# Properties

 Label 2-475-19.18-c0-0-2 Degree $2$ Conductor $475$ Sign $-1$ Analytic cond. $0.237055$ Root an. cond. $0.486883$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.41i·2-s − 1.41i·3-s − 1.00·4-s − 2.00·6-s − 1.00·9-s + 1.41i·12-s + 1.41i·13-s − 0.999·16-s + 1.41i·18-s + 19-s + 2.00·26-s + 1.41i·32-s + 1.00·36-s + 1.41i·37-s − 1.41i·38-s + 2.00·39-s + ⋯
 L(s)  = 1 − 1.41i·2-s − 1.41i·3-s − 1.00·4-s − 2.00·6-s − 1.00·9-s + 1.41i·12-s + 1.41i·13-s − 0.999·16-s + 1.41i·18-s + 19-s + 2.00·26-s + 1.41i·32-s + 1.00·36-s + 1.41i·37-s − 1.41i·38-s + 2.00·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$475$$    =    $$5^{2} \cdot 19$$ Sign: $-1$ Analytic conductor: $$0.237055$$ Root analytic conductor: $$0.486883$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{475} (151, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 475,\ (\ :0),\ -1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8608239639$$ $$L(\frac12)$$ $$\approx$$ $$0.8608239639$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
19 $$1 - T$$
good2 $$1 + 1.41iT - T^{2}$$
3 $$1 + 1.41iT - T^{2}$$
7 $$1 + T^{2}$$
11 $$1 + T^{2}$$
13 $$1 - 1.41iT - T^{2}$$
17 $$1 + T^{2}$$
23 $$1 + T^{2}$$
29 $$1 - T^{2}$$
31 $$1 - T^{2}$$
37 $$1 - 1.41iT - T^{2}$$
41 $$1 - T^{2}$$
43 $$1 + T^{2}$$
47 $$1 + T^{2}$$
53 $$1 + 1.41iT - T^{2}$$
59 $$1 - T^{2}$$
61 $$1 + T^{2}$$
67 $$1 + 1.41iT - T^{2}$$
71 $$1 - T^{2}$$
73 $$1 + T^{2}$$
79 $$1 - T^{2}$$
83 $$1 + T^{2}$$
89 $$1 - T^{2}$$
97 $$1 - 1.41iT - T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$