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Properties

Label 1-163-163.10-r0-0-0
Degree $1$
Conductor $163$
Sign $0.320 + 0.947i$
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  + (0.813 + 0.581i)2-s + (0.987 − 0.154i)3-s + (0.323 + 0.946i)4-s + (−0.993 + 0.116i)5-s + (0.893 + 0.448i)6-s + (0.249 + 0.968i)7-s + (−0.286 + 0.957i)8-s + (0.952 − 0.305i)9-s + (−0.875 − 0.483i)10-s + (−0.627 − 0.778i)11-s + (0.466 + 0.884i)12-s + (0.973 + 0.230i)13-s + (−0.360 + 0.932i)14-s + (−0.963 + 0.268i)15-s + (−0.790 + 0.612i)16-s + (−0.686 − 0.727i)17-s + ⋯
L(s)  = 1  + (0.813 + 0.581i)2-s + (0.987 − 0.154i)3-s + (0.323 + 0.946i)4-s + (−0.993 + 0.116i)5-s + (0.893 + 0.448i)6-s + (0.249 + 0.968i)7-s + (−0.286 + 0.957i)8-s + (0.952 − 0.305i)9-s + (−0.875 − 0.483i)10-s + (−0.627 − 0.778i)11-s + (0.466 + 0.884i)12-s + (0.973 + 0.230i)13-s + (−0.360 + 0.932i)14-s + (−0.963 + 0.268i)15-s + (−0.790 + 0.612i)16-s + (−0.686 − 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.320 + 0.947i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.320 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.657064334 + 1.188982080i\)
\(L(\frac12)\) \(\approx\) \(1.657064334 + 1.188982080i\)
\(L(1)\) \(\approx\) \(1.660666547 + 0.7546514565i\)
\(L(1)\) \(\approx\) \(1.660666547 + 0.7546514565i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (0.813 + 0.581i)T \)
3 \( 1 + (0.987 - 0.154i)T \)
5 \( 1 + (-0.993 + 0.116i)T \)
7 \( 1 + (0.249 + 0.968i)T \)
11 \( 1 + (-0.627 - 0.778i)T \)
13 \( 1 + (0.973 + 0.230i)T \)
17 \( 1 + (-0.686 - 0.727i)T \)
19 \( 1 + (-0.565 + 0.824i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (-0.910 - 0.413i)T \)
31 \( 1 + (0.396 - 0.918i)T \)
37 \( 1 + (-0.0581 + 0.998i)T \)
41 \( 1 + (0.323 - 0.946i)T \)
43 \( 1 + (0.0193 - 0.999i)T \)
47 \( 1 + (0.0968 - 0.995i)T \)
53 \( 1 + (0.173 + 0.984i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.286 - 0.957i)T \)
67 \( 1 + (-0.981 + 0.192i)T \)
71 \( 1 + (0.856 + 0.516i)T \)
73 \( 1 + (-0.981 - 0.192i)T \)
79 \( 1 + (-0.740 + 0.672i)T \)
83 \( 1 + (0.713 - 0.700i)T \)
89 \( 1 + (-0.627 + 0.778i)T \)
97 \( 1 + (-0.211 - 0.977i)T \)
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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

−27.65436755716210066484491559205, −26.594632667042162661206912130671, −25.65451539822285451358343560935, −24.322674230297318766148789160307, −23.56195074515208020349801414566, −22.87176510069072688453336999071, −21.385394254113158361747002439048, −20.644865163487097378631535396142, −19.85219713900470823562858291695, −19.30475449621951796691347960741, −17.94363127197436748777539506996, −16.102047359988522008847337389295, −15.30587577547375867760856882514, −14.57041303026561990635628187380, −13.238630383699708666879254884087, −12.84560907231847351898872005922, −11.13898654579482098669669230502, −10.55354901976152853199214810443, −9.10611157498460011167284121373, −7.81793495856323317510333407139, −6.82910168738239075640713032994, −4.780231088362939320260261276803, −4.04599701770390655889490714312, −3.04977015571685984813439544024, −1.48447287385163456590284907653, 2.38742342485597814079256272226, 3.417059331266548373134262229703, 4.48409125445556408033142821271, 5.97042073847483248828804730726, 7.24548647077303685229080162225, 8.33103805837217075321394824618, 8.79574599896063736771820958263, 10.98298045288289965874563021845, 12.00755107584405855472829979073, 13.0667351533251429492680835099, 13.96709475498019345175966577589, 15.183993815086652748765028775608, 15.507510051537552415359830020461, 16.57333643175077034299934895731, 18.46428175651768648013646249224, 18.89173105183024333358430994384, 20.48439763393742350488916944774, 20.99351489276634868596822534228, 22.164241107130131708652585852144, 23.252473566337377095599497924012, 24.21556439703914303137181003576, 24.81426127478754183993699697295, 25.89577725975096021027122813194, 26.662157194974476072650148807686, 27.61431631005072731123308632025

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