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Properties

Label 1-139-139.38-r0-0-0
Degree $1$
Conductor $139$
Sign $0.336 + 0.941i$
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  + (−0.949 + 0.313i)2-s + (−0.877 + 0.480i)3-s + (0.803 − 0.595i)4-s + (−0.648 − 0.761i)5-s + (0.682 − 0.730i)6-s + (−0.974 + 0.225i)7-s + (−0.576 + 0.816i)8-s + (0.538 − 0.842i)9-s + (0.854 + 0.519i)10-s + (0.613 + 0.789i)11-s + (−0.419 + 0.907i)12-s + (0.0227 − 0.999i)13-s + (0.854 − 0.519i)14-s + (0.934 + 0.356i)15-s + (0.291 − 0.956i)16-s + (0.898 + 0.439i)17-s + ⋯
L(s)  = 1  + (−0.949 + 0.313i)2-s + (−0.877 + 0.480i)3-s + (0.803 − 0.595i)4-s + (−0.648 − 0.761i)5-s + (0.682 − 0.730i)6-s + (−0.974 + 0.225i)7-s + (−0.576 + 0.816i)8-s + (0.538 − 0.842i)9-s + (0.854 + 0.519i)10-s + (0.613 + 0.789i)11-s + (−0.419 + 0.907i)12-s + (0.0227 − 0.999i)13-s + (0.854 − 0.519i)14-s + (0.934 + 0.356i)15-s + (0.291 − 0.956i)16-s + (0.898 + 0.439i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(139\)
Sign: $0.336 + 0.941i$
Analytic conductor: \(0.645513\)
Root analytic conductor: \(0.645513\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{139} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 139,\ (0:\ ),\ 0.336 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3168379494 + 0.2231939429i\)
\(L(\frac12)\) \(\approx\) \(0.3168379494 + 0.2231939429i\)
\(L(1)\) \(\approx\) \(0.4461118561 + 0.1222482952i\)
\(L(1)\) \(\approx\) \(0.4461118561 + 0.1222482952i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad139 \( 1 \)
good2 \( 1 + (-0.949 + 0.313i)T \)
3 \( 1 + (-0.877 + 0.480i)T \)
5 \( 1 + (-0.648 - 0.761i)T \)
7 \( 1 + (-0.974 + 0.225i)T \)
11 \( 1 + (0.613 + 0.789i)T \)
13 \( 1 + (0.0227 - 0.999i)T \)
17 \( 1 + (0.898 + 0.439i)T \)
19 \( 1 + (-0.829 + 0.557i)T \)
23 \( 1 + (0.682 + 0.730i)T \)
29 \( 1 + (0.113 + 0.993i)T \)
31 \( 1 + (0.538 + 0.842i)T \)
37 \( 1 + (0.934 - 0.356i)T \)
41 \( 1 + (-0.715 + 0.699i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.983 + 0.181i)T \)
53 \( 1 + (0.746 + 0.665i)T \)
59 \( 1 + (-0.917 + 0.398i)T \)
61 \( 1 + (-0.715 - 0.699i)T \)
67 \( 1 + (0.377 + 0.926i)T \)
71 \( 1 + (0.983 - 0.181i)T \)
73 \( 1 + (0.995 + 0.0909i)T \)
79 \( 1 + (0.962 - 0.269i)T \)
83 \( 1 + (0.377 - 0.926i)T \)
89 \( 1 + (-0.158 - 0.987i)T \)
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

−28.28347584079178221603233305272, −27.272475811838007645585512688195, −26.49618740989621797418275416832, −25.470323917570836890860668934729, −24.29545019565918099308120523037, −23.21978298114695524949753599860, −22.292189446569267876574688830157, −21.327728793605583466518846253570, −19.6419474668729032535910947370, −18.98690218021096841272286846881, −18.5096219682014575357481498197, −16.90556888410458878390967372424, −16.5648439922979545558423500035, −15.338507664250627028899652266750, −13.62838297331630067941471995018, −12.26847843101477038419005629865, −11.48611156832072725424476075753, −10.655677651048073108403838058141, −9.496981149469707578613843914923, −8.01683675412140707425612583372, −6.78968302381420990745604251388, −6.3529615652872062764462411672, −3.99178358511160752636341473736, −2.59255784916839854975853093786, −0.64596068874647849710281188742, 1.12342586761390641872580828779, 3.515409989446267409516279554305, 5.12686038287091742300356611999, 6.206808105826488598142035557043, 7.393742002494132810621021654717, 8.77725788267010975876503946187, 9.76918010585535705401171004515, 10.656874124401457822730341034885, 12.05660950984476557307491016972, 12.62287015009717516058343661848, 15.00294309505355822465437380418, 15.61042354405606887320654671192, 16.67970422907700094379355975730, 17.148989239376628370789777975174, 18.435500587158059327124767080319, 19.55936941556754127446889443624, 20.34714232258698279596072232732, 21.57953056653739793698135287954, 23.10804629585419096503141157550, 23.409158134218482633786110734452, 24.95916304668511624932367817002, 25.58010208360268974115905813499, 27.04295734056393006896961870062, 27.66219158866404397362769066923, 28.31294162733009893672075864441

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