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Properties

Label 1-163-163.144-r0-0-0
Degree $1$
Conductor $163$
Sign $0.728 + 0.685i$
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  + (−0.135 + 0.990i)2-s + (−0.910 + 0.413i)3-s + (−0.963 − 0.268i)4-s + (0.893 + 0.448i)5-s + (−0.286 − 0.957i)6-s + (0.466 − 0.884i)7-s + (0.396 − 0.918i)8-s + (0.657 − 0.753i)9-s + (−0.565 + 0.824i)10-s + (0.0968 − 0.995i)11-s + (0.987 − 0.154i)12-s + (0.597 − 0.802i)13-s + (0.813 + 0.581i)14-s + (−0.999 − 0.0387i)15-s + (0.856 + 0.516i)16-s + (−0.993 + 0.116i)17-s + ⋯
L(s)  = 1  + (−0.135 + 0.990i)2-s + (−0.910 + 0.413i)3-s + (−0.963 − 0.268i)4-s + (0.893 + 0.448i)5-s + (−0.286 − 0.957i)6-s + (0.466 − 0.884i)7-s + (0.396 − 0.918i)8-s + (0.657 − 0.753i)9-s + (−0.565 + 0.824i)10-s + (0.0968 − 0.995i)11-s + (0.987 − 0.154i)12-s + (0.597 − 0.802i)13-s + (0.813 + 0.581i)14-s + (−0.999 − 0.0387i)15-s + (0.856 + 0.516i)16-s + (−0.993 + 0.116i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.728 + 0.685i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.728 + 0.685i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8082837458 + 0.3204022327i\)
\(L(\frac12)\) \(\approx\) \(0.8082837458 + 0.3204022327i\)
\(L(1)\) \(\approx\) \(0.7887003828 + 0.3272963917i\)
\(L(1)\) \(\approx\) \(0.7887003828 + 0.3272963917i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (-0.135 + 0.990i)T \)
3 \( 1 + (-0.910 + 0.413i)T \)
5 \( 1 + (0.893 + 0.448i)T \)
7 \( 1 + (0.466 - 0.884i)T \)
11 \( 1 + (0.0968 - 0.995i)T \)
13 \( 1 + (0.597 - 0.802i)T \)
17 \( 1 + (-0.993 + 0.116i)T \)
19 \( 1 + (0.952 - 0.305i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (0.925 + 0.378i)T \)
31 \( 1 + (-0.0581 - 0.998i)T \)
37 \( 1 + (0.973 - 0.230i)T \)
41 \( 1 + (-0.963 + 0.268i)T \)
43 \( 1 + (-0.431 + 0.902i)T \)
47 \( 1 + (-0.790 - 0.612i)T \)
53 \( 1 + (0.766 + 0.642i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.396 + 0.918i)T \)
67 \( 1 + (0.249 - 0.968i)T \)
71 \( 1 + (0.996 - 0.0774i)T \)
73 \( 1 + (0.249 + 0.968i)T \)
79 \( 1 + (0.323 - 0.946i)T \)
83 \( 1 + (0.533 + 0.845i)T \)
89 \( 1 + (0.0968 + 0.995i)T \)
97 \( 1 + (-0.981 + 0.192i)T \)
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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

−28.145274116366870561655226434724, −27.0323649577054259339081257001, −25.63936174828848030657245017324, −24.6469362117966619069619727649, −23.62055953215345437368647762296, −22.43047958763307727628203193998, −21.78382338816190941885501351636, −20.920013734960103990313909054004, −19.857989596646597976484407276563, −18.342255105517612709137894024463, −18.08276120503547363154529567688, −17.16946127937986757662186269163, −15.94558717387936988911640418031, −14.18512677050109884506108815979, −13.25045363539430558660647548058, −12.20461603955971297067750819142, −11.64458961272557771752965775008, −10.36022610338894331361145053083, −9.39298920985659757235963561332, −8.288233163703270039597112418488, −6.54925367965270523082229016196, −5.29653551315719354822658293250, −4.469917897241258505279142828677, −2.24036648260127347973894224002, −1.491209122989141851164589815100, 1.01361877483844570186707639413, 3.64973241427246420250421734899, 4.97831968049780381106496074408, 5.96218368487880957271272364573, 6.73856224362096007611038509338, 8.09422580257330187474890464149, 9.51507795416294227942048996405, 10.43998637257763571945488228521, 11.29889300732808052497404205805, 13.22613490297036174128961860790, 13.83668605687425819153031325587, 15.059497218076611452289997810258, 16.124479386172609764740154635903, 16.95095147184626576241544369440, 17.90196255898406412706034121980, 18.25600935218279104713020667754, 20.05046927098548186769434896127, 21.487614284370710851861965553104, 22.156830990541673077112413668196, 23.07134732544432102798472909819, 24.01143950168599158580435475910, 24.80713454210245215960656816056, 26.18287304165034183867385295938, 26.71908485071507345591163450450, 27.610267703057428637533039577514

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