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Properties

Label 2-840-280.139-c1-0-71
Degree $2$
Conductor $840$
Sign $0.582 + 0.812i$
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  + (−0.716 + 1.21i)2-s + 3-s + (−0.973 − 1.74i)4-s + (0.151 − 2.23i)5-s + (−0.716 + 1.21i)6-s + (−2.00 + 1.72i)7-s + (2.82 + 0.0635i)8-s + 9-s + (2.61 + 1.78i)10-s + 2.25·11-s + (−0.973 − 1.74i)12-s − 6.09i·13-s + (−0.673 − 3.68i)14-s + (0.151 − 2.23i)15-s + (−2.10 + 3.40i)16-s − 2.64·17-s + ⋯
L(s)  = 1  + (−0.506 + 0.862i)2-s + 0.577·3-s + (−0.486 − 0.873i)4-s + (0.0678 − 0.997i)5-s + (−0.292 + 0.497i)6-s + (−0.756 + 0.653i)7-s + (0.999 + 0.0224i)8-s + 0.333·9-s + (0.825 + 0.563i)10-s + 0.680·11-s + (−0.281 − 0.504i)12-s − 1.68i·13-s + (−0.180 − 0.983i)14-s + (0.0391 − 0.576i)15-s + (−0.525 + 0.850i)16-s − 0.641·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.582 + 0.812i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.582 + 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.935085 - 0.480285i\)
\(L(\frac12)\) \(\approx\) \(0.935085 - 0.480285i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.716 - 1.21i)T \)
3 \( 1 - T \)
5 \( 1 + (-0.151 + 2.23i)T \)
7 \( 1 + (2.00 - 1.72i)T \)
good11 \( 1 - 2.25T + 11T^{2} \)
13 \( 1 + 6.09iT - 13T^{2} \)
17 \( 1 + 2.64T + 17T^{2} \)
19 \( 1 + 1.31iT - 19T^{2} \)
23 \( 1 + 6.62T + 23T^{2} \)
29 \( 1 + 6.44iT - 29T^{2} \)
31 \( 1 - 6.67T + 31T^{2} \)
37 \( 1 - 2.30T + 37T^{2} \)
41 \( 1 + 5.50iT - 41T^{2} \)
43 \( 1 + 1.26iT - 43T^{2} \)
47 \( 1 - 0.599iT - 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + 0.847iT - 59T^{2} \)
61 \( 1 - 11.7T + 61T^{2} \)
67 \( 1 + 14.2iT - 67T^{2} \)
71 \( 1 + 3.94iT - 71T^{2} \)
73 \( 1 - 8.34T + 73T^{2} \)
79 \( 1 - 0.674iT - 79T^{2} \)
83 \( 1 - 7.23T + 83T^{2} \)
89 \( 1 - 5.66iT - 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−9.658842946926921546175235111636, −9.209656346256166488059906893149, −8.234811129318424286833235399535, −7.930958823999348967580294191556, −6.51895712274173816480652415415, −5.87778002781443994163252115460, −4.89961492367360404070322771537, −3.79549818043147948450580652641, −2.23086483810926900472357909413, −0.57229991110130829338980565321, 1.65414982966384557420475445686, 2.73957845892599905862682530280, 3.78093085014017397054656823795, 4.32984302075907946228021064129, 6.46567090388507264690689401552, 6.89465347381389679031927144171, 7.905764857007975249253401113955, 8.890881385265117834477722589732, 9.699176497546518663229556321687, 10.10088198439171664294282560332

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