Properties

Label 1-4463-4463.2621-r0-0-0
Degree $1$
Conductor $4463$
Sign $0.441 + 0.897i$
Analytic cond. $20.7260$
Root an. cond. $20.7260$
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  + (0.924 + 0.381i)2-s + (−0.742 − 0.669i)3-s + (0.708 + 0.705i)4-s + (−0.337 − 0.941i)5-s + (−0.431 − 0.902i)6-s + (−0.906 + 0.421i)7-s + (0.386 + 0.922i)8-s + (0.103 + 0.994i)9-s + (0.0471 − 0.998i)10-s + (0.988 + 0.154i)11-s + (−0.0541 − 0.998i)12-s + (0.494 − 0.869i)13-s + (−0.999 + 0.0436i)14-s + (−0.379 + 0.925i)15-s + (0.00492 + 0.999i)16-s + (−0.924 − 0.380i)17-s + ⋯
L(s)  = 1  + (0.924 + 0.381i)2-s + (−0.742 − 0.669i)3-s + (0.708 + 0.705i)4-s + (−0.337 − 0.941i)5-s + (−0.431 − 0.902i)6-s + (−0.906 + 0.421i)7-s + (0.386 + 0.922i)8-s + (0.103 + 0.994i)9-s + (0.0471 − 0.998i)10-s + (0.988 + 0.154i)11-s + (−0.0541 − 0.998i)12-s + (0.494 − 0.869i)13-s + (−0.999 + 0.0436i)14-s + (−0.379 + 0.925i)15-s + (0.00492 + 0.999i)16-s + (−0.924 − 0.380i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4463 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4463 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4463\)
Sign: $0.441 + 0.897i$
Analytic conductor: \(20.7260\)
Root analytic conductor: \(20.7260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4463} (2621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4463,\ (0:\ ),\ 0.441 + 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.379598423 + 0.8587770586i\)
\(L(\frac12)\) \(\approx\) \(1.379598423 + 0.8587770586i\)
\(L(1)\) \(\approx\) \(1.222437738 + 0.09160019568i\)
\(L(1)\) \(\approx\) \(1.222437738 + 0.09160019568i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4463 \( 1 \)
good2 \( 1 + (0.924 + 0.381i)T \)
3 \( 1 + (-0.742 - 0.669i)T \)
5 \( 1 + (-0.337 - 0.941i)T \)
7 \( 1 + (-0.906 + 0.421i)T \)
11 \( 1 + (0.988 + 0.154i)T \)
13 \( 1 + (0.494 - 0.869i)T \)
17 \( 1 + (-0.924 - 0.380i)T \)
19 \( 1 + (-0.914 + 0.403i)T \)
23 \( 1 + (0.935 - 0.352i)T \)
29 \( 1 + (0.812 + 0.582i)T \)
31 \( 1 + (-0.994 - 0.108i)T \)
37 \( 1 + (-0.938 + 0.346i)T \)
41 \( 1 + (0.582 + 0.812i)T \)
43 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (0.913 - 0.407i)T \)
53 \( 1 + (0.600 - 0.799i)T \)
59 \( 1 + (-0.997 + 0.0689i)T \)
61 \( 1 + (-0.466 - 0.884i)T \)
67 \( 1 + (-0.945 + 0.324i)T \)
71 \( 1 + (0.833 + 0.552i)T \)
73 \( 1 + (0.861 + 0.507i)T \)
79 \( 1 + (0.489 - 0.871i)T \)
83 \( 1 + (-0.991 - 0.133i)T \)
89 \( 1 + (-0.727 + 0.686i)T \)
97 \( 1 + (0.999 - 0.0309i)T \)
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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

−18.24965593517531292498358630187, −17.19774271778484432571796723141, −16.78339320811149294692236509820, −15.85635214417918850036048703751, −15.45454915155999221111129127286, −14.89292498521828480618797022384, −13.976913116954218207725410501338, −13.60650094150910367018856797023, −12.48988645430756965089519080808, −12.09880683334144308230800345511, −11.13105576776900603816174903418, −10.887057530647889253008850804856, −10.36216653136233813241538069348, −9.33214293422162691602008677197, −8.92531594389519761887069950553, −7.135077963508090245058618735204, −6.84519425707179940566797611824, −6.25041142905771408220471457788, −5.67867134509323243046050438413, −4.353002350415556371939780915760, −4.11753046361167094027469169434, −3.49666868323024900005804129400, −2.675721962069674275805664606531, −1.6198454703209067584760108904, −0.410244728436551039917354856612, 0.90289568392305546294040248438, 1.82869533491183766805054121657, 2.77501054966595366387135867844, 3.689611915847362711065059036209, 4.46638170234258644924102758251, 5.1513006129537471289599213209, 5.83764403926863469929590345721, 6.53203287172856015081047228538, 6.9310338328798169627907216373, 7.918948652744449649189545423468, 8.61388965058313414416167224797, 9.23716964015950343012510487503, 10.529328954038750369663623690287, 11.148503327113412870912446806254, 11.94935675420265821235733565914, 12.43124269684376784499384259315, 12.97334432623758730882125597547, 13.29568714817271242593848765133, 14.25706851692149532000090492635, 15.173268913186206311307438663207, 15.7533881906750573052955523939, 16.38430565001450293990032096053, 16.87485626138064846818025563158, 17.46451033856365577484874105110, 18.25536507318890960551483262848

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