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Properties

Label 1-163-163.134-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: $\chi_{163} (134, \cdot )$
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

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

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