The classical explicit formula for the Riemann Zeta function states that $$ \psi(x)=x-\sum_{\rho} \frac{x^{\rho}}{\rho}+O(1), $$ where $\psi(x)=\sum_{n \leq x} \Lambda(n)$ and the sum is over all non-trivial zeroes of $\zeta(s)$.

Let $L/K$ be a finite Galois extension of number fields and let $V$ be an irreducible complex representation of $\textrm{Gal}(L/K)$. Attached to this piece of data, one can define the Artin $L$-function $L(V, s)$. Does a similar formula exist for $L(V, s)$?

I would also be interested in a similar formula for the slightly less general case of Hecke $L$-functions.


2 Answers 2


One proves explicit formulae essentially by integrating the logarithmic derivative of the $L$-function. For simplicity, let $\psi$ be a Schwartz function with $\psi(1) = 1$, and let $$ \widehat{\psi}(s) = \int_0^\infty \psi(t) t^{s} \frac{dt}{t} $$ be its Mellin transform. Then one can get an explicit formula by considering $$ \sum_{n \geq 1} \Lambda_V(n) \psi(n) = \frac{-1}{2\pi i} \int_{(2)} \frac{L'}{L}(V, s) \widehat{\psi}(s) ds. $$ Typically one would shift the line of integration to $\mathrm{Re}(s) = c < 0$, apply the functional equation, and get an expression of the form $$\begin{align} \sum_{n \geq 1} &\big( \Lambda_V(n) \psi(n) + \Lambda_\overline{V}(n) \tfrac{\psi(n^{-1})}{n} \big) = \\ &\{ \mathrm{polar\;data} \} - \sum_\rho \widehat{\psi}(\rho) + \frac{1}{2\pi i} \int_{(1/2)}\big( \tfrac{\gamma'}{\gamma}(V, s) + \tfrac{\gamma'}{\gamma}(V, 1-s) \big) \widehat{\psi}(s) ds + O(1). \end{align}$$ Here, I take $\overline{V}$ to be the conjugate representation and assume the functional equation is of the form $$ Q^{s/2} \gamma(V, s) L(V, s) = \varepsilon(V) Q^{(1-s)/2} \gamma(V, 1-s) L(\overline{V}, 1-s) $$ for collected gamma factors $\gamma(V, s)$ and a root number $\lvert \varepsilon(V) \rvert = 1$. (Brauer's On Artin's L-series with general group characters details the general functional equations, and this explicit formula is a in $\S$5.5 of Iwaniec–Kowalski).

But a major challenge in the face of being computationally useful is that we don't know the complete polar data for an arbitrary Artin $L$-function. Assuming Artin's conjecture and taking a nontrivial irreducible representation, there should be no polar contribution and we get a classical explicit formula.

  • $\begingroup$ Thanks for the answer! Can I just clarify what the $\Lambda_V$ appearing in your answer is? $\endgroup$
    – Dekimshita
    Apr 6 at 18:43
  • $\begingroup$ Those are the generalized von Mangoldt numbers associated to $L(V, s)$. They are the coefficients of the Dirichlet series $L'/L(V, s)$. For primes, one has $\Lambda_V(p) = a(p) \log p$ where $a(p)$ is the $p$th coefficient of $L(V, s)$. At prime powers, you'll have a slightly more complicated relation that can be written down in terms of the factors in the Euler product. See 5.26 in Iwaniec&ndash;Kowalski for more. $\endgroup$ Apr 6 at 19:07

Yes, a similar formula exists for $L(V,s)$. See : Weil André, Sur les formules explicites de la théorie des nombres. Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 3–18.


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