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Properties

Label 2-840-21.17-c1-0-18
Degree $2$
Conductor $840$
Sign $0.606 + 0.795i$
Analytic cond. $6.70743$
Root an. cond. $2.58987$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.71 − 0.262i)3-s + (0.5 + 0.866i)5-s + (2.06 − 1.65i)7-s + (2.86 + 0.897i)9-s + (−3.50 − 2.02i)11-s − 0.519i·13-s + (−0.629 − 1.61i)15-s + (−2.57 + 4.45i)17-s + (5.57 − 3.22i)19-s + (−3.96 + 2.29i)21-s + (3.81 − 2.20i)23-s + (−0.499 + 0.866i)25-s + (−4.66 − 2.28i)27-s − 1.19i·29-s + (0.891 + 0.514i)31-s + ⋯
L(s)  = 1  + (−0.988 − 0.151i)3-s + (0.223 + 0.387i)5-s + (0.779 − 0.625i)7-s + (0.954 + 0.299i)9-s + (−1.05 − 0.609i)11-s − 0.144i·13-s + (−0.162 − 0.416i)15-s + (−0.624 + 1.08i)17-s + (1.27 − 0.738i)19-s + (−0.865 + 0.500i)21-s + (0.794 − 0.458i)23-s + (−0.0999 + 0.173i)25-s + (−0.897 − 0.440i)27-s − 0.222i·29-s + (0.160 + 0.0924i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.606 + 0.795i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.606 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03024 - 0.510284i\)
\(L(\frac12)\) \(\approx\) \(1.03024 - 0.510284i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 0.262i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.06 + 1.65i)T \)
good11 \( 1 + (3.50 + 2.02i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.519iT - 13T^{2} \)
17 \( 1 + (2.57 - 4.45i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.57 + 3.22i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.81 + 2.20i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.19iT - 29T^{2} \)
31 \( 1 + (-0.891 - 0.514i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.84 + 10.1i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.5T + 41T^{2} \)
43 \( 1 - 0.147T + 43T^{2} \)
47 \( 1 + (1.01 + 1.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.65 - 2.11i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.94 + 5.10i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-13.0 + 7.55i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.35 + 12.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.19iT - 71T^{2} \)
73 \( 1 + (1.01 + 0.584i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.457 + 0.792i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.49T + 83T^{2} \)
89 \( 1 + (-4.06 - 7.04i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.4iT - 97T^{2} \)
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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

−10.51489605560744734065810435186, −9.371514850065848554706402955986, −8.170677564573249446688066490314, −7.42152614133139873971479071160, −6.63093245257370231821172373430, −5.56545354603409226540991712304, −4.95997016687198977463011635270, −3.78117998720004984061711981143, −2.26406049113263336942440112086, −0.74045429528802914092069313311, 1.25660468283967254591775680597, 2.65457800607873318002491848311, 4.35271510439880978509220684071, 5.24730059403585424434953129057, 5.52440424392668238164915264127, 6.92383894658327214269581245299, 7.63433187679713201686023074373, 8.716045210961202035137296156259, 9.659153533329911328651684490750, 10.26873710395669219861293281499

Graph of the $Z$-function along the critical line