Properties

 Degree $2$ Conductor $1470$ Sign $0.605 + 0.795i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 0.999·6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.499 − 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.499 − 0.866i)12-s − 13-s + 0.999·15-s + (−0.5 − 0.866i)16-s + (0.499 − 0.866i)18-s + (−1.5 − 2.59i)19-s + 0.999·20-s + ⋯
 L(s)  = 1 + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s − 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (0.150 − 0.261i)11-s + (−0.144 − 0.249i)12-s − 0.277·13-s + 0.258·15-s + (−0.125 − 0.216i)16-s + (0.117 − 0.204i)18-s + (−0.344 − 0.596i)19-s + 0.223·20-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$1470$$    =    $$2 \cdot 3 \cdot 5 \cdot 7^{2}$$ Sign: $0.605 + 0.795i$ Motivic weight: $$1$$ Character: $\chi_{1470} (361, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1470,\ (\ :1/2),\ 0.605 + 0.795i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.8801287083$$ $$L(\frac12)$$ $$\approx$$ $$0.8801287083$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.5 - 0.866i)T$$
3 $$1 + (0.5 - 0.866i)T$$
5 $$1 + (0.5 + 0.866i)T$$
7 $$1$$
good11 $$1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + T + 13T^{2}$$
17 $$1 + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (3.5 + 6.06i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + 8T + 29T^{2}$$
31 $$1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 - 11T + 41T^{2}$$
43 $$1 - 8T + 43T^{2}$$
47 $$1 + (2.5 + 4.33i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-5.5 + 9.52i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 6T + 71T^{2}$$
73 $$1 + (3 - 5.19i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + 8T + 83T^{2}$$
89 $$1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 - 16T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$