Properties

 Degree $3$ Conductor $327$ Sign $1$ Motivic weight $0$ Arithmetic yes Primitive no Self-dual yes

Related objects

(not yet available)

Dirichlet series

 $\zeta_K(s)$  = 1 + 2·3-s + 5-s + 8-s + 3·9-s + 3·11-s + 13-s + 2·15-s + 19-s + 2·24-s + 2·25-s + 4·27-s + 29-s + 6·33-s + 37-s + 2·39-s + 40-s + 3·45-s + 3·53-s + 3·55-s + 2·57-s + 64-s + 65-s + 67-s + 71-s + 3·72-s + 4·75-s + 79-s + ⋯

Functional equation

\begin{aligned}\Lambda_K(s)=\mathstrut & 327 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}

Invariants

 Degree: $$3$$ Conductor: $$327$$    =    $$3 \cdot 109$$ Sign: $1$ Arithmetic: yes Primitive: no Self-dual: yes Selberg data: $$(3,\ 327,\ (0:0),\ 1)$$

Particular Values

$\zeta_K(1/2) \approx -0.9015133868$
Pole at $$s=1$$

Euler product

$$\zeta_K(s) = \displaystyle\prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

Factorization

$$\zeta_K(s) =$$ $$\zeta(s)$$$$\;\cdot$$$$L(s, \rho_{2.327.3t2.a.a})$$