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Properties

Label 1-129-129.17-r1-0-0
Degree $1$
Conductor $129$
Sign $0.231 - 0.972i$
Analytic cond. $13.8629$
Root an. cond. $13.8629$
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  + (0.900 − 0.433i)2-s + (0.623 − 0.781i)4-s + (0.733 + 0.680i)5-s + (−0.5 − 0.866i)7-s + (0.222 − 0.974i)8-s + (0.955 + 0.294i)10-s + (−0.623 − 0.781i)11-s + (0.955 − 0.294i)13-s + (−0.826 − 0.563i)14-s + (−0.222 − 0.974i)16-s + (0.733 − 0.680i)17-s + (0.365 + 0.930i)19-s + (0.988 − 0.149i)20-s + (−0.900 − 0.433i)22-s + (0.988 − 0.149i)23-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)2-s + (0.623 − 0.781i)4-s + (0.733 + 0.680i)5-s + (−0.5 − 0.866i)7-s + (0.222 − 0.974i)8-s + (0.955 + 0.294i)10-s + (−0.623 − 0.781i)11-s + (0.955 − 0.294i)13-s + (−0.826 − 0.563i)14-s + (−0.222 − 0.974i)16-s + (0.733 − 0.680i)17-s + (0.365 + 0.930i)19-s + (0.988 − 0.149i)20-s + (−0.900 − 0.433i)22-s + (0.988 − 0.149i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $0.231 - 0.972i$
Analytic conductor: \(13.8629\)
Root analytic conductor: \(13.8629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 129,\ (1:\ ),\ 0.231 - 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.590181757 - 2.047037316i\)
\(L(\frac12)\) \(\approx\) \(2.590181757 - 2.047037316i\)
\(L(1)\) \(\approx\) \(1.834611264 - 0.7746185789i\)
\(L(1)\) \(\approx\) \(1.834611264 - 0.7746185789i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.900 - 0.433i)T \)
5 \( 1 + (0.733 + 0.680i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.623 - 0.781i)T \)
13 \( 1 + (0.955 - 0.294i)T \)
17 \( 1 + (0.733 - 0.680i)T \)
19 \( 1 + (0.365 + 0.930i)T \)
23 \( 1 + (0.988 - 0.149i)T \)
29 \( 1 + (-0.826 - 0.563i)T \)
31 \( 1 + (0.0747 - 0.997i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (-0.955 - 0.294i)T \)
59 \( 1 + (0.222 + 0.974i)T \)
61 \( 1 + (0.0747 + 0.997i)T \)
67 \( 1 + (0.365 + 0.930i)T \)
71 \( 1 + (0.988 + 0.149i)T \)
73 \( 1 + (0.955 - 0.294i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.826 + 0.563i)T \)
89 \( 1 + (-0.826 + 0.563i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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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

−28.664328733780679043579520628204, −28.1563093570716234672006701461, −26.14758085592555296133829263979, −25.5717529285981831361806384638, −24.76342612440724685982514157450, −23.68644710052116547152146524900, −22.78370372232270697863397021147, −21.539500632466258312225486105495, −21.083884903642603171840880867885, −19.90143680623880492762102613620, −18.337487121310354697276395468659, −17.26284219327431275430638507355, −16.15027129818613733413606115349, −15.384884334428472228660747648882, −14.149526695414848278365191125786, −12.92406886923018822910894573584, −12.56268843821936086832425042182, −11.05447617787805106501427060230, −9.455989414650965276640095468466, −8.39179888890778104802409922157, −6.86165441322577907038313733994, −5.70672070423607980146511566235, −4.91462144994936071181839141715, −3.25947400406633281090682271186, −1.86740269703603704866022150774, 1.07058815499734581014054799415, 2.82965822168164781872485324965, 3.70095853193405065995134983917, 5.45641748764346545412174625444, 6.31324882481063005475567302015, 7.571654454838169375749653337719, 9.63915650242089704109857200016, 10.531184014882864805997633602051, 11.35002343311255459280369002022, 13.00454415202675758219255817618, 13.620686604950288123710728045154, 14.484253064246364860603105777955, 15.805767881786962849690088788427, 16.818260630685121681100574884379, 18.46403658255806021090516344529, 19.106872294591624149301022356958, 20.73153476537070928442365421656, 20.995251770767980846520936714217, 22.54599609870286072901094898529, 22.89174176819524495395286163874, 24.08406591501154404880999280652, 25.2684672206636034148112842897, 26.13586493179979246394271183540, 27.342861055658934545790245250300, 28.79918205554504676176556746648

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