Properties

 Label 2-570-1.1-c5-0-10 Degree $2$ Conductor $570$ Sign $1$ Analytic cond. $91.4187$ Root an. cond. $9.56131$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 4·2-s + 9·3-s + 16·4-s + 25·5-s − 36·6-s − 70.9·7-s − 64·8-s + 81·9-s − 100·10-s − 424.·11-s + 144·12-s + 434.·13-s + 283.·14-s + 225·15-s + 256·16-s − 1.58e3·17-s − 324·18-s + 361·19-s + 400·20-s − 638.·21-s + 1.69e3·22-s − 59.0·23-s − 576·24-s + 625·25-s − 1.73e3·26-s + 729·27-s − 1.13e3·28-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.547·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 1.05·11-s + 0.288·12-s + 0.713·13-s + 0.386·14-s + 0.258·15-s + 0.250·16-s − 1.33·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.315·21-s + 0.747·22-s − 0.0232·23-s − 0.204·24-s + 0.200·25-s − 0.504·26-s + 0.192·27-s − 0.273·28-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$570$$    =    $$2 \cdot 3 \cdot 5 \cdot 19$$ Sign: $1$ Analytic conductor: $$91.4187$$ Root analytic conductor: $$9.56131$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{570} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 570,\ (\ :5/2),\ 1)$$

Particular Values

 $$L(3)$$ $$\approx$$ $$1.670823032$$ $$L(\frac12)$$ $$\approx$$ $$1.670823032$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 4T$$
3 $$1 - 9T$$
5 $$1 - 25T$$
19 $$1 - 361T$$
good7 $$1 + 70.9T + 1.68e4T^{2}$$
11 $$1 + 424.T + 1.61e5T^{2}$$
13 $$1 - 434.T + 3.71e5T^{2}$$
17 $$1 + 1.58e3T + 1.41e6T^{2}$$
23 $$1 + 59.0T + 6.43e6T^{2}$$
29 $$1 + 3.14e3T + 2.05e7T^{2}$$
31 $$1 - 1.19e3T + 2.86e7T^{2}$$
37 $$1 - 6.11e3T + 6.93e7T^{2}$$
41 $$1 - 1.73e4T + 1.15e8T^{2}$$
43 $$1 - 3.18e3T + 1.47e8T^{2}$$
47 $$1 - 1.35e4T + 2.29e8T^{2}$$
53 $$1 - 1.63e4T + 4.18e8T^{2}$$
59 $$1 + 3.87e3T + 7.14e8T^{2}$$
61 $$1 - 3.71e4T + 8.44e8T^{2}$$
67 $$1 - 1.07e4T + 1.35e9T^{2}$$
71 $$1 - 4.55e4T + 1.80e9T^{2}$$
73 $$1 + 2.33e4T + 2.07e9T^{2}$$
79 $$1 - 3.04e4T + 3.07e9T^{2}$$
83 $$1 + 5.15e4T + 3.93e9T^{2}$$
89 $$1 + 6.18e4T + 5.58e9T^{2}$$
97 $$1 - 3.36e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$