# Properties

 Degree 3 Conductor $13^{2}$ Sign $1$ Motivic weight 0 Primitive no Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $\zeta_K(s)$  = 1 + 3·5-s + 8-s + 13-s + 6·25-s + 27-s + 3·31-s + 3·40-s + 3·47-s + 3·53-s + 64-s + 3·65-s + 3·73-s + 3·79-s + 3·83-s + 3·103-s + 104-s + 3·109-s + 10·125-s + 3·131-s + 3·135-s + 3·151-s + 9·155-s + 3·157-s + 169-s + 3·181-s + 6·200-s + 216-s + ⋯

## Functional equation

\begin{aligned}\Lambda_K(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s)^{3} \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}

## Invariants

 $$d$$ = $$3$$ $$N$$ = $$169$$    =    $$13^{2}$$ $$\varepsilon$$ = $1$ primitive : no self-dual : yes Selberg data = $$(3,\ 169,\ (0, 0, 0:\ ),\ 1)$$

## Euler product

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

## Factorization

$$\zeta_K(s) =$$ $$\zeta(s)$$$$\;\cdot$$ $$L(s,\chi_{13}(3, \cdot))$$$$\;\cdot$$ $$L(s,\chi_{13}(9, \cdot))$$

## Particular Values

$\zeta_K(1/2) \approx -0.2674316037$
Pole at $$s=1$$