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Properties

Label 1-133-133.24-r1-0-0
Degree $1$
Conductor $133$
Sign $0.182 + 0.983i$
Analytic cond. $14.2928$
Root an. cond. $14.2928$
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  + (0.173 + 0.984i)2-s + (−0.173 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)5-s + (0.939 − 0.342i)6-s + (−0.5 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.173 + 0.984i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (0.939 − 0.342i)13-s + (0.173 − 0.984i)15-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + (−0.5 − 0.866i)18-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (−0.173 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)5-s + (0.939 − 0.342i)6-s + (−0.5 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.173 + 0.984i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (0.939 − 0.342i)13-s + (0.173 − 0.984i)15-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + (−0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.182 + 0.983i$
Analytic conductor: \(14.2928\)
Root analytic conductor: \(14.2928\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (1:\ ),\ 0.182 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.406588966 + 1.169749106i\)
\(L(\frac12)\) \(\approx\) \(1.406588966 + 1.169749106i\)
\(L(1)\) \(\approx\) \(1.111238801 + 0.4620576370i\)
\(L(1)\) \(\approx\) \(1.111238801 + 0.4620576370i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.173 + 0.984i)T \)
3 \( 1 + (-0.173 - 0.984i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.939 - 0.342i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (0.766 + 0.642i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (-0.939 + 0.342i)T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (-0.173 - 0.984i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.173 + 0.984i)T \)
97 \( 1 + (-0.766 + 0.642i)T \)
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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

−28.40576855598352201237164365073, −27.34786878694372035213476486798, −26.44209070242743384419534208458, −25.41233572391101025986274084708, −23.84352104804741565551893282243, −22.83524104074113518132856495088, −21.81658540526803815165258131510, −20.98402742517627563925250785424, −20.68534532327383296653688724138, −19.13075317466549158066921783358, −18.08198845462624298659601893308, −16.96323009176365755647202137696, −15.97148743588705245481229045349, −14.39906405207714979250742539194, −13.69071567586417636920201358751, −12.4637233080103799588079668754, −11.116297928493708326674086228366, −10.41427543680361754618812847156, −9.316379545943084317702787932546, −8.55038424204779162545849843126, −5.93729962801974786030643104867, −5.1840883134748989989872747819, −3.83502965342819376620077621183, −2.61462348888341501987109868123, −0.81787208418998526845972343377, 1.33488543592467418841061045530, 3.098800577525847018001227606526, 5.16038199377228479945348774193, 6.03580648120954728861271071694, 7.03604096910581422284687534892, 8.02959102850192081373726702225, 9.32550290170175049696302091965, 10.68563952724201359476220201736, 12.45009145525283392851056972497, 13.20068145494794971560783988221, 14.09372148431234963933508578861, 15.07975303308047963121326571961, 16.49985255750093791505908729491, 17.56441396932794599086867043243, 18.10564072570723427811165486521, 19.0136153914974425072857347362, 20.689148488606076733376460949629, 21.861573447311575052555622107528, 23.04478361625535323865953225791, 23.50043692371146919014490487374, 24.79280569279168390755776521329, 25.59393667838030982217661645775, 25.937059578601203075881432251546, 27.60045766080581984265705261958, 28.6053999989082232822300240355

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