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Properties

Label 1-139-139.81-r0-0-0
Degree $1$
Conductor $139$
Sign $-0.632 + 0.774i$
Analytic cond. $0.645513$
Root an. cond. $0.645513$
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  + (0.377 + 0.926i)2-s + (−0.158 − 0.987i)3-s + (−0.715 + 0.699i)4-s + (0.291 + 0.956i)5-s + (0.854 − 0.519i)6-s + (−0.877 + 0.480i)7-s + (−0.917 − 0.398i)8-s + (−0.949 + 0.313i)9-s + (−0.775 + 0.631i)10-s + (−0.419 + 0.907i)11-s + (0.803 + 0.595i)12-s + (0.934 + 0.356i)13-s + (−0.775 − 0.631i)14-s + (0.898 − 0.439i)15-s + (0.0227 − 0.999i)16-s + (0.538 + 0.842i)17-s + ⋯
L(s)  = 1  + (0.377 + 0.926i)2-s + (−0.158 − 0.987i)3-s + (−0.715 + 0.699i)4-s + (0.291 + 0.956i)5-s + (0.854 − 0.519i)6-s + (−0.877 + 0.480i)7-s + (−0.917 − 0.398i)8-s + (−0.949 + 0.313i)9-s + (−0.775 + 0.631i)10-s + (−0.419 + 0.907i)11-s + (0.803 + 0.595i)12-s + (0.934 + 0.356i)13-s + (−0.775 − 0.631i)14-s + (0.898 − 0.439i)15-s + (0.0227 − 0.999i)16-s + (0.538 + 0.842i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(139\)
Sign: $-0.632 + 0.774i$
Analytic conductor: \(0.645513\)
Root analytic conductor: \(0.645513\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{139} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 139,\ (0:\ ),\ -0.632 + 0.774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3912070154 + 0.8247909310i\)
\(L(\frac12)\) \(\approx\) \(0.3912070154 + 0.8247909310i\)
\(L(1)\) \(\approx\) \(0.7951803554 + 0.5532898590i\)
\(L(1)\) \(\approx\) \(0.7951803554 + 0.5532898590i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad139 \( 1 \)
good2 \( 1 + (0.377 + 0.926i)T \)
3 \( 1 + (-0.158 - 0.987i)T \)
5 \( 1 + (0.291 + 0.956i)T \)
7 \( 1 + (-0.877 + 0.480i)T \)
11 \( 1 + (-0.419 + 0.907i)T \)
13 \( 1 + (0.934 + 0.356i)T \)
17 \( 1 + (0.538 + 0.842i)T \)
19 \( 1 + (-0.998 + 0.0455i)T \)
23 \( 1 + (0.854 + 0.519i)T \)
29 \( 1 + (-0.247 - 0.968i)T \)
31 \( 1 + (-0.949 - 0.313i)T \)
37 \( 1 + (0.898 + 0.439i)T \)
41 \( 1 + (0.983 + 0.181i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.974 + 0.225i)T \)
53 \( 1 + (0.613 - 0.789i)T \)
59 \( 1 + (0.962 - 0.269i)T \)
61 \( 1 + (0.983 - 0.181i)T \)
67 \( 1 + (0.995 + 0.0909i)T \)
71 \( 1 + (-0.974 - 0.225i)T \)
73 \( 1 + (0.113 + 0.993i)T \)
79 \( 1 + (-0.334 + 0.942i)T \)
83 \( 1 + (0.995 - 0.0909i)T \)
89 \( 1 + (-0.829 - 0.557i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−28.1302697265018388550916745346, −27.42634446552828448243578250913, −26.35469845806549446733693636866, −25.19103609267531715556336322575, −23.59995702911147200069819413714, −23.01845881807751488826704757208, −21.83522312035781791711143947529, −21.02890030640426425368360496064, −20.37672108675055173639683694640, −19.39546662226832320840482134622, −18.07459046974979755411628977415, −16.6236917514389198221808106194, −16.07471120381395883742664452449, −14.59478825271289392487402838979, −13.392705050753362834150021671738, −12.70182388984854973703059684525, −11.22241696886359231554330609421, −10.43468886768791420634805697943, −9.35034773101149942050625930162, −8.5839655645758219939173383158, −6.06301795602881904337491914769, −5.15328564161231030786773435974, −3.949803853308057942425061360527, −2.95095680114805054956555999973, −0.740241002521091117761921140, 2.2967904804468263246928200569, 3.653782195558928716385389987850, 5.64299805143313537520931306471, 6.40055333222331634701385696587, 7.21290900609787961527351824687, 8.4034175569457740094610024278, 9.782256435819880850773151523093, 11.37732999189266124387913867983, 12.79743753143060912593835040021, 13.228887324766669918218732320167, 14.56172465140246222421836321249, 15.32625254808476001495168993927, 16.69491767813355792825147046828, 17.69123312355918901097029899044, 18.58196915710042786637366526997, 19.23582549037908424319581552157, 21.12157388161914300639324781454, 22.22063933487145146993071429849, 23.1508740126251144546546283466, 23.54340692211177006845228999053, 25.07681398132079302477122135875, 25.69879685535522084595759419921, 26.12788542994830219214503157552, 27.82108430917335067983077667590, 28.885938868762458499018944154831

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