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Properties

Label 1-129-129.110-r1-0-0
Degree $1$
Conductor $129$
Sign $-0.228 - 0.973i$
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  + (0.222 + 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.365 + 0.930i)5-s + (−0.5 + 0.866i)7-s + (−0.623 − 0.781i)8-s + (−0.988 − 0.149i)10-s + (0.900 + 0.433i)11-s + (−0.988 + 0.149i)13-s + (−0.955 − 0.294i)14-s + (0.623 − 0.781i)16-s + (−0.365 − 0.930i)17-s + (0.826 + 0.563i)19-s + (−0.0747 − 0.997i)20-s + (−0.222 + 0.974i)22-s + (−0.0747 − 0.997i)23-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.365 + 0.930i)5-s + (−0.5 + 0.866i)7-s + (−0.623 − 0.781i)8-s + (−0.988 − 0.149i)10-s + (0.900 + 0.433i)11-s + (−0.988 + 0.149i)13-s + (−0.955 − 0.294i)14-s + (0.623 − 0.781i)16-s + (−0.365 − 0.930i)17-s + (0.826 + 0.563i)19-s + (−0.0747 − 0.997i)20-s + (−0.222 + 0.974i)22-s + (−0.0747 − 0.997i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $-0.228 - 0.973i$
Analytic conductor: \(13.8629\)
Root analytic conductor: \(13.8629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 129,\ (1:\ ),\ -0.228 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3452663458 + 0.4355327127i\)
\(L(\frac12)\) \(\approx\) \(-0.3452663458 + 0.4355327127i\)
\(L(1)\) \(\approx\) \(0.5173294298 + 0.5952457389i\)
\(L(1)\) \(\approx\) \(0.5173294298 + 0.5952457389i\)

Euler product

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

−27.826827748048817228416661611697, −27.16306587746456815346513551058, −26.07029405469853666940585108673, −24.35608769787859636174317843626, −23.81494160308064811688236958, −22.526347101816138719720759628933, −21.79082901581474532068609955991, −20.423355153798966108660666266104, −19.82159071299293938362993201157, −19.14395382148724199195454247707, −17.43566370690844236468268609188, −16.77887226975215119836873599846, −15.264505957322003050762458539055, −13.91362507367754288636921487447, −13.046883568370014681687697474337, −12.08225970713739933418460096600, −11.06726088437442875528644733280, −9.74114518400380191167199470679, −8.9286932487007613884481226082, −7.43394430100661135412887098666, −5.63846826616940233184552024456, −4.333476467630513141762572035640, −3.43890294644465593964439116123, −1.510406336940686791968034172930, −0.21299695267678337585801197204, 2.68695432985338289534688264623, 4.02561282605457459393255763832, 5.48385423184470399496577950963, 6.71507681376741781235613270317, 7.43931340133276433681456588900, 8.97285450345643122868920588098, 9.88057017406331326109592037157, 11.71182187846422274690689005264, 12.547962179644811200606891788850, 14.12698365638871370923969401230, 14.78217960381908667634481435795, 15.73409570803069103753631414964, 16.74322043428634909576829304592, 18.061036140930905843131779925378, 18.72921604016884745512390180853, 19.92462513426997727385866772368, 21.72088058354105448505601638643, 22.451295379190240340068591365048, 22.959867205154869536471600641897, 24.53831124493996021179383917530, 25.033624552562854856096916287875, 26.22817836261754052372535315574, 26.94889528871850811461071813194, 27.910919278003314118942100391706, 29.29924525521451548795367942613

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