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Properties

Label 2-840-840.149-c0-0-2
Degree $2$
Conductor $840$
Sign $0.553 + 0.832i$
Analytic cond. $0.419214$
Root an. cond. $0.647467$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s − i·5-s − 0.999·6-s + (0.5 + 0.866i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.866 + 1.5i)11-s + (0.866 + 0.499i)12-s − 0.999i·14-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.499i)18-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s − i·5-s − 0.999·6-s + (0.5 + 0.866i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.866 + 1.5i)11-s + (0.866 + 0.499i)12-s − 0.999i·14-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.499i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.553 + 0.832i$
Analytic conductor: \(0.419214\)
Root analytic conductor: \(0.647467\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :0),\ 0.553 + 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9564929575\)
\(L(\frac12)\) \(\approx\) \(0.9564929575\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + iT \)
7 \( 1 + (-0.5 - 0.866i)T \)
good11 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + 1.73T + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - iT - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + 1.73iT - T^{2} \)
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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

−9.736071728033145693709177936444, −9.368545620203741964185541502463, −8.778402598286576862517730616380, −7.87999885386450777794618484255, −7.31912537166990524109629335804, −6.13342334060492676729729662235, −4.65919921179108091808855108845, −3.68220185592599898367091644439, −2.15147541620097819104548521194, −1.60111316472822602150173439195, 1.65628045306500383703109608875, 3.11031713814213868060518998039, 3.99459923711770873784317693961, 5.43578485794763381387852745434, 6.50290135782636562221341284744, 7.34924213062891873335044696507, 7.993207112860179285967838423045, 8.867194457503399674980107539615, 9.548136342536584559994636800212, 10.54729315821396148477768536510

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