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Properties

Label 1-181-181.52-r0-0-0
Degree $1$
Conductor $181$
Sign $-0.444 - 0.895i$
Analytic cond. $0.840560$
Root an. cond. $0.840560$
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  + (0.882 − 0.469i)2-s + (−0.615 − 0.788i)3-s + (0.559 − 0.829i)4-s + (0.669 − 0.743i)5-s + (−0.913 − 0.406i)6-s + (0.5 − 0.866i)7-s + (0.104 − 0.994i)8-s + (−0.241 + 0.970i)9-s + (0.241 − 0.970i)10-s + (0.438 + 0.898i)11-s + (−0.997 + 0.0697i)12-s + (0.0348 + 0.999i)13-s + (0.0348 − 0.999i)14-s + (−0.997 − 0.0697i)15-s + (−0.374 − 0.927i)16-s + (−0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (0.882 − 0.469i)2-s + (−0.615 − 0.788i)3-s + (0.559 − 0.829i)4-s + (0.669 − 0.743i)5-s + (−0.913 − 0.406i)6-s + (0.5 − 0.866i)7-s + (0.104 − 0.994i)8-s + (−0.241 + 0.970i)9-s + (0.241 − 0.970i)10-s + (0.438 + 0.898i)11-s + (−0.997 + 0.0697i)12-s + (0.0348 + 0.999i)13-s + (0.0348 − 0.999i)14-s + (−0.997 − 0.0697i)15-s + (−0.374 − 0.927i)16-s + (−0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 181 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 181 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(181\)
Sign: $-0.444 - 0.895i$
Analytic conductor: \(0.840560\)
Root analytic conductor: \(0.840560\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{181} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 181,\ (0:\ ),\ -0.444 - 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9302824715 - 1.500053506i\)
\(L(\frac12)\) \(\approx\) \(0.9302824715 - 1.500053506i\)
\(L(1)\) \(\approx\) \(1.220483364 - 0.9939786443i\)
\(L(1)\) \(\approx\) \(1.220483364 - 0.9939786443i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad181 \( 1 \)
good2 \( 1 + (0.882 - 0.469i)T \)
3 \( 1 + (-0.615 - 0.788i)T \)
5 \( 1 + (0.669 - 0.743i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.438 + 0.898i)T \)
13 \( 1 + (0.0348 + 0.999i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 - T \)
23 \( 1 + (0.719 - 0.694i)T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.990 - 0.139i)T \)
41 \( 1 + (-0.961 - 0.275i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (0.997 - 0.0697i)T \)
53 \( 1 + (-0.961 + 0.275i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (0.913 + 0.406i)T \)
71 \( 1 + (0.978 - 0.207i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (0.961 - 0.275i)T \)
83 \( 1 + (-0.848 - 0.529i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (-0.438 + 0.898i)T \)
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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

−27.37563301901835045665213271519, −26.6944569489251539151185592384, −25.45246160501607164397221109726, −24.83209097310376145882994769703, −23.612741689022494290012216488050, −22.579025135656079809045953877238, −21.92638191951106576808652562747, −21.40271242288213199803015329298, −20.39714206102214965339991183703, −18.63250698487728281831455792993, −17.52996796636644703025684730196, −16.8853446723447090696561339521, −15.4127192576486350620596298132, −15.13099236453308685627649517354, −13.97402398347158861543566741826, −12.854963668832481749393775569198, −11.41428965775941885220486511895, −11.05802014937559637766114048665, −9.508265586067616985141617806037, −8.26299660031449930506882931136, −6.59338329478234530841151208017, −5.83403928329138899144984221000, −5.0121041444196548843544907110, −3.57878483291041707768045072405, −2.44732828501741545863620192501, 1.35551248726318785883597739309, 2.062569819481676149561509836420, 4.303023596184163751510172725542, 4.94317631793293088757017957665, 6.37182739999342935161784401926, 7.04037996232068271849705703677, 8.793015057092227651917890607078, 10.2994765787048326750911337998, 11.14968915856040140295640575515, 12.31820794090602992104083745362, 12.98401244739574382779997275412, 13.86910819912568413359662117920, 14.787435850173224638820080698553, 16.51135431413430555412613178700, 17.14501229945493178519164457942, 18.23416639228087954596089179177, 19.55678278136208776720689003305, 20.26883187213591086114886320538, 21.34262263048485070552270613702, 22.1789334601670059677474692463, 23.46494189091615218752299804051, 23.78398347712487714807049781117, 24.77723992551528956341609381057, 25.60364555078232926657761290799, 27.35946055195861267713867037475

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