# Properties

 Label 2-2070-1.1-c3-0-18 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 − 35.4·7-s + 8·8-s + 10·10-s + 16.6·11-s − 79.9·13-s − 70.8·14-s + 16·16-s + 46.8·17-s − 110.·19-s + 20·20-s + 33.2·22-s + 23·23-s + 25·25-s − 159.·26-s − 141.·28-s + 0.836·29-s − 119.·31-s + 32·32-s + 93.6·34-s − 177.·35-s + 368.·37-s − 221.·38-s + 40·40-s + 95.7·41-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.5·4-s + 0.447·5-s − 1.91·7-s + 0.353·8-s + 0.316·10-s + 0.455·11-s − 1.70·13-s − 1.35·14-s + 0.250·16-s + 0.667·17-s − 1.33·19-s + 0.223·20-s + 0.322·22-s + 0.208·23-s + 0.200·25-s − 1.20·26-s − 0.956·28-s + 0.00535·29-s − 0.694·31-s + 0.176·32-s + 0.472·34-s − 0.855·35-s + 1.63·37-s − 0.944·38-s + 0.158·40-s + 0.364·41-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

 $$L(2)$$ $$\approx$$ $$2.333141510$$ $$L(\frac12)$$ $$\approx$$ $$2.333141510$$ $$L(\frac{5}{2})$$ 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 + 35.4T + 343T^{2}$$
11 $$1 - 16.6T + 1.33e3T^{2}$$
13 $$1 + 79.9T + 2.19e3T^{2}$$
17 $$1 - 46.8T + 4.91e3T^{2}$$
19 $$1 + 110.T + 6.85e3T^{2}$$
29 $$1 - 0.836T + 2.43e4T^{2}$$
31 $$1 + 119.T + 2.97e4T^{2}$$
37 $$1 - 368.T + 5.06e4T^{2}$$
41 $$1 - 95.7T + 6.89e4T^{2}$$
43 $$1 - 331.T + 7.95e4T^{2}$$
47 $$1 - 535.T + 1.03e5T^{2}$$
53 $$1 + 409.T + 1.48e5T^{2}$$
59 $$1 - 352.T + 2.05e5T^{2}$$
61 $$1 + 507.T + 2.26e5T^{2}$$
67 $$1 + 820.T + 3.00e5T^{2}$$
71 $$1 - 733.T + 3.57e5T^{2}$$
73 $$1 + 91.4T + 3.89e5T^{2}$$
79 $$1 - 329.T + 4.93e5T^{2}$$
83 $$1 - 753.T + 5.71e5T^{2}$$
89 $$1 - 1.05e3T + 7.04e5T^{2}$$
97 $$1 + 271.T + 9.12e5T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−9.124105540437047103299145200312, −7.67282084948072386443599003765, −7.03488983972451791985354318786, −6.23629001997672875113452356278, −5.79399871380085064876019243066, −4.66025198416688089646774880672, −3.82820270861479506528859621878, −2.86229806421714376891858656027, −2.24959710240942508583289448674, −0.59165253085540314681533273102, 0.59165253085540314681533273102, 2.24959710240942508583289448674, 2.86229806421714376891858656027, 3.82820270861479506528859621878, 4.66025198416688089646774880672, 5.79399871380085064876019243066, 6.23629001997672875113452356278, 7.03488983972451791985354318786, 7.67282084948072386443599003765, 9.124105540437047103299145200312