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Properties

Label 1-139-139.4-r0-0-0
Degree $1$
Conductor $139$
Sign $0.999 + 0.0200i$
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  + (0.995 + 0.0909i)2-s + (−0.829 − 0.557i)3-s + (0.983 + 0.181i)4-s + (0.0227 + 0.999i)5-s + (−0.775 − 0.631i)6-s + (−0.158 − 0.987i)7-s + (0.962 + 0.269i)8-s + (0.377 + 0.926i)9-s + (−0.0682 + 0.997i)10-s + (0.803 + 0.595i)11-s + (−0.715 − 0.699i)12-s + (0.898 − 0.439i)13-s + (−0.0682 − 0.997i)14-s + (0.538 − 0.842i)15-s + (0.934 + 0.356i)16-s + (−0.949 − 0.313i)17-s + ⋯
L(s)  = 1  + (0.995 + 0.0909i)2-s + (−0.829 − 0.557i)3-s + (0.983 + 0.181i)4-s + (0.0227 + 0.999i)5-s + (−0.775 − 0.631i)6-s + (−0.158 − 0.987i)7-s + (0.962 + 0.269i)8-s + (0.377 + 0.926i)9-s + (−0.0682 + 0.997i)10-s + (0.803 + 0.595i)11-s + (−0.715 − 0.699i)12-s + (0.898 − 0.439i)13-s + (−0.0682 − 0.997i)14-s + (0.538 − 0.842i)15-s + (0.934 + 0.356i)16-s + (−0.949 − 0.313i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(139\)
Sign: $0.999 + 0.0200i$
Analytic conductor: \(0.645513\)
Root analytic conductor: \(0.645513\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{139} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 139,\ (0:\ ),\ 0.999 + 0.0200i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.611694594 + 0.01615789162i\)
\(L(\frac12)\) \(\approx\) \(1.611694594 + 0.01615789162i\)
\(L(1)\) \(\approx\) \(1.509341077 + 0.001192726315i\)
\(L(1)\) \(\approx\) \(1.509341077 + 0.001192726315i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad139 \( 1 \)
good2 \( 1 + (0.995 + 0.0909i)T \)
3 \( 1 + (-0.829 - 0.557i)T \)
5 \( 1 + (0.0227 + 0.999i)T \)
7 \( 1 + (-0.158 - 0.987i)T \)
11 \( 1 + (0.803 + 0.595i)T \)
13 \( 1 + (0.898 - 0.439i)T \)
17 \( 1 + (-0.949 - 0.313i)T \)
19 \( 1 + (0.746 - 0.665i)T \)
23 \( 1 + (-0.775 + 0.631i)T \)
29 \( 1 + (-0.648 + 0.761i)T \)
31 \( 1 + (0.377 - 0.926i)T \)
37 \( 1 + (0.538 + 0.842i)T \)
41 \( 1 + (-0.974 + 0.225i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.877 + 0.480i)T \)
53 \( 1 + (-0.419 - 0.907i)T \)
59 \( 1 + (-0.334 + 0.942i)T \)
61 \( 1 + (-0.974 - 0.225i)T \)
67 \( 1 + (0.113 + 0.993i)T \)
71 \( 1 + (-0.877 - 0.480i)T \)
73 \( 1 + (-0.247 - 0.968i)T \)
79 \( 1 + (0.682 - 0.730i)T \)
83 \( 1 + (0.113 - 0.993i)T \)
89 \( 1 + (-0.998 - 0.0455i)T \)
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

−28.49940981501065410495226765124, −27.93433935664658460439003452300, −26.47289878631669698145764232392, −24.94326897439319667571921437067, −24.40224606013759219612616514546, −23.35065899947517376462327500914, −22.31894137916711726151461133133, −21.60380151504438383477412915789, −20.81569503181635214534533372870, −19.727668238178677877018999543295, −18.28729665288421675039027185477, −16.80870327373296711809021951348, −16.12287733554434732426323298085, −15.371591044565872650755708139467, −13.98921162151055250503046605828, −12.74800929115822107561912194113, −11.90311017226026446921453290643, −11.1967949615032539131723536667, −9.66379106051241042725748215886, −8.50926224118527485470347753717, −6.37495249571406172013999835546, −5.78431927890863159332830970915, −4.6109707153585832616563244587, −3.61491245717819180058964096398, −1.58611830349401960803575613571, 1.68224916598069191139195960554, 3.36208655972352319199695506771, 4.58823258401541791010091843836, 6.09183306554849806885544173259, 6.84638333362538291899729047113, 7.61794468251507147173911855068, 10.08951833608479591243786706043, 11.136706093613268852402054731, 11.734976859600649880744292692070, 13.276921412481192502567207903284, 13.72629557108301220607803072716, 15.072881440713524689725778988225, 16.137724270219059894249128820006, 17.28885069346363905008094852818, 18.16012712475416803668097563157, 19.59192699387945734685302270665, 20.41501233273755914670303386877, 22.14137614546905451291298998984, 22.40540515792823460234261931985, 23.3533016856645279772301302760, 24.103863823697197986918543649032, 25.314695765702182735575843622433, 26.18763330802079782469849608115, 27.56324266112143899537247093129, 28.8151497705632339738837073480

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