# Properties

 Degree 3 Conductor $3^{3} \cdot 13$ Sign $1$ Motivic weight 0 Primitive no Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $\zeta_K(s)$  = 1 + 3-s + 7-s + 8-s + 9-s + 3·11-s + 2·13-s + 17-s + 19-s + 21-s + 23-s + 24-s + 27-s + 29-s + 31-s + 3·33-s + 37-s + 2·39-s + 3·41-s + 2·49-s + 51-s + 53-s + 56-s + 57-s + 63-s + 64-s + 67-s + 69-s + ⋯

## Functional equation

\begin{aligned}\Lambda_K(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}

## Invariants

 $$d$$ = $$3$$ $$N$$ = $$351$$    =    $$3^{3} \cdot 13$$ $$\varepsilon$$ = $1$ primitive : no self-dual : yes Selberg data = $$(3,\ 351,\ (0:0),\ 1)$$

## Euler product

\begin{aligned}\zeta_K(s) = \prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}

## Factorization

$$\zeta_K(s) =$$ $$\zeta(s)$$$$\;\cdot$$$$L(s, \rho_{2.351.3t2.b.a})$$

## Particular Values

$\zeta_K(1/2) \approx -0.6517273998$
Pole at $$s=1$$