# Properties

 Label 2-475-5.4-c1-0-20 Degree $2$ Conductor $475$ Sign $0.447 + 0.894i$ Analytic cond. $3.79289$ Root an. cond. $1.94753$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2i·3-s + 2·4-s + i·7-s − 9-s + 3·11-s − 4i·12-s − 4i·13-s + 4·16-s + 3i·17-s − 19-s + 2·21-s − 4i·27-s + 2i·28-s − 6·29-s − 4·31-s + ⋯
 L(s)  = 1 − 1.15i·3-s + 4-s + 0.377i·7-s − 0.333·9-s + 0.904·11-s − 1.15i·12-s − 1.10i·13-s + 16-s + 0.727i·17-s − 0.229·19-s + 0.436·21-s − 0.769i·27-s + 0.377i·28-s − 1.11·29-s − 0.718·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$475$$    =    $$5^{2} \cdot 19$$ Sign: $0.447 + 0.894i$ Analytic conductor: $$3.79289$$ Root analytic conductor: $$1.94753$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{475} (324, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 475,\ (\ :1/2),\ 0.447 + 0.894i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.57003 - 0.970337i$$ $$L(\frac12)$$ $$\approx$$ $$1.57003 - 0.970337i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
19 $$1 + T$$
good2 $$1 - 2T^{2}$$
3 $$1 + 2iT - 3T^{2}$$
7 $$1 - iT - 7T^{2}$$
11 $$1 - 3T + 11T^{2}$$
13 $$1 + 4iT - 13T^{2}$$
17 $$1 - 3iT - 17T^{2}$$
23 $$1 - 23T^{2}$$
29 $$1 + 6T + 29T^{2}$$
31 $$1 + 4T + 31T^{2}$$
37 $$1 + 2iT - 37T^{2}$$
41 $$1 + 6T + 41T^{2}$$
43 $$1 + iT - 43T^{2}$$
47 $$1 - 3iT - 47T^{2}$$
53 $$1 - 12iT - 53T^{2}$$
59 $$1 - 6T + 59T^{2}$$
61 $$1 + T + 61T^{2}$$
67 $$1 - 4iT - 67T^{2}$$
71 $$1 - 6T + 71T^{2}$$
73 $$1 + 7iT - 73T^{2}$$
79 $$1 + 8T + 79T^{2}$$
83 $$1 - 12iT - 83T^{2}$$
89 $$1 + 12T + 89T^{2}$$
97 $$1 + 8iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$