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Properties

Label 1-163-163.118-r0-0-0
Degree $1$
Conductor $163$
Sign $-0.971 + 0.236i$
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  + (0.533 − 0.845i)2-s + (0.249 − 0.968i)3-s + (−0.431 − 0.902i)4-s + (−0.835 − 0.549i)5-s + (−0.686 − 0.727i)6-s + (−0.211 + 0.977i)7-s + (−0.993 − 0.116i)8-s + (−0.875 − 0.483i)9-s + (−0.910 + 0.413i)10-s + (−0.963 + 0.268i)11-s + (−0.981 + 0.192i)12-s + (0.396 − 0.918i)13-s + (0.713 + 0.700i)14-s + (−0.740 + 0.672i)15-s + (−0.627 + 0.778i)16-s + (0.597 − 0.802i)17-s + ⋯
L(s)  = 1  + (0.533 − 0.845i)2-s + (0.249 − 0.968i)3-s + (−0.431 − 0.902i)4-s + (−0.835 − 0.549i)5-s + (−0.686 − 0.727i)6-s + (−0.211 + 0.977i)7-s + (−0.993 − 0.116i)8-s + (−0.875 − 0.483i)9-s + (−0.910 + 0.413i)10-s + (−0.963 + 0.268i)11-s + (−0.981 + 0.192i)12-s + (0.396 − 0.918i)13-s + (0.713 + 0.700i)14-s + (−0.740 + 0.672i)15-s + (−0.627 + 0.778i)16-s + (0.597 − 0.802i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $-0.971 + 0.236i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ -0.971 + 0.236i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1195004672 - 0.9956401192i\)
\(L(\frac12)\) \(\approx\) \(-0.1195004672 - 0.9956401192i\)
\(L(1)\) \(\approx\) \(0.5804954616 - 0.8752128264i\)
\(L(1)\) \(\approx\) \(0.5804954616 - 0.8752128264i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (0.533 - 0.845i)T \)
3 \( 1 + (0.249 - 0.968i)T \)
5 \( 1 + (-0.835 - 0.549i)T \)
7 \( 1 + (-0.211 + 0.977i)T \)
11 \( 1 + (-0.963 + 0.268i)T \)
13 \( 1 + (0.396 - 0.918i)T \)
17 \( 1 + (0.597 - 0.802i)T \)
19 \( 1 + (0.925 - 0.378i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.466 - 0.884i)T \)
31 \( 1 + (0.893 - 0.448i)T \)
37 \( 1 + (-0.286 - 0.957i)T \)
41 \( 1 + (-0.431 + 0.902i)T \)
43 \( 1 + (0.813 - 0.581i)T \)
47 \( 1 + (-0.999 - 0.0387i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.993 + 0.116i)T \)
67 \( 1 + (0.996 - 0.0774i)T \)
71 \( 1 + (0.0968 + 0.995i)T \)
73 \( 1 + (0.996 + 0.0774i)T \)
79 \( 1 + (0.0193 - 0.999i)T \)
83 \( 1 + (0.952 - 0.305i)T \)
89 \( 1 + (-0.963 - 0.268i)T \)
97 \( 1 + (0.856 + 0.516i)T \)
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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

−27.84864276248605742291209976089, −26.901197921188001770664999647, −26.20969545786136720966670272313, −25.83164415855569815075952831118, −24.06820638189118509443216642195, −23.391075150918061903103265401371, −22.6198710971252668178007817509, −21.57645637791083453092357818219, −20.70948949359259077311434495666, −19.56225451930198592149069403542, −18.30001238438761111718730530834, −16.89062695897544833245276487401, −16.085513799362749115028852347901, −15.51725890250877325758883391431, −14.25831734341506294879209246992, −13.77158513282050441429944264928, −12.1565295807803345293067719124, −10.943253590624215782264305897220, −9.91703696248434885840985298043, −8.37007321728195941706432295910, −7.63146072797771077458813422483, −6.34221105963265758910972821727, −4.92120397477294838866517367079, −3.84316400328451454070996167587, −3.183383667175900018312644342864, 0.68709607559957574594692858511, 2.394321635871355252628411345259, 3.30237033678120388022708306810, 5.01551530346581467070900233791, 5.95167569528457842977999994245, 7.673771822750461769650187430411, 8.59890137125509495065298302078, 9.85572606379561378760727384461, 11.4578813891110601300206288406, 12.15361239355930865323993710931, 12.877685087047319762713885699050, 13.80493372527725253812514023509, 15.16528051580199755818452347637, 15.910428504721496985322257720680, 17.93588266939221968228018140810, 18.51198405860134102040532700279, 19.482412834660168829068199960516, 20.30999189685765009747085510514, 21.081632812750213718841921761219, 22.68425168947358516549291442777, 23.11920394573799099772184840577, 24.3004015420403722447004613490, 24.849797649468527527590545562402, 26.234017881027445759316193741878, 27.66562751658851153794858325174

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