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Properties

Label 1-588-588.383-r1-0-0
Degree $1$
Conductor $588$
Sign $0.945 + 0.325i$
Analytic cond. $63.1893$
Root an. cond. $63.1893$
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  + (−0.733 + 0.680i)5-s + (0.826 − 0.563i)11-s + (0.900 + 0.433i)13-s + (0.365 + 0.930i)17-s + (−0.5 − 0.866i)19-s + (0.365 − 0.930i)23-s + (0.0747 − 0.997i)25-s + (−0.623 + 0.781i)29-s + (−0.5 + 0.866i)31-s + (−0.988 − 0.149i)37-s + (−0.222 − 0.974i)41-s + (0.222 − 0.974i)43-s + (−0.0747 − 0.997i)47-s + (0.988 − 0.149i)53-s + (−0.222 + 0.974i)55-s + ⋯
L(s)  = 1  + (−0.733 + 0.680i)5-s + (0.826 − 0.563i)11-s + (0.900 + 0.433i)13-s + (0.365 + 0.930i)17-s + (−0.5 − 0.866i)19-s + (0.365 − 0.930i)23-s + (0.0747 − 0.997i)25-s + (−0.623 + 0.781i)29-s + (−0.5 + 0.866i)31-s + (−0.988 − 0.149i)37-s + (−0.222 − 0.974i)41-s + (0.222 − 0.974i)43-s + (−0.0747 − 0.997i)47-s + (0.988 − 0.149i)53-s + (−0.222 + 0.974i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.945 + 0.325i$
Analytic conductor: \(63.1893\)
Root analytic conductor: \(63.1893\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 588,\ (1:\ ),\ 0.945 + 0.325i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.818464686 + 0.3039737067i\)
\(L(\frac12)\) \(\approx\) \(1.818464686 + 0.3039737067i\)
\(L(1)\) \(\approx\) \(1.054928617 + 0.1001919774i\)
\(L(1)\) \(\approx\) \(1.054928617 + 0.1001919774i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.733 + 0.680i)T \)
11 \( 1 + (0.826 - 0.563i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
17 \( 1 + (0.365 + 0.930i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.365 - 0.930i)T \)
29 \( 1 + (-0.623 + 0.781i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.988 - 0.149i)T \)
41 \( 1 + (-0.222 - 0.974i)T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (-0.0747 - 0.997i)T \)
53 \( 1 + (0.988 - 0.149i)T \)
59 \( 1 + (0.733 + 0.680i)T \)
61 \( 1 + (0.988 + 0.149i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (-0.0747 + 0.997i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.900 - 0.433i)T \)
89 \( 1 + (0.826 + 0.563i)T \)
97 \( 1 - T \)
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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

−23.01673598040658171956627149137, −22.28769603342006174653612847479, −20.849791967312750669012174096725, −20.63868900305736765326313553518, −19.57776419708445791820996218585, −18.90802513333510217830562294545, −17.87592771470531798800297784519, −16.943543024778167345599023224506, −16.24466403772536528257807787440, −15.380711938161639204560363741974, −14.61982355702956789736484550753, −13.47063197885565044010301116178, −12.69967404153441002172970438076, −11.774222725150713537570142142109, −11.19294588999545538811029094496, −9.84856902674075276250863502860, −9.09990011438355640087848799884, −8.099180628749977852803517458205, −7.377947065232468702973161445996, −6.177559188049450833196436084230, −5.159905094074023567127792727186, −4.10425583865917031423308394831, −3.38221185204061937183294512578, −1.74912909931247839653377257792, −0.71672508270329126168107419769, 0.73289890666790787905038636258, 2.110384434414660678557190386327, 3.52534072684410986598295974600, 3.940819888830496491896835139609, 5.38195312253973682617739820254, 6.592149753811704856659660316739, 7.04676222982370221957274878572, 8.503640089047678423390881452950, 8.84407099935317165777555030975, 10.437844627861599523310108571695, 10.96048000653913065244591913447, 11.830663489764250275119947565632, 12.731098640589339052640101255156, 13.8656750039924478456291961302, 14.59806426455615893704016596718, 15.3777672083349864653659053027, 16.294891567102582532087409272881, 17.05817195525669451834777845009, 18.16065575721267179757742792888, 18.974441231937410790125616569411, 19.48315729283306033538411153659, 20.4430522426006169051273249746, 21.5160348121350848019938656077, 22.14748167622159978678181621056, 23.04216118062903318513187970153

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