# Properties

 Degree $2$ Conductor $155$ Sign $1$ Motivic weight $0$ Arithmetic yes Primitive no Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $\zeta_K(s)$  = 1 + 2·3-s + 4-s + 5-s + 3·9-s + 2·12-s + 2·13-s + 2·15-s + 16-s + 2·17-s + 2·19-s + 20-s + 2·23-s + 25-s + 4·27-s + 31-s + 3·36-s + 2·37-s + 4·39-s + 2·41-s + 2·43-s + 3·45-s + 2·48-s + 49-s + 4·51-s + 2·52-s + 2·53-s + 4·57-s + ⋯

## Functional equation

\begin{aligned}\Lambda_K(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$155$$    =    $$5 \cdot 31$$ Sign: $1$ Arithmetic: yes Primitive: no Self-dual: yes Selberg data: $$(2,\ 155,\ (\ :0),\ 1)$$

## Particular Values

$\zeta_K(1/2) \approx -2.314517204$
Pole at $$s=1$$

## Euler product

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

## Factorization

$$\zeta_K(s) =$$ $$\zeta(s)$$$$\;\cdot$$ $$L(s,\chi_{155}(154, \cdot))$$