# Properties

 Degree 2 Conductor $2^{3} \cdot 137$ Sign $1$ Motivic weight 0 Primitive no Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $\zeta_K(s)$  = 1 + 2-s + 2·3-s + 4-s + 2·5-s + 2·6-s + 2·7-s + 8-s + 3·9-s + 2·10-s + 2·12-s + 2·13-s + 2·14-s + 4·15-s + 16-s + 2·17-s + 3·18-s + 2·20-s + 4·21-s + 2·24-s + 3·25-s + 2·26-s + 4·27-s + 2·28-s + 2·29-s + 4·30-s + 32-s + 2·34-s + ⋯

## Functional equation

\begin{aligned} \Lambda_K(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1096$$    =    $$2^{3} \cdot 137$$ $$\varepsilon$$ = $1$ primitive : no self-dual : yes Selberg data = $(2,\ 1096,\ (0, 0:\ ),\ 1)$

## Euler product

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

## Factorization

$$\zeta_K(s) =$$ $$\zeta(s)$$$$\;\cdot$$ $$L(s,\chi_{1096}(821, \cdot))$$

## Particular Values

$\zeta_K(1/2) \approx -4.417328031$
Pole at $$s=1$$