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Properties

Label 1-163-163.140-r0-0-0
Degree $1$
Conductor $163$
Sign $-0.392 + 0.919i$
Analytic cond. $0.756968$
Root an. cond. $0.756968$
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  + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)3-s + (0.766 + 0.642i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.939 − 0.342i)7-s + (−0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (0.766 − 0.642i)10-s + (0.766 + 0.642i)11-s + (0.173 + 0.984i)12-s + (−0.5 + 0.866i)13-s + (0.766 + 0.642i)14-s + (−0.939 + 0.342i)15-s + (0.173 + 0.984i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)3-s + (0.766 + 0.642i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.939 − 0.342i)7-s + (−0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (0.766 − 0.642i)10-s + (0.766 + 0.642i)11-s + (0.173 + 0.984i)12-s + (−0.5 + 0.866i)13-s + (0.766 + 0.642i)14-s + (−0.939 + 0.342i)15-s + (0.173 + 0.984i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $-0.392 + 0.919i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ -0.392 + 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3582272040 + 0.5422494801i\)
\(L(\frac12)\) \(\approx\) \(0.3582272040 + 0.5422494801i\)
\(L(1)\) \(\approx\) \(0.6493650095 + 0.2821969724i\)
\(L(1)\) \(\approx\) \(0.6493650095 + 0.2821969724i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 + (0.766 + 0.642i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.939 + 0.342i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (-0.939 - 0.342i)T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (-0.939 - 0.342i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−27.38565533561640693860374193424, −26.354405440546056118624561776028, −25.474625712539224573514456690, −24.6297660701211062934463215478, −24.12164417596378585347323988387, −22.86776503208035922464293646282, −21.224889327326173815200916559843, −20.02422683599301425193468390855, −19.47027641284879871193564986340, −18.91148935038970894430209123407, −17.47137086375056048006051237018, −16.677834188510753083062377192248, −15.46281112180336781724866153103, −14.81192521373885309018482485438, −13.146131052896768394707744823315, −12.4416504048980973117787623789, −11.09654862954664386410099558004, −9.47567990640543107310649574193, −8.859730548746169481425007517767, −7.977636830965427147744024293489, −6.82765371485404365024744838933, −5.75659990801188176362025550522, −3.69299860699619945069309693939, −2.2378056321984276433504338198, −0.64608406991196230071569875533, 2.16227955780593406484305586571, 3.274378905110734989240849365272, 4.2255324655649507143860064384, 6.78510372529813028300921262591, 7.287117387049403499375036730936, 8.833529988146474074055894322959, 9.591075841746675605854514412770, 10.50462253512816088002904489738, 11.50155812852065697675515224575, 12.81337206201535811864182141124, 14.30479545005077416178261703238, 15.22465633886750584819456014223, 16.20945385070653665991918820203, 17.05927681540653215162723134198, 18.54702276571631195170813825361, 19.41579899491755645295200241159, 19.83427432613134698993419585906, 20.99960703443137310885039086984, 22.07982637251422448081680146757, 22.862159569316867840972338925051, 24.59273875340096677798649268010, 25.71818411327574994772054046385, 26.10904772655811017837301465324, 27.13653352187156886822355502702, 27.59512644847496431114589807039

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