# Properties

 Label 2-840-1.1-c1-0-0 Degree $2$ Conductor $840$ Sign $1$ Analytic cond. $6.70743$ Root an. cond. $2.58987$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 3-s − 5-s + 7-s + 9-s − 2·13-s + 15-s + 6·17-s − 4·19-s − 21-s + 4·23-s + 25-s − 27-s + 6·29-s − 35-s + 6·37-s + 2·39-s + 6·41-s − 4·43-s − 45-s + 8·47-s + 49-s − 6·51-s + 14·53-s + 4·57-s − 4·59-s − 2·61-s + 63-s + ⋯
 L(s)  = 1 − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s + 1.45·17-s − 0.917·19-s − 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.169·35-s + 0.986·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.840·51-s + 1.92·53-s + 0.529·57-s − 0.520·59-s − 0.256·61-s + 0.125·63-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$840$$    =    $$2^{3} \cdot 3 \cdot 5 \cdot 7$$ Sign: $1$ Analytic conductor: $$6.70743$$ Root analytic conductor: $$2.58987$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{840} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 840,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.221288344$$ $$L(\frac12)$$ $$\approx$$ $$1.221288344$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + T$$
5 $$1 + T$$
7 $$1 - T$$
good11 $$1 + p T^{2}$$
13 $$1 + 2 T + p T^{2}$$
17 $$1 - 6 T + p T^{2}$$
19 $$1 + 4 T + p T^{2}$$
23 $$1 - 4 T + p T^{2}$$
29 $$1 - 6 T + p T^{2}$$
31 $$1 + p T^{2}$$
37 $$1 - 6 T + p T^{2}$$
41 $$1 - 6 T + p T^{2}$$
43 $$1 + 4 T + p T^{2}$$
47 $$1 - 8 T + p T^{2}$$
53 $$1 - 14 T + p T^{2}$$
59 $$1 + 4 T + p T^{2}$$
61 $$1 + 2 T + p T^{2}$$
67 $$1 - 12 T + p T^{2}$$
71 $$1 + 12 T + p T^{2}$$
73 $$1 - 10 T + p T^{2}$$
79 $$1 - 8 T + p T^{2}$$
83 $$1 + 4 T + p T^{2}$$
89 $$1 - 6 T + p T^{2}$$
97 $$1 + 14 T + p T^{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

−10.35899536139237859657452871452, −9.450754894645541234345596216334, −8.393189724041127383030805741896, −7.62422934296928339340232971391, −6.79350883637272350613856089385, −5.74083335944297837978899468723, −4.88159926035910944720765525241, −3.96302542474933435088975798400, −2.62319993876382304656802984913, −0.954082357122462532643032272715, 0.954082357122462532643032272715, 2.62319993876382304656802984913, 3.96302542474933435088975798400, 4.88159926035910944720765525241, 5.74083335944297837978899468723, 6.79350883637272350613856089385, 7.62422934296928339340232971391, 8.393189724041127383030805741896, 9.450754894645541234345596216334, 10.35899536139237859657452871452