# Properties

 Label 2-840-5.4-c1-0-11 Degree $2$ Conductor $840$ Sign $0.241 + 0.970i$ Analytic cond. $6.70743$ Root an. cond. $2.58987$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − i·3-s + (−2.17 + 0.539i)5-s + i·7-s − 9-s + 5.41·11-s − 4.34i·13-s + (0.539 + 2.17i)15-s + 1.07i·17-s − 4.34·19-s + 21-s − 6.34i·23-s + (4.41 − 2.34i)25-s + i·27-s + 8.83·29-s − 4.34·31-s + ⋯
 L(s)  = 1 − 0.577i·3-s + (−0.970 + 0.241i)5-s + 0.377i·7-s − 0.333·9-s + 1.63·11-s − 1.20i·13-s + (0.139 + 0.560i)15-s + 0.261i·17-s − 0.995·19-s + 0.218·21-s − 1.32i·23-s + (0.883 − 0.468i)25-s + 0.192i·27-s + 1.64·29-s − 0.779·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$840$$    =    $$2^{3} \cdot 3 \cdot 5 \cdot 7$$ Sign: $0.241 + 0.970i$ Analytic conductor: $$6.70743$$ Root analytic conductor: $$2.58987$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{840} (169, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 840,\ (\ :1/2),\ 0.241 + 0.970i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.970710 - 0.759037i$$ $$L(\frac12)$$ $$\approx$$ $$0.970710 - 0.759037i$$ $$L(\frac{3}{2})$$ 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 + iT$$
5 $$1 + (2.17 - 0.539i)T$$
7 $$1 - iT$$
good11 $$1 - 5.41T + 11T^{2}$$
13 $$1 + 4.34iT - 13T^{2}$$
17 $$1 - 1.07iT - 17T^{2}$$
19 $$1 + 4.34T + 19T^{2}$$
23 $$1 + 6.34iT - 23T^{2}$$
29 $$1 - 8.83T + 29T^{2}$$
31 $$1 + 4.34T + 31T^{2}$$
37 $$1 + 8.68iT - 37T^{2}$$
41 $$1 - 8.34T + 41T^{2}$$
43 $$1 + 6.15iT - 43T^{2}$$
47 $$1 + 6.83iT - 47T^{2}$$
53 $$1 + 6.18iT - 53T^{2}$$
59 $$1 - 6.83T + 59T^{2}$$
61 $$1 + 4.52T + 61T^{2}$$
67 $$1 - 67T^{2}$$
71 $$1 + 14.0T + 71T^{2}$$
73 $$1 - 11.1iT - 73T^{2}$$
79 $$1 + 0.680T + 79T^{2}$$
83 $$1 - 6.83iT - 83T^{2}$$
89 $$1 + 6.49T + 89T^{2}$$
97 $$1 + 10.4iT - 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

−10.18101122581583643800849374524, −8.718727264241038011214500505918, −8.537433700570052842278717990491, −7.38621827139592882771916611727, −6.64933764406660740135415111004, −5.84632556479999142047508828970, −4.46110931238619617887673748912, −3.57790458223595874851368343121, −2.36629133819046856623897589246, −0.68923602224855809126287836310, 1.34773931793172283447907851778, 3.21806668371009817783939205664, 4.25650789469641968859104270442, 4.54299887212047800787024304945, 6.16022267313455681960729853672, 6.91631365615558338856309102956, 7.87619455763515309666097304942, 8.942727611892243809611659591649, 9.299862808756148442222395347590, 10.40798959369547643222047473067