# Properties

 Label 2-1840-460.459-c1-0-70 Degree $2$ Conductor $1840$ Sign $0.147 + 0.989i$ Analytic cond. $14.6924$ Root an. cond. $3.83307$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 3.16·3-s − 2.23i·5-s − 4.24i·7-s + 7.00·9-s − 7.07i·15-s − 13.4i·21-s + (−4.74 + 0.707i)23-s − 5.00·25-s + 12.6·27-s − 6·29-s − 9.48·35-s + 12·41-s + 12.7i·43-s − 15.6i·45-s + 9.48·47-s + ⋯
 L(s)  = 1 + 1.82·3-s − 0.999i·5-s − 1.60i·7-s + 2.33·9-s − 1.82i·15-s − 2.92i·21-s + (−0.989 + 0.147i)23-s − 1.00·25-s + 2.43·27-s − 1.11·29-s − 1.60·35-s + 1.87·41-s + 1.94i·43-s − 2.33i·45-s + 1.38·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.147 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.147 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1840$$    =    $$2^{4} \cdot 5 \cdot 23$$ Sign: $0.147 + 0.989i$ Analytic conductor: $$14.6924$$ Root analytic conductor: $$3.83307$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1840} (1839, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1840,\ (\ :1/2),\ 0.147 + 0.989i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$3.312105968$$ $$L(\frac12)$$ $$\approx$$ $$3.312105968$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + 2.23iT$$
23 $$1 + (4.74 - 0.707i)T$$
good3 $$1 - 3.16T + 3T^{2}$$
7 $$1 + 4.24iT - 7T^{2}$$
11 $$1 + 11T^{2}$$
13 $$1 - 13T^{2}$$
17 $$1 + 17T^{2}$$
19 $$1 + 19T^{2}$$
29 $$1 + 6T + 29T^{2}$$
31 $$1 - 31T^{2}$$
37 $$1 + 37T^{2}$$
41 $$1 - 12T + 41T^{2}$$
43 $$1 - 12.7iT - 43T^{2}$$
47 $$1 - 9.48T + 47T^{2}$$
53 $$1 + 53T^{2}$$
59 $$1 - 59T^{2}$$
61 $$1 - 13.4iT - 61T^{2}$$
67 $$1 + 4.24iT - 67T^{2}$$
71 $$1 - 71T^{2}$$
73 $$1 - 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 + 15.5iT - 83T^{2}$$
89 $$1 - 17.8iT - 89T^{2}$$
97 $$1 + 97T^{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

−9.105697671966544422037816504857, −8.206154656231066002290505884728, −7.66981167545775576475447695974, −7.19058096652757856726012202484, −5.87828133552865796052554823830, −4.32511251589903727092955923506, −4.22353258160984330676185182317, −3.20672562077088951890911286649, −1.98267923644825071147193256303, −0.992719158316320777705041565287, 2.12864841305706741144718164091, 2.35789569964417640388409082685, 3.36165500083336533606413305160, 4.07293377587954730748701419088, 5.49144701961845764894916238254, 6.33830228616134470610451434397, 7.37557812733734881293790762204, 7.892227914853925266771239993353, 8.770174630115810353971532412091, 9.215403497678866252495955612694