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Properties

Label 1-129-129.68-r1-0-0
Degree $1$
Conductor $129$
Sign $-0.861 + 0.507i$
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  + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (−0.0747 + 0.997i)5-s + (−0.5 − 0.866i)7-s + (0.900 − 0.433i)8-s + (0.826 − 0.563i)10-s + (0.222 + 0.974i)11-s + (0.826 + 0.563i)13-s + (−0.365 + 0.930i)14-s + (−0.900 − 0.433i)16-s + (−0.0747 − 0.997i)17-s + (−0.733 − 0.680i)19-s + (−0.955 − 0.294i)20-s + (0.623 − 0.781i)22-s + (−0.955 − 0.294i)23-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (−0.0747 + 0.997i)5-s + (−0.5 − 0.866i)7-s + (0.900 − 0.433i)8-s + (0.826 − 0.563i)10-s + (0.222 + 0.974i)11-s + (0.826 + 0.563i)13-s + (−0.365 + 0.930i)14-s + (−0.900 − 0.433i)16-s + (−0.0747 − 0.997i)17-s + (−0.733 − 0.680i)19-s + (−0.955 − 0.294i)20-s + (0.623 − 0.781i)22-s + (−0.955 − 0.294i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $-0.861 + 0.507i$
Analytic conductor: \(13.8629\)
Root analytic conductor: \(13.8629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 129,\ (1:\ ),\ -0.861 + 0.507i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02580069162 + 0.09468498935i\)
\(L(\frac12)\) \(\approx\) \(0.02580069162 + 0.09468498935i\)
\(L(1)\) \(\approx\) \(0.5735550455 - 0.08386564089i\)
\(L(1)\) \(\approx\) \(0.5735550455 - 0.08386564089i\)

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 \( 1 + (-0.0747 + 0.997i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.222 + 0.974i)T \)
13 \( 1 + (0.826 + 0.563i)T \)
17 \( 1 + (-0.0747 - 0.997i)T \)
19 \( 1 + (-0.733 - 0.680i)T \)
23 \( 1 + (-0.955 - 0.294i)T \)
29 \( 1 + (-0.365 + 0.930i)T \)
31 \( 1 + (-0.988 + 0.149i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (0.222 - 0.974i)T \)
53 \( 1 + (-0.826 + 0.563i)T \)
59 \( 1 + (0.900 + 0.433i)T \)
61 \( 1 + (-0.988 - 0.149i)T \)
67 \( 1 + (-0.733 - 0.680i)T \)
71 \( 1 + (-0.955 + 0.294i)T \)
73 \( 1 + (0.826 + 0.563i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.365 - 0.930i)T \)
89 \( 1 + (-0.365 - 0.930i)T \)
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

−28.07041438232936259234855523324, −27.199797812654913788426987443152, −25.87945547405675448960720785106, −25.16507742381980480872092624420, −24.272122123266622821765535697628, −23.44921182106964555396242635591, −22.16078110818882209483783699432, −20.935367450286470594378488875626, −19.640124644285044694216921902439, −18.90785753699813115809487813195, −17.75401335661256642158928202101, −16.62143605146632677115031564439, −15.96079295712488401561489505688, −14.98321100134750864614089978419, −13.56843202115845139429847052041, −12.545745705927091523205130574125, −11.06983004377612757712498927239, −9.70383597518219374364331707857, −8.637878129242466773569256776373, −8.08472068615058780566828307457, −6.11573860310374822287054408153, −5.65965649541101956526236509513, −3.90633551891681787715363591433, −1.65584204353358037685217934076, −0.04777417957884826195214730216, 1.85761750297907544102756571630, 3.277889175766907110051282577244, 4.32167233865279347492154338981, 6.71292122615091379848432458996, 7.36910823162325756362204016998, 8.970923455783176652297190101609, 10.090010000117180258110606092957, 10.86440964615091951700001877926, 11.929013247006728091826283745170, 13.24421543377901726652550827606, 14.204586571465862589884383939966, 15.738343093036453327672328725515, 16.863356908879617423073752531999, 17.9688536221304102112009955502, 18.72357934723702770784454723592, 19.84496226271870745066252857765, 20.54335884786236743039214380631, 21.90346405594935379582943348202, 22.67583842549863688428980177459, 23.61541949244925769280894671588, 25.66299933480824099833365023440, 25.89961369309500106278939417471, 26.99632656886048366278279857026, 27.89135102421147788120537991553, 29.0154774762039128775146442623

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