# Properties

 Degree 3 Conductor $2^{2} \cdot 37$ Sign $1$ Motivic weight 0 Primitive no Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $\zeta_K(s)$  = 1 + 2-s + 4-s + 5-s + 8-s + 10-s + 13-s + 16-s + 17-s + 19-s + 20-s + 23-s + 2·25-s + 26-s + 27-s + 29-s + 31-s + 32-s + 34-s + 2·37-s + 38-s + 40-s + 43-s + 46-s + 2·50-s + 52-s + 54-s + 58-s + ⋯

## Functional equation

\begin{aligned}\Lambda_K(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\R}(s)^{3} \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}

## Invariants

 $$d$$ = $$3$$ $$N$$ = $$148$$    =    $$2^{2} \cdot 37$$ $$\varepsilon$$ = $1$ primitive : no self-dual : yes Selberg data = $$(3,\ 148,\ (0, 0, 0:\ ),\ 1)$$

## Euler product

\begin{aligned}\zeta_K(s) = \prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}

## Factorization

$$\zeta_K(s) =$$ $$\zeta(s)$$$$\;\cdot$$$$L(s, \rho_{2.148.3t2.a.a})$$

## Particular Values

$\zeta_K(1/2) \approx -0.3439602502$
Pole at $$s=1$$