Properties

Label 1-129-129.41-r1-0-0
Degree $1$
Conductor $129$
Sign $0.973 + 0.228i$
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 + (0.900 + 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

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

Invariants

Degree: \(1\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $0.973 + 0.228i$
Analytic conductor: \(13.8629\)
Root analytic conductor: \(13.8629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 129,\ (1:\ ),\ 0.973 + 0.228i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.616033921 + 0.1870955074i\)
\(L(\frac12)\) \(\approx\) \(1.616033921 + 0.1870955074i\)
\(L(1)\) \(\approx\) \(1.029832489 + 0.2145966697i\)
\(L(1)\) \(\approx\) \(1.029832489 + 0.2145966697i\)

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

−28.39854057327208709133529551859, −27.678810659341374318640709001175, −26.69944216517144342561127157837, −25.473104157477184787240538351171, −24.83802575258026099346815865483, −23.39727405779111087116532576661, −22.00634819871238810818785135235, −21.16669020427632491003007245325, −20.5438120932773623419855278870, −19.36631624287544643507970547938, −18.18603106378656041501029424119, −17.30054646400707892218595196889, −16.762620493836608961048052619617, −14.874129055292618565380612028194, −13.79127062567440634138224507184, −12.504649661241041710279832481475, −11.752288273477654745462009542796, −10.26630693216135368381327453414, −9.594590849636385585945928626711, −8.3596882740364917769287378748, −7.24742866042267803367210569153, −5.35226187507210662573679993730, −4.141039243887400920788461980234, −2.21975975589602503755316415133, −1.371038252499007130902923608265, 0.933531392474831685974139767, 2.534390413877834785973825224163, 4.86429270448090103888958000599, 5.81604128977806452578214800932, 7.04391765779169154505738551967, 8.18559632195301585423902954932, 9.32219269961366614717112212283, 10.41040001271665545046015405489, 11.39220476642680141167836077106, 13.30499763595741418001918928707, 14.40319500688996005414888611085, 14.907621425406411482415023656853, 16.51206322486857169244390502848, 17.286515860710764091362668460092, 18.182328917685134821754543807390, 19.01949837905657241030559981930, 20.373917779643710969910611664111, 21.58968058100168220351613316779, 22.55337321512655653756783176866, 23.96341877384156328622995857005, 24.62545097783314760913362035514, 25.51119263910882154719438839386, 26.62158872945276778057106639742, 27.27300176221597957485359818086, 28.35766846226878881205441588002

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