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Properties

Label 1-163-163.115-r0-0-0
Degree $1$
Conductor $163$
Sign $0.296 + 0.954i$
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  + (0.597 + 0.802i)2-s + (0.973 − 0.230i)3-s + (−0.286 + 0.957i)4-s + (0.173 + 0.984i)5-s + (0.766 + 0.642i)6-s + (0.396 − 0.918i)7-s + (−0.939 + 0.342i)8-s + (0.893 − 0.448i)9-s + (−0.686 + 0.727i)10-s + (0.973 − 0.230i)11-s + (−0.0581 + 0.998i)12-s + (−0.939 − 0.342i)13-s + (0.973 − 0.230i)14-s + (0.396 + 0.918i)15-s + (−0.835 − 0.549i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.597 + 0.802i)2-s + (0.973 − 0.230i)3-s + (−0.286 + 0.957i)4-s + (0.173 + 0.984i)5-s + (0.766 + 0.642i)6-s + (0.396 − 0.918i)7-s + (−0.939 + 0.342i)8-s + (0.893 − 0.448i)9-s + (−0.686 + 0.727i)10-s + (0.973 − 0.230i)11-s + (−0.0581 + 0.998i)12-s + (−0.939 − 0.342i)13-s + (0.973 − 0.230i)14-s + (0.396 + 0.918i)15-s + (−0.835 − 0.549i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.296 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.296 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.296 + 0.954i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.296 + 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.619704702 + 1.192565069i\)
\(L(\frac12)\) \(\approx\) \(1.619704702 + 1.192565069i\)
\(L(1)\) \(\approx\) \(1.599625543 + 0.8014879411i\)
\(L(1)\) \(\approx\) \(1.599625543 + 0.8014879411i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (0.597 + 0.802i)T \)
3 \( 1 + (0.973 - 0.230i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (0.396 - 0.918i)T \)
11 \( 1 + (0.973 - 0.230i)T \)
13 \( 1 + (-0.939 - 0.342i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (-0.993 - 0.116i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.597 - 0.802i)T \)
31 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (-0.286 - 0.957i)T \)
43 \( 1 + (-0.686 - 0.727i)T \)
47 \( 1 + (0.597 + 0.802i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (-0.286 - 0.957i)T \)
71 \( 1 + (-0.686 - 0.727i)T \)
73 \( 1 + (-0.286 + 0.957i)T \)
79 \( 1 + (0.893 + 0.448i)T \)
83 \( 1 + (0.396 - 0.918i)T \)
89 \( 1 + (0.973 + 0.230i)T \)
97 \( 1 + (0.893 + 0.448i)T \)
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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

−27.7031167256248702318918403089, −26.88746659691517312617004184471, −25.2179579779706652706766227344, −24.64704286537272756778318639377, −23.88308881627262393042241643919, −22.13504155076473661689724613931, −21.67843868529178687825425618271, −20.63780418513565228472894141231, −19.934379292209630894701702221080, −19.14451306225970085921234919545, −17.93345105369196302650874578286, −16.45464689731450163811055074866, −15.095006291045342969378759601, −14.55336017231226742562062863019, −13.3707097731812121719697287817, −12.46442333964047349554012054285, −11.60653987020775665660001610372, −9.98004574358029796405138999384, −9.13879352933554116730473038496, −8.41176304275110561880469713435, −6.43492606461368435600980237213, −4.8229132619773589864626968854, −4.275294663870996883831066122397, −2.55079026239543819247911250391, −1.714981219937836119612165842006, 2.21303016489752405196372015602, 3.54065487491819500932861701885, 4.418147402604122476584250473467, 6.32905262062204052511865725563, 7.1213493398562366617383439464, 7.98605618011146668152957159588, 9.23068628099675234077273997905, 10.57533994481905212742709438274, 12.03103738159125062254925491725, 13.36454606317621790559646956134, 14.08166449842109752573203174122, 14.75226419801690172906719060187, 15.58402306949048855186458120633, 17.206566981730740275861212177654, 17.752381848059978972757206807953, 19.234611721466691360303060443, 20.04455438139411907366877009539, 21.41108995974674262310689983837, 22.06408416437083849857409015835, 23.264246274265724753592351829983, 24.15062925049830547932854973533, 25.0807578841377879493425601399, 25.83861539461151886673146105890, 26.94422627392545706059997385632, 27.065060057724366295382747692026

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