# Properties

 Degree 2 Conductor $2^{2} \cdot 251$ Sign $1$ Motivic weight 0 Primitive no Self-dual yes

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## Dirichlet series

 $\zeta_K(s)$  = 1 + 2-s + 4-s + 2·5-s + 8-s + 9-s + 2·10-s + 2·11-s + 2·13-s + 16-s + 2·17-s + 18-s + 2·19-s + 2·20-s + 2·22-s + 3·25-s + 2·26-s + 32-s + 2·34-s + 36-s + 2·38-s + 2·40-s + 2·41-s + 2·43-s + 2·44-s + 2·45-s + 2·47-s + 49-s + ⋯

## Functional equation

\begin{aligned} \Lambda_K(s)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1004$$    =    $$2^{2} \cdot 251$$ $$\varepsilon$$ = $1$ primitive : no self-dual : yes Selberg data = $(2,\ 1004,\ (0, 0:\ ),\ 1)$

## Euler product

\begin{aligned} \zeta_K(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}

## Factorization

$$\zeta_K(s) =$$ $$\zeta(s)$$$$\;\cdot$$ $$L(s,\chi_{1004}(1003, \cdot))$$

## Particular Values

$\zeta_K(1/2) \approx -1.972875207$
Pole at $$s=1$$