Properties

Degree 1
Conductor 139
Sign $0.336 + 0.941i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(139\)
\( \varepsilon \)  =  $0.336 + 0.941i$
motivic weight  =  \(0\)
character  :  $\chi_{139} (38, \cdot )$
Sato-Tate  :  $\mu(69)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 139,\ (0:\ ),\ 0.336 + 0.941i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3168379494 + 0.2231939429i$
$L(\frac12,\chi)$  $\approx$  $0.3168379494 + 0.2231939429i$
$L(\chi,1)$  $\approx$  0.4461118561 + 0.1222482952i
$L(1,\chi)$  $\approx$  0.4461118561 + 0.1222482952i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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