# Properties

 Label 2-1425-5.4-c1-0-14 Degree $2$ Conductor $1425$ Sign $-0.894 + 0.447i$ Analytic cond. $11.3786$ Root an. cond. $3.37323$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.73i·2-s + i·3-s − 0.999·4-s − 1.73·6-s + 2.73i·7-s + 1.73i·8-s − 9-s + 4.73·11-s − 0.999i·12-s + 0.732i·13-s − 4.73·14-s − 5·16-s − 1.73i·18-s − 19-s − 2.73·21-s + 8.19i·22-s + ⋯
 L(s)  = 1 + 1.22i·2-s + 0.577i·3-s − 0.499·4-s − 0.707·6-s + 1.03i·7-s + 0.612i·8-s − 0.333·9-s + 1.42·11-s − 0.288i·12-s + 0.203i·13-s − 1.26·14-s − 1.25·16-s − 0.408i·18-s − 0.229·19-s − 0.596·21-s + 1.74i·22-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1425$$    =    $$3 \cdot 5^{2} \cdot 19$$ Sign: $-0.894 + 0.447i$ Analytic conductor: $$11.3786$$ Root analytic conductor: $$3.37323$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1425} (799, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1425,\ (\ :1/2),\ -0.894 + 0.447i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.681800650$$ $$L(\frac12)$$ $$\approx$$ $$1.681800650$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - iT$$
5 $$1$$
19 $$1 + T$$
good2 $$1 - 1.73iT - 2T^{2}$$
7 $$1 - 2.73iT - 7T^{2}$$
11 $$1 - 4.73T + 11T^{2}$$
13 $$1 - 0.732iT - 13T^{2}$$
17 $$1 - 17T^{2}$$
23 $$1 - 3.46iT - 23T^{2}$$
29 $$1 + 8.19T + 29T^{2}$$
31 $$1 - 8.92T + 31T^{2}$$
37 $$1 - 6.19iT - 37T^{2}$$
41 $$1 - 1.26T + 41T^{2}$$
43 $$1 - 4.19iT - 43T^{2}$$
47 $$1 - 3.46iT - 47T^{2}$$
53 $$1 + 9.46iT - 53T^{2}$$
59 $$1 + 2.53T + 59T^{2}$$
61 $$1 + 6.53T + 61T^{2}$$
67 $$1 + 8iT - 67T^{2}$$
71 $$1 + 4.39T + 71T^{2}$$
73 $$1 + 16.9iT - 73T^{2}$$
79 $$1 - 10.9T + 79T^{2}$$
83 $$1 + 12.9iT - 83T^{2}$$
89 $$1 + 10.7T + 89T^{2}$$
97 $$1 - 6.19iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$