# Properties

 Degree $2$ Conductor $1470$ Sign $0.0332 + 0.999i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·2-s − i·3-s − 4-s + (0.0743 + 2.23i)5-s + 6-s − i·8-s − 9-s + (−2.23 + 0.0743i)10-s − 3.05·11-s + i·12-s − 1.64i·13-s + (2.23 − 0.0743i)15-s + 16-s − 2.90i·17-s − i·18-s − 2.21·19-s + ⋯
 L(s)  = 1 + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.0332 + 0.999i)5-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s + (−0.706 + 0.0234i)10-s − 0.921·11-s + 0.288i·12-s − 0.455i·13-s + (0.577 − 0.0191i)15-s + 0.250·16-s − 0.705i·17-s − 0.235i·18-s − 0.507·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0332 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0332 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1470$$    =    $$2 \cdot 3 \cdot 5 \cdot 7^{2}$$ Sign: $0.0332 + 0.999i$ Motivic weight: $$1$$ Character: $\chi_{1470} (589, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1470,\ (\ :1/2),\ 0.0332 + 0.999i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.5692516406$$ $$L(\frac12)$$ $$\approx$$ $$0.5692516406$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - iT$$
3 $$1 + iT$$
5 $$1 + (-0.0743 - 2.23i)T$$
7 $$1$$
good11 $$1 + 3.05T + 11T^{2}$$
13 $$1 + 1.64iT - 13T^{2}$$
17 $$1 + 2.90iT - 17T^{2}$$
19 $$1 + 2.21T + 19T^{2}$$
23 $$1 + 1.78iT - 23T^{2}$$
29 $$1 + 5.58T + 29T^{2}$$
31 $$1 - 0.944T + 31T^{2}$$
37 $$1 + 10.7iT - 37T^{2}$$
41 $$1 - 6.67T + 41T^{2}$$
43 $$1 + 10.5iT - 43T^{2}$$
47 $$1 - 11.3iT - 47T^{2}$$
53 $$1 + 5.78iT - 53T^{2}$$
59 $$1 - 5.97T + 59T^{2}$$
61 $$1 - 0.445T + 61T^{2}$$
67 $$1 - 1.26iT - 67T^{2}$$
71 $$1 + 14.8T + 71T^{2}$$
73 $$1 + 7.91iT - 73T^{2}$$
79 $$1 + 14.2T + 79T^{2}$$
83 $$1 + 0.874iT - 83T^{2}$$
89 $$1 + 17.5T + 89T^{2}$$
97 $$1 + 10.4iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$