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Properties

Label 1-129-129.65-r0-0-0
Degree $1$
Conductor $129$
Sign $-0.932 + 0.361i$
Analytic cond. $0.599073$
Root an. cond. $0.599073$
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  + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)4-s + (−0.900 + 0.433i)5-s − 7-s + (−0.900 + 0.433i)8-s + (−0.900 − 0.433i)10-s + (0.222 + 0.974i)11-s + (−0.900 + 0.433i)13-s + (−0.623 − 0.781i)14-s + (−0.900 − 0.433i)16-s + (0.900 + 0.433i)17-s + (0.222 − 0.974i)19-s + (−0.222 − 0.974i)20-s + (−0.623 + 0.781i)22-s + (0.222 + 0.974i)23-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)4-s + (−0.900 + 0.433i)5-s − 7-s + (−0.900 + 0.433i)8-s + (−0.900 − 0.433i)10-s + (0.222 + 0.974i)11-s + (−0.900 + 0.433i)13-s + (−0.623 − 0.781i)14-s + (−0.900 − 0.433i)16-s + (0.900 + 0.433i)17-s + (0.222 − 0.974i)19-s + (−0.222 − 0.974i)20-s + (−0.623 + 0.781i)22-s + (0.222 + 0.974i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $-0.932 + 0.361i$
Analytic conductor: \(0.599073\)
Root analytic conductor: \(0.599073\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 129,\ (0:\ ),\ -0.932 + 0.361i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1660569663 + 0.8878841524i\)
\(L(\frac12)\) \(\approx\) \(0.1660569663 + 0.8878841524i\)
\(L(1)\) \(\approx\) \(0.7222301670 + 0.6836656598i\)
\(L(1)\) \(\approx\) \(0.7222301670 + 0.6836656598i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 - T \)
11 \( 1 + (0.222 + 0.974i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (0.900 + 0.433i)T \)
19 \( 1 + (0.222 - 0.974i)T \)
23 \( 1 + (0.222 + 0.974i)T \)
29 \( 1 + (0.623 + 0.781i)T \)
31 \( 1 + (0.623 + 0.781i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (0.222 - 0.974i)T \)
53 \( 1 + (0.900 + 0.433i)T \)
59 \( 1 + (0.900 + 0.433i)T \)
61 \( 1 + (-0.623 + 0.781i)T \)
67 \( 1 + (-0.222 + 0.974i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (0.900 - 0.433i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.623 + 0.781i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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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

−28.535268545461508335778292622363, −27.40906547791761837072849928684, −26.73372617876703750056935174101, −24.9597866863013223604610245971, −24.18483283135839014046954877510, −22.93961092047314440677694128748, −22.51468305644375777434813874466, −21.17937624134210699472480803398, −20.209438356788738467239219410728, −19.25136920870137174664397080105, −18.755069744947963108602922900314, −16.79774581199327142438526664414, −15.87573704519737403055544647879, −14.73418067444545853276622655081, −13.57118687414827740012594644773, −12.40878597568651186092718448370, −11.86641534755322560400424869695, −10.461510476640958244094265188571, −9.44780210663274233801934793390, −8.04637351575014603748452471849, −6.40486488692253600198699312383, −5.1406703956502234552387555913, −3.79045111738572660228401799757, −2.87575964152632790988973530779, −0.68233408589952739045036077033, 2.87518713075557114190723921095, 3.94207531410691438080073600317, 5.201197411623620239008464649777, 6.88304188195997463625018521819, 7.24414482710193832376831729846, 8.79033218428749440681729023338, 10.113659874515995815450274007897, 11.894020614399880042006727070508, 12.452884171565654308100691496812, 13.80699313577644972222955314895, 14.95305066117712554820625599415, 15.631916391430740287902963699863, 16.6713732540854892352674636206, 17.73428844500516239881037995802, 19.15760074296278527941584010685, 19.969769825927874009538794542656, 21.53498818842335217863518731283, 22.43469917583242676022728639807, 23.20132114678175608364434620248, 23.97442579182033209911726941082, 25.31005065658464758969519610843, 26.02413619465220198087288253705, 26.931558445460947860641045926503, 28.024838350429192620449426304563, 29.44861364882335544314183811117

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