Properties

 Label 2-8470-1.1-c1-0-35 Degree $2$ Conductor $8470$ Sign $1$ Analytic cond. $67.6332$ Root an. cond. $8.22394$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + 2-s − 0.406·3-s + 4-s − 5-s − 0.406·6-s + 7-s + 8-s − 2.83·9-s − 10-s − 0.406·12-s + 0.813·13-s + 14-s + 0.406·15-s + 16-s − 3.83·17-s − 2.83·18-s − 5.42·19-s − 20-s − 0.406·21-s − 5.42·23-s − 0.406·24-s + 25-s + 0.813·26-s + 2.37·27-s + 28-s − 6.61·29-s + 0.406·30-s + ⋯
 L(s)  = 1 + 0.707·2-s − 0.234·3-s + 0.5·4-s − 0.447·5-s − 0.166·6-s + 0.377·7-s + 0.353·8-s − 0.944·9-s − 0.316·10-s − 0.117·12-s + 0.225·13-s + 0.267·14-s + 0.105·15-s + 0.250·16-s − 0.930·17-s − 0.668·18-s − 1.24·19-s − 0.223·20-s − 0.0887·21-s − 1.13·23-s − 0.0830·24-s + 0.200·25-s + 0.159·26-s + 0.456·27-s + 0.188·28-s − 1.22·29-s + 0.0742·30-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$8470$$    =    $$2 \cdot 5 \cdot 7 \cdot 11^{2}$$ Sign: $1$ Analytic conductor: $$67.6332$$ Root analytic conductor: $$8.22394$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{8470} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 8470,\ (\ :1/2),\ 1)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$2.077886330$$ $$L(\frac12)$$ $$\approx$$ $$2.077886330$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - T$$
5 $$1 + T$$
7 $$1 - T$$
11 $$1$$
good3 $$1 + 0.406T + 3T^{2}$$
13 $$1 - 0.813T + 13T^{2}$$
17 $$1 + 3.83T + 17T^{2}$$
19 $$1 + 5.42T + 19T^{2}$$
23 $$1 + 5.42T + 23T^{2}$$
29 $$1 + 6.61T + 29T^{2}$$
31 $$1 - 6T + 31T^{2}$$
37 $$1 - 8.24T + 37T^{2}$$
41 $$1 - 5.59T + 41T^{2}$$
43 $$1 - 3.02T + 43T^{2}$$
47 $$1 + 5.02T + 47T^{2}$$
53 $$1 - 6.61T + 53T^{2}$$
59 $$1 - 10.6T + 59T^{2}$$
61 $$1 + 0.978T + 61T^{2}$$
67 $$1 - 8T + 67T^{2}$$
71 $$1 - 14.8T + 71T^{2}$$
73 $$1 - 6.20T + 73T^{2}$$
79 $$1 + 17.0T + 79T^{2}$$
83 $$1 - 8.81T + 83T^{2}$$
89 $$1 + 1.18T + 89T^{2}$$
97 $$1 + 3.42T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$