# Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 23$ Sign $1$ Motivic weight 0 Primitive no Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $\zeta_K(s)$  = 1 + 2-s + 3-s + 4-s + 2·5-s + 6-s + 2·7-s + 8-s + 9-s + 2·10-s + 12-s + 2·13-s + 2·14-s + 2·15-s + 16-s + 2·17-s + 18-s + 2·19-s + 2·20-s + 2·21-s + 23-s + 24-s + 3·25-s + 2·26-s + 27-s + 2·28-s + 2·30-s + 32-s + ⋯

## Functional equation

\begin{aligned}\Lambda_K(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$276$$    =    $$2^{2} \cdot 3 \cdot 23$$ $$\varepsilon$$ = $1$ primitive : no self-dual : yes Selberg data = $(2,\ 276,\ (\ :0),\ 1)$

## Euler product

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

## Factorization

$$\zeta_K(s) =$$ $$\zeta(s)$$$$\;\cdot$$ $$L(s,\chi_{276}(275, \cdot))$$

## Particular Values

$\zeta_K(1/2) \approx -4.057018688$
Pole at $$s=1$$