Properties

Label 1-129-129.38-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: $\chi_{129} (38, \cdot )$
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

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

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