\mathbb R^+}f_1’(t)t^{s+1}\,\frac{dt}t.\] The left term vanishes at both endpoints (for \(\operatorname{Re}s>1\)) because \(f_1\) has compact support, so we are left with \[D(f_1,s)=-D(f_1’,s+1).\] This equation then provides a continuation of \(D(f_1,s)\) to the region \(\operatorname{Re}(s+1)>1\), and one can inductively continue the process.

The above two points complete the proof of the continuation. Additionally, the inductive process in the second point implies that it is enough to note that \(D(f_1,1)=\int_{\mathbb R^+}f_1(t)\,dt\) and then induct downwards to the nonpositive integers. \(\blacksquare\)

The above proposition gives us access to special values of \(L\)-functions for nonpositive integers. To access special values of positive integers, we will require a trick.

Definition 9. Let \(\chi\pmod N\) be an odd primitive Dirichlet character. Then we define the completed \(L\)-function by \[\Lambda(s,\chi)=\pi^{-\frac{s+1}2}\Gamma\left(\frac{s+1}2\right)L(s,\chi).\] If \(\chi\) is instead of even, we have \(\Lambda(s,\chi)=\pi^{-s}\Gamma(s/2)L(s,\chi)\).

Theorem 10. Let \(\chi\pmod N\) be an odd primitive Dirichlet character. Then \[\Lambda(s,\chi)=-i\tau(\chi)N^{-s}\Lambda(1-s,\overline\chi).\] Here, \(\tau(\chi)=\sum_n\chi(n)e^{2\pi in/N}\) is the Gauss sum.

Proof. We omit this proof, but we say that one basically applies Mellin inversion to the expression of \(L(s,\chi)\) given in Proposition 8. \(\blacksquare\)

Remark 11. If \(\chi\) is even, then we instead have \(\Lambda(s,\chi)=\pi^{-s}\Gamma(s/2)L(s,\chi)\) and \[\Lambda(s,\chi)=\tau(\chi)N^{-s}\Lambda(1-s,\overline\chi).\]

Proof of Theorem 4. Define \[f(t)=\frac1{1-e^{-Nt}}\sum_{n=0}^{N-1}\overline\chi(n)e^{-nt}.\] By combining Lemma 7 and Proposition 8, we see that \[L(-n,\overline\chi)=(-1)^nf^{(n)}(0)\] for all nonnegative integers \(n\). In particular, because \(f\) is some rational polynomial in \(e^{-t}\) with coefficients in \(\mathbb Q(e^{2\pi i/N})\), the same is true for all derivatives of \(f\), so it follows that \(L(-n,\chi)\) is algebraic.

From here, the functional equation implies that \[L(s,\chi)=-i\tau(\chi)N^{-s}\pi^{s-\frac12}\cdot\frac{\Gamma\left(\frac{2-s}2\right)}{\Gamma\left(\frac{1+s}2\right)}\cdot L(1-s,\overline\chi).\] If we let \(s\) be a positive integer, then the previous paragraph tells us that \(L(1-s,\overline\chi)\) is algebraic. Similarly, \(-i\tau(\chi)N^{-s}\) are also automatically algebraic, so we are left to deal with the multiple of \(\pi\) and the ratio of \(\Gamma\). Well, \(\Gamma(1/2)=\sqrt\pi\). Thus, when \(s\) is odd, the numerator outputs a rational multiple of \(\sqrt\pi\), and the denominator outputs a rational number. This cmopletes the proof of theorem. \(\blacksquare\)

Remark 12. It is not hard to see that \(f\) defined in the proof is even: the main point is that \[\sum_{n=0}^{N-1}\overline\chi(n)e^{nt}=\chi(-1)e^{Nt}\sum_{n=0}^{N-1}\overline\chi(n)e^{-nt}\] by reindexing the sum (sending \(n\mapsto-n\)). Thus, \(L(-n,\overline\chi)\) vanishes for all odd nonnegative \(n\), so \(L(n,\chi)\) vanishes for all positive even \(n\). This explains why Theorem 4 says nothing about positive even special values.

Example 13. Let \(\chi\pmod3\) be the nontrivial Dirichlet character. For example, in this case, \(N=3\) and \(\tau(\chi)=e^{2\pi i/3}-e^{4\pi i/3}=i\sqrt3\). Thus, we may unwind the proof of Theorem 4 to find that \[L(n,\chi)=-i\cdot i\sqrt3\cdot N^{-n}\pi^{n-\frac12}\cdot\frac{\Gamma\left(\frac{2-n}2\right)}{\Gamma\left(\frac{1+n}2\right)}\cdot(-1)^{n-1}f^{(n-1)}(0),\] where \[f(t)=\frac{e^{-t}-e^{-2t}}{1-e^{-3t}}.\] Thus, when \(n=2m+1\) is some odd positive integer, we find \[L(2m+1,\chi)=\sqrt3\cdot N^{-(2m+1)}\cdot\pi^{2m+1-\frac12}\cdot\frac{\Gamma\left(\frac12-m\right)}{\Gamma\left(m+1\right)}\cdot f^{(2m)}(0).\] For example, at \(m=0\), we get \(L(1,\chi)=\frac{\pi}{3\sqrt3}\). Similarly, at \(m=1\), we get \(L(3,\chi)=\frac{4\pi^3}{81\sqrt3}\). It is possible to simplify this final formula more, but we will not bother.