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Properties

Label 1-139-139.69-r0-0-0
Degree $1$
Conductor $139$
Sign $-0.906 - 0.421i$
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  + (−0.998 + 0.0455i)2-s + (0.291 + 0.956i)3-s + (0.995 − 0.0909i)4-s + (−0.715 + 0.699i)5-s + (−0.334 − 0.942i)6-s + (−0.648 − 0.761i)7-s + (−0.990 + 0.136i)8-s + (−0.829 + 0.557i)9-s + (0.682 − 0.730i)10-s + (−0.949 + 0.313i)11-s + (0.377 + 0.926i)12-s + (−0.974 − 0.225i)13-s + (0.682 + 0.730i)14-s + (−0.877 − 0.480i)15-s + (0.983 − 0.181i)16-s + (−0.158 − 0.987i)17-s + ⋯
L(s)  = 1  + (−0.998 + 0.0455i)2-s + (0.291 + 0.956i)3-s + (0.995 − 0.0909i)4-s + (−0.715 + 0.699i)5-s + (−0.334 − 0.942i)6-s + (−0.648 − 0.761i)7-s + (−0.990 + 0.136i)8-s + (−0.829 + 0.557i)9-s + (0.682 − 0.730i)10-s + (−0.949 + 0.313i)11-s + (0.377 + 0.926i)12-s + (−0.974 − 0.225i)13-s + (0.682 + 0.730i)14-s + (−0.877 − 0.480i)15-s + (0.983 − 0.181i)16-s + (−0.158 − 0.987i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(139\)
Sign: $-0.906 - 0.421i$
Analytic conductor: \(0.645513\)
Root analytic conductor: \(0.645513\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{139} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 139,\ (0:\ ),\ -0.906 - 0.421i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03439797969 + 0.1556232714i\)
\(L(\frac12)\) \(\approx\) \(-0.03439797969 + 0.1556232714i\)
\(L(1)\) \(\approx\) \(0.3926153956 + 0.1959369863i\)
\(L(1)\) \(\approx\) \(0.3926153956 + 0.1959369863i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad139 \( 1 \)
good2 \( 1 + (-0.998 + 0.0455i)T \)
3 \( 1 + (0.291 + 0.956i)T \)
5 \( 1 + (-0.715 + 0.699i)T \)
7 \( 1 + (-0.648 - 0.761i)T \)
11 \( 1 + (-0.949 + 0.313i)T \)
13 \( 1 + (-0.974 - 0.225i)T \)
17 \( 1 + (-0.158 - 0.987i)T \)
19 \( 1 + (0.934 + 0.356i)T \)
23 \( 1 + (-0.334 + 0.942i)T \)
29 \( 1 + (-0.419 + 0.907i)T \)
31 \( 1 + (-0.829 - 0.557i)T \)
37 \( 1 + (-0.877 + 0.480i)T \)
41 \( 1 + (0.113 - 0.993i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.247 + 0.968i)T \)
53 \( 1 + (0.538 + 0.842i)T \)
59 \( 1 + (-0.576 + 0.816i)T \)
61 \( 1 + (0.113 + 0.993i)T \)
67 \( 1 + (0.746 - 0.665i)T \)
71 \( 1 + (-0.247 - 0.968i)T \)
73 \( 1 + (0.613 + 0.789i)T \)
79 \( 1 + (-0.917 - 0.398i)T \)
83 \( 1 + (0.746 + 0.665i)T \)
89 \( 1 + (0.0227 + 0.999i)T \)
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

−28.215750079251080720806859727844, −26.6279961350143117573777524370, −26.08603087152993302763963192057, −24.64779328165545154732304650993, −24.44730733627790002376583707806, −23.258692305933494130943729295585, −21.65670772124904080854214265620, −20.35513124995449149456633556304, −19.57990585482671355698897170811, −18.896808322869289664134465037782, −17.99511271900885305913577106134, −16.78465717865481773130302014882, −15.84012204942421392926010746203, −14.82403986684328540733811281146, −12.98012088154423929697561682376, −12.325652680821315169672861787952, −11.36686633362513555524152780272, −9.74207058399650059702416000141, −8.63933185078720141254547422292, −7.93036069841879198265799867342, −6.82793814900891938978025253484, −5.505717173871253369452584854293, −3.20002908685600318079977045869, −2.00893905382940960876610436873, −0.1710380292832629261432636100, 2.65547411582317598784791825922, 3.59324614315434049549977434602, 5.333167636085938768683298183586, 7.19911329133240745115915968664, 7.715234878407651554891165561381, 9.32042193863819720798893826817, 10.14696144962385960983681065749, 10.88201136882073218172180765453, 12.07085801407299145942167728820, 13.920663704801550944950126950811, 15.14900124250811185268285086663, 15.85288898304217435465111995894, 16.65898644808942283409121272768, 17.924748782679261420224784865030, 19.046647830460264131305379616255, 20.03914154511856929434111353516, 20.51344546747003922169095989932, 22.02311632402806210098979484107, 22.924737223471869073828487898303, 24.13882507085357205868239590295, 25.67794161728310887862647176101, 26.14937267426972093405006483178, 27.09351817514618743144404720589, 27.49825268540034673969652090209, 28.9289086718720793891336002181

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