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Properties

Label 1-139-139.31-r0-0-0
Degree $1$
Conductor $139$
Sign $0.828 - 0.560i$
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  + (−0.829 + 0.557i)2-s + (−0.648 − 0.761i)3-s + (0.377 − 0.926i)4-s + (0.803 − 0.595i)5-s + (0.962 + 0.269i)6-s + (−0.247 + 0.968i)7-s + (0.203 + 0.979i)8-s + (−0.158 + 0.987i)9-s + (−0.334 + 0.942i)10-s + (0.538 − 0.842i)11-s + (−0.949 + 0.313i)12-s + (0.983 − 0.181i)13-s + (−0.334 − 0.942i)14-s + (−0.974 − 0.225i)15-s + (−0.715 − 0.699i)16-s + (−0.877 + 0.480i)17-s + ⋯
L(s)  = 1  + (−0.829 + 0.557i)2-s + (−0.648 − 0.761i)3-s + (0.377 − 0.926i)4-s + (0.803 − 0.595i)5-s + (0.962 + 0.269i)6-s + (−0.247 + 0.968i)7-s + (0.203 + 0.979i)8-s + (−0.158 + 0.987i)9-s + (−0.334 + 0.942i)10-s + (0.538 − 0.842i)11-s + (−0.949 + 0.313i)12-s + (0.983 − 0.181i)13-s + (−0.334 − 0.942i)14-s + (−0.974 − 0.225i)15-s + (−0.715 − 0.699i)16-s + (−0.877 + 0.480i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(139\)
Sign: $0.828 - 0.560i$
Analytic conductor: \(0.645513\)
Root analytic conductor: \(0.645513\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{139} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 139,\ (0:\ ),\ 0.828 - 0.560i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6564983689 - 0.2013011320i\)
\(L(\frac12)\) \(\approx\) \(0.6564983689 - 0.2013011320i\)
\(L(1)\) \(\approx\) \(0.6975047211 - 0.08117942817i\)
\(L(1)\) \(\approx\) \(0.6975047211 - 0.08117942817i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad139 \( 1 \)
good2 \( 1 + (-0.829 + 0.557i)T \)
3 \( 1 + (-0.648 - 0.761i)T \)
5 \( 1 + (0.803 - 0.595i)T \)
7 \( 1 + (-0.247 + 0.968i)T \)
11 \( 1 + (0.538 - 0.842i)T \)
13 \( 1 + (0.983 - 0.181i)T \)
17 \( 1 + (-0.877 + 0.480i)T \)
19 \( 1 + (0.0227 - 0.999i)T \)
23 \( 1 + (0.962 - 0.269i)T \)
29 \( 1 + (0.613 + 0.789i)T \)
31 \( 1 + (-0.158 - 0.987i)T \)
37 \( 1 + (-0.974 + 0.225i)T \)
41 \( 1 + (0.995 - 0.0909i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.113 - 0.993i)T \)
53 \( 1 + (0.898 + 0.439i)T \)
59 \( 1 + (-0.990 - 0.136i)T \)
61 \( 1 + (0.995 + 0.0909i)T \)
67 \( 1 + (-0.998 + 0.0455i)T \)
71 \( 1 + (0.113 + 0.993i)T \)
73 \( 1 + (0.746 - 0.665i)T \)
79 \( 1 + (-0.576 + 0.816i)T \)
83 \( 1 + (-0.998 - 0.0455i)T \)
89 \( 1 + (0.291 + 0.956i)T \)
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

−28.62822492869591790722906711573, −27.47231492788707287690375506048, −26.6958619605110219044698753351, −25.93449881263960458074497830156, −24.98103598461746438054581395913, −23.027469179446155855718597993812, −22.55993781184789877748078637402, −21.24474716093376402638413515998, −20.7123463849578461272375610436, −19.53986253439961772455592603847, −18.13000298621885835860950124955, −17.50583973008804795403533187293, −16.6608242057610417021083522722, −15.6190778574324470504879342672, −14.14180044159253683208912010919, −12.86291425546429729052586506749, −11.47080877966989161697363682685, −10.63257879620361607334751724340, −9.89529581205501361539079454, −8.993927767991027980669889545140, −7.1251320250249216359950615031, −6.29158246359149110226924178071, −4.385887167065032367915872508779, −3.244261520752057459093209769045, −1.424135742119968711614467936725, 1.02090318544597527865432710499, 2.323927505927401992544787997625, 5.16853006678720859654769732030, 6.03639358348435842340285500900, 6.764477173374369220378222751286, 8.57838049414720396765162876087, 8.95153371165468850705086743906, 10.61015953011868522098489617962, 11.56693300789305631463016476413, 12.95324530548079712238526590245, 13.843701640975677388203783488230, 15.41871736814933896518227798402, 16.43275473636926940883297038290, 17.28901061420853646881860716850, 18.12757284289457545678765332593, 18.93129914221810256150339858168, 19.95000712694060724343912934656, 21.44470442290739749594282404485, 22.46923367766283457853445906654, 23.83542344428184464459838355412, 24.54772779250943172083792557352, 25.195790517955958335901334712097, 26.11034322657179816320808782056, 27.67081108708496841392002611026, 28.33974231874748407588953085083

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