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Properties

Label 1-640-640.587-r0-0-0
Degree $1$
Conductor $640$
Sign $-0.918 - 0.395i$
Analytic cond. $2.97214$
Root an. cond. $2.97214$
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  + (−0.195 − 0.980i)3-s + (−0.382 + 0.923i)7-s + (−0.923 + 0.382i)9-s + (0.195 − 0.980i)11-s + (0.831 + 0.555i)13-s + (−0.707 − 0.707i)17-s + (−0.831 − 0.555i)19-s + (0.980 + 0.195i)21-s + (−0.923 + 0.382i)23-s + (0.555 + 0.831i)27-s + (−0.195 − 0.980i)29-s − i·31-s − 33-s + (0.555 + 0.831i)37-s + (0.382 − 0.923i)39-s + ⋯
L(s)  = 1  + (−0.195 − 0.980i)3-s + (−0.382 + 0.923i)7-s + (−0.923 + 0.382i)9-s + (0.195 − 0.980i)11-s + (0.831 + 0.555i)13-s + (−0.707 − 0.707i)17-s + (−0.831 − 0.555i)19-s + (0.980 + 0.195i)21-s + (−0.923 + 0.382i)23-s + (0.555 + 0.831i)27-s + (−0.195 − 0.980i)29-s − i·31-s − 33-s + (0.555 + 0.831i)37-s + (0.382 − 0.923i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.918 - 0.395i$
Analytic conductor: \(2.97214\)
Root analytic conductor: \(2.97214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 640,\ (0:\ ),\ -0.918 - 0.395i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1258505937 - 0.6109313215i\)
\(L(\frac12)\) \(\approx\) \(0.1258505937 - 0.6109313215i\)
\(L(1)\) \(\approx\) \(0.7210346330 - 0.3046117896i\)
\(L(1)\) \(\approx\) \(0.7210346330 - 0.3046117896i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.195 - 0.980i)T \)
7 \( 1 + (-0.382 + 0.923i)T \)
11 \( 1 + (0.195 - 0.980i)T \)
13 \( 1 + (0.831 + 0.555i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
19 \( 1 + (-0.831 - 0.555i)T \)
23 \( 1 + (-0.923 + 0.382i)T \)
29 \( 1 + (-0.195 - 0.980i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.555 + 0.831i)T \)
41 \( 1 + (-0.382 - 0.923i)T \)
43 \( 1 + (0.195 - 0.980i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (-0.980 - 0.195i)T \)
59 \( 1 + (-0.555 - 0.831i)T \)
61 \( 1 + (0.980 - 0.195i)T \)
67 \( 1 + (-0.195 - 0.980i)T \)
71 \( 1 + (-0.923 - 0.382i)T \)
73 \( 1 + (0.382 + 0.923i)T \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 + (-0.555 + 0.831i)T \)
89 \( 1 + (-0.382 + 0.923i)T \)
97 \( 1 - T \)
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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

−23.17408708843221861303511506716, −22.48920416963255583883230044099, −21.663544396440019602144309797238, −20.71669942783471860211298305634, −20.11933737991477779004097129256, −19.494673559225762036407085278045, −17.98058550698091161670793417666, −17.50413752070602266194548673615, −16.45595368062702632090554529221, −15.98602870687805996948264581062, −14.91075651666834785550664323623, −14.353033094246232331230777295722, −13.13107496748756347828045486012, −12.423964921176476805663451588792, −11.12381971421821426647888306016, −10.51235345194608648675011727806, −9.873902178866521859175468997804, −8.84665061752993889576940611527, −7.92705025186879948429132522224, −6.64930002890760301959084827831, −5.93869959729979055060971764144, −4.577184008573927228302406137933, −4.0505672391919358616485415453, −3.07044724995618576202746552752, −1.52409824920698533131337117929, 0.30835771518377138546298198069, 1.85024225682721278942679931851, 2.66134868801604980105538182445, 3.88514054665082699054605472917, 5.332220484893433971245483377, 6.21890366799529458221491749640, 6.684852573761296769616379736580, 8.06287157502889902744918766730, 8.686826425006042700021893372081, 9.56346531511070564916051248150, 11.13009369097811144536957437344, 11.52369996971514206411270370349, 12.441486182781358872557572627768, 13.48570619899232095130538406735, 13.78407651279100020912264644490, 15.12061365428310871141356810873, 15.95369760378384587225251226100, 16.823432927395358803593928459677, 17.73275191457518093222772871900, 18.68343358012327878555491619101, 18.94969614362827378822860184096, 19.88802235161002281865442965639, 20.92557175679069022825061629633, 21.97539376154135014593694564551, 22.44255233618657262789720477798

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