# Properties

 Degree $2$ Conductor $1123$ Sign $1$ Motivic weight $0$ Arithmetic yes Primitive no Self-dual yes

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## Dirichlet series

 $\zeta_K(s)$  = 1 + 4-s + 2·7-s + 9-s + 16-s + 2·17-s + 2·19-s + 2·23-s + 25-s + 2·28-s + 36-s + 2·41-s + 2·43-s + 3·49-s + 2·53-s + 2·59-s + 2·61-s + 2·63-s + 64-s + 2·67-s + 2·68-s + 2·76-s + 2·79-s + 81-s + 2·89-s + 2·92-s + 100-s + 2·112-s + ⋯

## Functional equation

\begin{aligned}\Lambda_K(s)=\mathstrut & 1123 ^{s/2} \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1123$$ Sign: $1$ Arithmetic: yes Primitive: no Self-dual: yes Selberg data: $$(2,\ 1123,\ (\ :0),\ 1)$$

## Particular Values

$\zeta_K(1/2) \approx -0.9084341914$
Pole at $$s=1$$

## Euler product

$$\zeta_K(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Factorization

$$\zeta_K(s) =$$ $$\zeta(s)$$$$\;\cdot$$ $$L(s,\chi_{1123}(1122, \cdot))$$