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Properties

Label 1-709-709.56-r1-0-0
Degree $1$
Conductor $709$
Sign $-0.999 + 0.0131i$
Analytic cond. $76.1926$
Root an. cond. $76.1926$
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  + (0.883 + 0.468i)2-s + (−0.645 + 0.764i)3-s + (0.560 + 0.828i)4-s + (0.996 + 0.0886i)5-s + (−0.928 + 0.372i)6-s + (0.969 + 0.245i)7-s + (0.106 + 0.994i)8-s + (−0.167 − 0.985i)9-s + (0.838 + 0.545i)10-s + (0.954 + 0.297i)11-s + (−0.994 − 0.106i)12-s + (−0.716 + 0.697i)13-s + (0.740 + 0.671i)14-s + (−0.710 + 0.703i)15-s + (−0.372 + 0.928i)16-s + (−0.975 − 0.220i)17-s + ⋯
L(s)  = 1  + (0.883 + 0.468i)2-s + (−0.645 + 0.764i)3-s + (0.560 + 0.828i)4-s + (0.996 + 0.0886i)5-s + (−0.928 + 0.372i)6-s + (0.969 + 0.245i)7-s + (0.106 + 0.994i)8-s + (−0.167 − 0.985i)9-s + (0.838 + 0.545i)10-s + (0.954 + 0.297i)11-s + (−0.994 − 0.106i)12-s + (−0.716 + 0.697i)13-s + (0.740 + 0.671i)14-s + (−0.710 + 0.703i)15-s + (−0.372 + 0.928i)16-s + (−0.975 − 0.220i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.999 + 0.0131i$
Analytic conductor: \(76.1926\)
Root analytic conductor: \(76.1926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (1:\ ),\ -0.999 + 0.0131i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02136951060 + 3.254054178i\)
\(L(\frac12)\) \(\approx\) \(0.02136951060 + 3.254054178i\)
\(L(1)\) \(\approx\) \(1.270521037 + 1.223125757i\)
\(L(1)\) \(\approx\) \(1.270521037 + 1.223125757i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.883 + 0.468i)T \)
3 \( 1 + (-0.645 + 0.764i)T \)
5 \( 1 + (0.996 + 0.0886i)T \)
7 \( 1 + (0.969 + 0.245i)T \)
11 \( 1 + (0.954 + 0.297i)T \)
13 \( 1 + (-0.716 + 0.697i)T \)
17 \( 1 + (-0.975 - 0.220i)T \)
19 \( 1 + (-0.589 - 0.807i)T \)
23 \( 1 + (-0.330 - 0.943i)T \)
29 \( 1 + (-0.202 + 0.979i)T \)
31 \( 1 + (-0.967 + 0.254i)T \)
37 \( 1 + (-0.245 + 0.969i)T \)
41 \( 1 + (-0.993 + 0.115i)T \)
43 \( 1 + (-0.842 - 0.537i)T \)
47 \( 1 + (-0.185 + 0.982i)T \)
53 \( 1 + (-0.263 + 0.964i)T \)
59 \( 1 + (0.994 - 0.106i)T \)
61 \( 1 + (0.429 + 0.903i)T \)
67 \( 1 + (0.603 + 0.797i)T \)
71 \( 1 + (0.838 - 0.545i)T \)
73 \( 1 + (-0.364 + 0.931i)T \)
79 \( 1 + (-0.610 + 0.792i)T \)
83 \( 1 + (0.999 + 0.0266i)T \)
89 \( 1 + (0.691 + 0.722i)T \)
97 \( 1 + (0.857 - 0.515i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.92047616883179226248292843617, −21.53239824267459170274034536387, −20.40402593209752449644414627214, −19.73713791653049361378348875922, −18.80141240896201607985360181704, −17.80479876987309044339949007420, −17.28283088174637350092974056870, −16.45849308080564204889649877380, −15.02151337775454261451473045572, −14.36803412940035692713687985240, −13.5549404310687638407293234131, −12.97012434719028386326827474323, −12.03012150555153135215305207621, −11.321581163679518016078721097369, −10.56550033375630244886635620795, −9.67122169766581897341826179682, −8.319349151274368176894195918828, −7.1618795068156170966165862275, −6.31207920460010087790761575853, −5.54996475101158869251856769082, −4.86083232430691395141429158310, −3.694041138833138598387530860989, −1.96593502518237070856245375183, −1.87423914691860403636424342613, −0.52155237452117562441110715064, 1.64961442497501255983674155101, 2.60959996826432191466125805626, 4.0692214105815686009375050824, 4.79424906910875450020248362337, 5.34682440181932043528991087168, 6.58987337142155632902082139339, 6.85920075049322283390781525420, 8.63095915728824688675250457234, 9.21684293037466981417300305509, 10.46782196104753186439113456424, 11.28588396736453412631280482991, 11.96775522172174894851091440532, 12.87834464788900442128693895445, 14.03632933562545291474363676866, 14.63274866360322625331298514533, 15.16979247126674268949592289783, 16.32589461531748676893312274301, 17.10138480810611303833180245283, 17.479686814464996726369364028458, 18.35737827476143542174822785083, 20.08394303506236059092942112333, 20.63257234329710721395045971872, 21.679264236346037241117714325675, 21.94015412682525619406034558261, 22.447576329012268705838745046288

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