# Properties

 Label 2-2070-1.1-c3-0-61 Degree $2$ Conductor $2070$ Sign $1$ Analytic cond. $122.133$ Root an. cond. $11.0514$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2·2-s + 4·4-s + 5·5-s + 23.5·7-s + 8·8-s + 10·10-s + 32.1·11-s + 40.0·13-s + 47.1·14-s + 16·16-s − 126.·17-s + 0.232·19-s + 20·20-s + 64.2·22-s + 23·23-s + 25·25-s + 80.1·26-s + 94.2·28-s + 137.·29-s + 112.·31-s + 32·32-s − 252.·34-s + 117.·35-s + 45.7·37-s + 0.465·38-s + 40·40-s + 135.·41-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.27·7-s + 0.353·8-s + 0.316·10-s + 0.880·11-s + 0.855·13-s + 0.899·14-s + 0.250·16-s − 1.79·17-s + 0.00281·19-s + 0.223·20-s + 0.622·22-s + 0.208·23-s + 0.200·25-s + 0.604·26-s + 0.636·28-s + 0.878·29-s + 0.653·31-s + 0.176·32-s − 1.27·34-s + 0.568·35-s + 0.203·37-s + 0.00198·38-s + 0.158·40-s + 0.515·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2070$$    =    $$2 \cdot 3^{2} \cdot 5 \cdot 23$$ Sign: $1$ Analytic conductor: $$122.133$$ Root analytic conductor: $$11.0514$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{2070} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 2070,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$5.449887161$$ $$L(\frac12)$$ $$\approx$$ $$5.449887161$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 2T$$
3 $$1$$
5 $$1 - 5T$$
23 $$1 - 23T$$
good7 $$1 - 23.5T + 343T^{2}$$
11 $$1 - 32.1T + 1.33e3T^{2}$$
13 $$1 - 40.0T + 2.19e3T^{2}$$
17 $$1 + 126.T + 4.91e3T^{2}$$
19 $$1 - 0.232T + 6.85e3T^{2}$$
29 $$1 - 137.T + 2.43e4T^{2}$$
31 $$1 - 112.T + 2.97e4T^{2}$$
37 $$1 - 45.7T + 5.06e4T^{2}$$
41 $$1 - 135.T + 6.89e4T^{2}$$
43 $$1 - 543.T + 7.95e4T^{2}$$
47 $$1 + 26.4T + 1.03e5T^{2}$$
53 $$1 + 43.6T + 1.48e5T^{2}$$
59 $$1 + 202.T + 2.05e5T^{2}$$
61 $$1 - 150.T + 2.26e5T^{2}$$
67 $$1 + 420.T + 3.00e5T^{2}$$
71 $$1 + 667.T + 3.57e5T^{2}$$
73 $$1 - 602.T + 3.89e5T^{2}$$
79 $$1 + 1.37e3T + 4.93e5T^{2}$$
83 $$1 - 485.T + 5.71e5T^{2}$$
89 $$1 - 1.12e3T + 7.04e5T^{2}$$
97 $$1 + 1.48e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$