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Properties

Label 1-139-139.54-r0-0-0
Degree $1$
Conductor $139$
Sign $0.0897 - 0.995i$
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  + (0.803 − 0.595i)2-s + (0.538 − 0.842i)3-s + (0.291 − 0.956i)4-s + (−0.158 + 0.987i)5-s + (−0.0682 − 0.997i)6-s + (0.898 − 0.439i)7-s + (−0.334 − 0.942i)8-s + (−0.419 − 0.907i)9-s + (0.460 + 0.887i)10-s + (−0.247 + 0.968i)11-s + (−0.648 − 0.761i)12-s + (−0.998 − 0.0455i)13-s + (0.460 − 0.887i)14-s + (0.746 + 0.665i)15-s + (−0.829 − 0.557i)16-s + (0.613 + 0.789i)17-s + ⋯
L(s)  = 1  + (0.803 − 0.595i)2-s + (0.538 − 0.842i)3-s + (0.291 − 0.956i)4-s + (−0.158 + 0.987i)5-s + (−0.0682 − 0.997i)6-s + (0.898 − 0.439i)7-s + (−0.334 − 0.942i)8-s + (−0.419 − 0.907i)9-s + (0.460 + 0.887i)10-s + (−0.247 + 0.968i)11-s + (−0.648 − 0.761i)12-s + (−0.998 − 0.0455i)13-s + (0.460 − 0.887i)14-s + (0.746 + 0.665i)15-s + (−0.829 − 0.557i)16-s + (0.613 + 0.789i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0897 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0897 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(139\)
Sign: $0.0897 - 0.995i$
Analytic conductor: \(0.645513\)
Root analytic conductor: \(0.645513\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{139} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 139,\ (0:\ ),\ 0.0897 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.411788996 - 1.290312734i\)
\(L(\frac12)\) \(\approx\) \(1.411788996 - 1.290312734i\)
\(L(1)\) \(\approx\) \(1.513374286 - 0.8943426477i\)
\(L(1)\) \(\approx\) \(1.513374286 - 0.8943426477i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad139 \( 1 \)
good2 \( 1 + (0.803 - 0.595i)T \)
3 \( 1 + (0.538 - 0.842i)T \)
5 \( 1 + (-0.158 + 0.987i)T \)
7 \( 1 + (0.898 - 0.439i)T \)
11 \( 1 + (-0.247 + 0.968i)T \)
13 \( 1 + (-0.998 - 0.0455i)T \)
17 \( 1 + (0.613 + 0.789i)T \)
19 \( 1 + (0.377 - 0.926i)T \)
23 \( 1 + (-0.0682 + 0.997i)T \)
29 \( 1 + (-0.974 + 0.225i)T \)
31 \( 1 + (-0.419 + 0.907i)T \)
37 \( 1 + (0.746 - 0.665i)T \)
41 \( 1 + (0.0227 - 0.999i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.934 + 0.356i)T \)
53 \( 1 + (0.113 + 0.993i)T \)
59 \( 1 + (0.682 - 0.730i)T \)
61 \( 1 + (0.0227 + 0.999i)T \)
67 \( 1 + (-0.715 + 0.699i)T \)
71 \( 1 + (0.934 - 0.356i)T \)
73 \( 1 + (0.983 + 0.181i)T \)
79 \( 1 + (0.854 - 0.519i)T \)
83 \( 1 + (-0.715 - 0.699i)T \)
89 \( 1 + (-0.949 + 0.313i)T \)
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

−28.59007158370695455488721507622, −27.23829235978664861985004924118, −26.861412995671286903871041161930, −25.39303287244299316772128127084, −24.61499927337157034614639870367, −24.01245907743533329304048207264, −22.55809605914660421785188625718, −21.614650889285703628153562010524, −20.832687778763181322959146905759, −20.21913123139086075685040001145, −18.5742823544416834361943399308, −16.80810541929062647079067300969, −16.502376116353442285131123136157, −15.27139386838039665003351257367, −14.47004834357408155753993278588, −13.53664944144736030457946684539, −12.212254352580887491391774411020, −11.28240839380516757420076765420, −9.54444072638759715503337351139, −8.35658241782107236118284874501, −7.772903299054669820474628000317, −5.57985886000318892118915061069, −4.970877205646735019933620040485, −3.84240483549782871374358835527, −2.41007913998896658696077946532, 1.65705436003574808650226261797, 2.68788033663593975488925260459, 3.964193539730353039384016530455, 5.46155746419035602487486681662, 7.04862386702552549615133442242, 7.58426654015897465630111618809, 9.53579849729619996316747630452, 10.725880643612041555323566336460, 11.75744496979757101466459765155, 12.72406598577688072099120673353, 13.88755332768895314429987753017, 14.67327013227971588481902927134, 15.23979573228203800989537388191, 17.48933699226329654464107710260, 18.283825950671646159062546310703, 19.451950188957509508120374129822, 20.03898209906092049717133218082, 21.188388260121921916570263560975, 22.23238512129994171899740909080, 23.435310316472725592182333177209, 23.86168726372505094288157133969, 25.059353734575768593905391230641, 26.1026657231648835776980185251, 27.24756097394962786560680883635, 28.45966186966632381977174386196

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