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Properties

Label 1-139-139.13-r0-0-0
Degree $1$
Conductor $139$
Sign $0.954 + 0.296i$
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  + (−0.974 + 0.225i)2-s + (0.995 + 0.0909i)3-s + (0.898 − 0.439i)4-s + (0.746 − 0.665i)5-s + (−0.990 + 0.136i)6-s + (0.377 + 0.926i)7-s + (−0.775 + 0.631i)8-s + (0.983 + 0.181i)9-s + (−0.576 + 0.816i)10-s + (0.0227 + 0.999i)11-s + (0.934 − 0.356i)12-s + (−0.419 − 0.907i)13-s + (−0.576 − 0.816i)14-s + (0.803 − 0.595i)15-s + (0.613 − 0.789i)16-s + (−0.715 − 0.699i)17-s + ⋯
L(s)  = 1  + (−0.974 + 0.225i)2-s + (0.995 + 0.0909i)3-s + (0.898 − 0.439i)4-s + (0.746 − 0.665i)5-s + (−0.990 + 0.136i)6-s + (0.377 + 0.926i)7-s + (−0.775 + 0.631i)8-s + (0.983 + 0.181i)9-s + (−0.576 + 0.816i)10-s + (0.0227 + 0.999i)11-s + (0.934 − 0.356i)12-s + (−0.419 − 0.907i)13-s + (−0.576 − 0.816i)14-s + (0.803 − 0.595i)15-s + (0.613 − 0.789i)16-s + (−0.715 − 0.699i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(139\)
Sign: $0.954 + 0.296i$
Analytic conductor: \(0.645513\)
Root analytic conductor: \(0.645513\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{139} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 139,\ (0:\ ),\ 0.954 + 0.296i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.110874054 + 0.1686769978i\)
\(L(\frac12)\) \(\approx\) \(1.110874054 + 0.1686769978i\)
\(L(1)\) \(\approx\) \(1.058176104 + 0.1122163036i\)
\(L(1)\) \(\approx\) \(1.058176104 + 0.1122163036i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad139 \( 1 \)
good2 \( 1 + (-0.974 + 0.225i)T \)
3 \( 1 + (0.995 + 0.0909i)T \)
5 \( 1 + (0.746 - 0.665i)T \)
7 \( 1 + (0.377 + 0.926i)T \)
11 \( 1 + (0.0227 + 0.999i)T \)
13 \( 1 + (-0.419 - 0.907i)T \)
17 \( 1 + (-0.715 - 0.699i)T \)
19 \( 1 + (-0.247 + 0.968i)T \)
23 \( 1 + (-0.990 - 0.136i)T \)
29 \( 1 + (-0.829 - 0.557i)T \)
31 \( 1 + (0.983 - 0.181i)T \)
37 \( 1 + (0.803 + 0.595i)T \)
41 \( 1 + (0.538 - 0.842i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.949 + 0.313i)T \)
53 \( 1 + (0.291 - 0.956i)T \)
59 \( 1 + (-0.0682 - 0.997i)T \)
61 \( 1 + (0.538 + 0.842i)T \)
67 \( 1 + (-0.877 + 0.480i)T \)
71 \( 1 + (-0.949 - 0.313i)T \)
73 \( 1 + (-0.158 - 0.987i)T \)
79 \( 1 + (0.460 - 0.887i)T \)
83 \( 1 + (-0.877 - 0.480i)T \)
89 \( 1 + (0.113 + 0.993i)T \)
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

−28.37351770672632143529967638735, −26.90400241302534062430506047074, −26.44024923553769434037302984543, −25.87822988502202669681567939090, −24.576266388011546968434446775936, −23.942473830548415575845470527769, −21.76464185151849804383982968117, −21.36144518416706305042066521892, −20.07043273227780026881105375005, −19.388582805935584215543702412788, −18.39403770107213134227075121457, −17.45520784448379375099751106615, −16.40424697635065414982318230513, −15.04167157787758011196687678311, −14.02595373555705500890736021356, −13.15204284438502410501901489584, −11.36052680176179449344453241613, −10.45175281696695835874079834804, −9.45126082954562008887244540224, −8.455075553408606177621403848243, −7.28881327321678840612687599280, −6.41281308541514114362735303943, −4.0058454669549138378819733309, −2.6780736895494978970418030562, −1.58674354848707385734785667708, 1.76850381250506486658739637628, 2.54847865149090192969384753115, 4.77776272944364190855609404218, 6.11159966132040688948097669186, 7.69318509936211705055386543662, 8.48528690309937940719745048862, 9.5407381306204556317458403775, 10.09972463816579531272526750042, 11.926610109148771054462388638005, 13.006366101642511669791191427807, 14.48951266520094819011989102080, 15.25289437101440823024875203658, 16.24828026650619198900636325158, 17.6485320532182889687248195754, 18.21372652149547067460883650801, 19.46322943946412883425630541346, 20.51157057788260752180287767600, 20.90561337912693492445775690975, 22.30162729240933849488337399079, 24.21123212908068152080617462640, 24.9432968672167049409788011609, 25.334469179411279498332407519121, 26.387776843687370749495370014117, 27.57004172109346972809792954313, 28.14846139187889205063538290318

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