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Properties

Label 1-532-532.423-r0-0-0
Degree $1$
Conductor $532$
Sign $0.507 + 0.861i$
Analytic cond. $2.47059$
Root an. cond. $2.47059$
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)3-s + (0.939 + 0.342i)5-s + (−0.939 + 0.342i)9-s + (0.5 − 0.866i)11-s + (0.939 − 0.342i)13-s + (−0.173 + 0.984i)15-s + (0.939 + 0.342i)17-s + (−0.766 − 0.642i)23-s + (0.766 + 0.642i)25-s + (−0.5 − 0.866i)27-s + (0.766 + 0.642i)29-s + 31-s + (0.939 + 0.342i)33-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)3-s + (0.939 + 0.342i)5-s + (−0.939 + 0.342i)9-s + (0.5 − 0.866i)11-s + (0.939 − 0.342i)13-s + (−0.173 + 0.984i)15-s + (0.939 + 0.342i)17-s + (−0.766 − 0.642i)23-s + (0.766 + 0.642i)25-s + (−0.5 − 0.866i)27-s + (0.766 + 0.642i)29-s + 31-s + (0.939 + 0.342i)33-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $0.507 + 0.861i$
Analytic conductor: \(2.47059\)
Root analytic conductor: \(2.47059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (0:\ ),\ 0.507 + 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.586647929 + 0.9067567899i\)
\(L(\frac12)\) \(\approx\) \(1.586647929 + 0.9067567899i\)
\(L(1)\) \(\approx\) \(1.300214232 + 0.4533471774i\)
\(L(1)\) \(\approx\) \(1.300214232 + 0.4533471774i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.173 + 0.984i)T \)
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

−23.2128806975419400707224143304, −22.823073601446613504244180531, −21.450780249732134322566519223502, −20.82772401661748917115008702292, −19.92874651341835448419516407289, −19.10803715920919779865846246001, −18.0910585591072029911574280972, −17.64340605403215181478997224244, −16.781809442308121317922620780932, −15.73504383361977291066175280663, −14.36941309165206687253179229595, −13.953133919956541788297603492293, −13.05300492677448290464835513790, −12.257384203864445052667189365929, −11.47964369177246159772064450241, −10.097981547370744040679573635138, −9.32008889132622625140632787933, −8.38397595907315471183046949781, −7.411957417410369758472430922847, −6.366549775411303324469321468064, −5.775751277339564386476534458343, −4.47612369149235465810939496492, −3.07827173149436260051314667274, −1.90157759723260153375816230253, −1.1837230841628511127920012343, 1.34815918910743153026187997797, 2.85427941364525203985863058684, 3.53674294765620319031022169176, 4.7729108963943283017745750868, 5.87351190581363626725992800108, 6.37633970258887641834490961099, 8.099282988579963394255720191169, 8.79837667597383404730586025303, 9.80546583307008751467448806046, 10.480773616551431228970184174179, 11.21330120678977828558137258829, 12.40301965954946586755837569494, 13.784029558192534388952015079025, 14.0716949461040619949146351343, 15.05976795120507851457308974813, 16.05105744374788171801405314180, 16.73882402032964044305563637325, 17.57638332651192545834652981195, 18.54885138552552213530586711978, 19.445661286543140875393449392733, 20.5530772186367873829518456564, 21.13619741033536676572316122674, 21.870577696294884663937560614, 22.50706990255534780161249491298, 23.43588096826652152835997082939

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