# Explicit formula for Artin L-functions

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.

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.
• Thanks for the answer! Can I just clarify what the $\Lambda_V$ appearing in your answer is? Apr 6 at 18:43
• 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. 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.