# Properties

 Label 2-475-95.94-c2-0-32 Degree $2$ Conductor $475$ Sign $0.447 + 0.894i$ Analytic cond. $12.9428$ Root an. cond. $3.59761$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 4·4-s + 5i·7-s − 9·9-s + 3·11-s + 16·16-s − 15i·17-s + 19·19-s − 30i·23-s − 20i·28-s + 36·36-s − 85i·43-s − 12·44-s − 75i·47-s + 24·49-s + 103·61-s + ⋯
 L(s)  = 1 − 4-s + 0.714i·7-s − 9-s + 0.272·11-s + 16-s − 0.882i·17-s + 19-s − 1.30i·23-s − 0.714i·28-s + 36-s − 1.97i·43-s − 0.272·44-s − 1.59i·47-s + 0.489·49-s + 1.68·61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, 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: $$12.9428$$ Root analytic conductor: $$3.59761$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{475} (474, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 475,\ (\ :1),\ 0.447 + 0.894i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.9715583521$$ $$L(\frac12)$$ $$\approx$$ $$0.9715583521$$ $$L(2)$$ 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 - 19T$$
good2 $$1 + 4T^{2}$$
3 $$1 + 9T^{2}$$
7 $$1 - 5iT - 49T^{2}$$
11 $$1 - 3T + 121T^{2}$$
13 $$1 + 169T^{2}$$
17 $$1 + 15iT - 289T^{2}$$
23 $$1 + 30iT - 529T^{2}$$
29 $$1 - 841T^{2}$$
31 $$1 - 961T^{2}$$
37 $$1 + 1.36e3T^{2}$$
41 $$1 - 1.68e3T^{2}$$
43 $$1 + 85iT - 1.84e3T^{2}$$
47 $$1 + 75iT - 2.20e3T^{2}$$
53 $$1 + 2.80e3T^{2}$$
59 $$1 - 3.48e3T^{2}$$
61 $$1 - 103T + 3.72e3T^{2}$$
67 $$1 + 4.48e3T^{2}$$
71 $$1 - 5.04e3T^{2}$$
73 $$1 + 25iT - 5.32e3T^{2}$$
79 $$1 - 6.24e3T^{2}$$
83 $$1 - 90iT - 6.88e3T^{2}$$
89 $$1 - 7.92e3T^{2}$$
97 $$1 + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$