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Properties

Label 1-139-139.51-r0-0-0
Degree $1$
Conductor $139$
Sign $0.661 + 0.750i$
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  + (0.898 + 0.439i)2-s + (0.983 − 0.181i)3-s + (0.613 + 0.789i)4-s + (0.113 + 0.993i)5-s + (0.962 + 0.269i)6-s + (−0.715 − 0.699i)7-s + (0.203 + 0.979i)8-s + (0.934 − 0.356i)9-s + (−0.334 + 0.942i)10-s + (−0.998 − 0.0455i)11-s + (0.746 + 0.665i)12-s + (−0.648 − 0.761i)13-s + (−0.334 − 0.942i)14-s + (0.291 + 0.956i)15-s + (−0.247 + 0.968i)16-s + (0.0227 − 0.999i)17-s + ⋯
L(s)  = 1  + (0.898 + 0.439i)2-s + (0.983 − 0.181i)3-s + (0.613 + 0.789i)4-s + (0.113 + 0.993i)5-s + (0.962 + 0.269i)6-s + (−0.715 − 0.699i)7-s + (0.203 + 0.979i)8-s + (0.934 − 0.356i)9-s + (−0.334 + 0.942i)10-s + (−0.998 − 0.0455i)11-s + (0.746 + 0.665i)12-s + (−0.648 − 0.761i)13-s + (−0.334 − 0.942i)14-s + (0.291 + 0.956i)15-s + (−0.247 + 0.968i)16-s + (0.0227 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(139\)
Sign: $0.661 + 0.750i$
Analytic conductor: \(0.645513\)
Root analytic conductor: \(0.645513\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{139} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 139,\ (0:\ ),\ 0.661 + 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.993701086 + 0.9000447968i\)
\(L(\frac12)\) \(\approx\) \(1.993701086 + 0.9000447968i\)
\(L(1)\) \(\approx\) \(1.895548210 + 0.5961960944i\)
\(L(1)\) \(\approx\) \(1.895548210 + 0.5961960944i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad139 \( 1 \)
good2 \( 1 + (0.898 + 0.439i)T \)
3 \( 1 + (0.983 - 0.181i)T \)
5 \( 1 + (0.113 + 0.993i)T \)
7 \( 1 + (-0.715 - 0.699i)T \)
11 \( 1 + (-0.998 - 0.0455i)T \)
13 \( 1 + (-0.648 - 0.761i)T \)
17 \( 1 + (0.0227 - 0.999i)T \)
19 \( 1 + (-0.877 + 0.480i)T \)
23 \( 1 + (0.962 - 0.269i)T \)
29 \( 1 + (0.377 - 0.926i)T \)
31 \( 1 + (0.934 + 0.356i)T \)
37 \( 1 + (0.291 - 0.956i)T \)
41 \( 1 + (-0.419 + 0.907i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.803 + 0.595i)T \)
53 \( 1 + (-0.829 + 0.557i)T \)
59 \( 1 + (-0.990 - 0.136i)T \)
61 \( 1 + (-0.419 - 0.907i)T \)
67 \( 1 + (0.538 + 0.842i)T \)
71 \( 1 + (0.803 - 0.595i)T \)
73 \( 1 + (-0.949 - 0.313i)T \)
79 \( 1 + (-0.576 + 0.816i)T \)
83 \( 1 + (0.538 - 0.842i)T \)
89 \( 1 + (-0.974 - 0.225i)T \)
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

−28.605001343650373737208885496366, −27.51530684108949903446137501403, −25.97713062919968283131994914687, −25.2546147881959199697552257124, −24.22933333409941547286013848621, −23.52213346765864267044640353637, −21.84765814983525228994356932257, −21.38740315820656589521558991249, −20.44172675685614818466785339313, −19.440168677302216380814289484668, −18.84161041450468234776490237462, −16.812329775344767731807433440180, −15.62534090724250459554274823910, −15.07143468769064376207510927748, −13.65174798437939299583555180775, −12.92223064628146131140118348138, −12.193204522017538321998960615141, −10.46586105258420301439080036314, −9.44386731919021078569810959980, −8.44139938464758074967783000227, −6.80238301790951871011691927201, −5.30139809679657332228390589089, −4.31958732713607719039084112011, −2.92630993041758036513704016188, −1.88376697205674392253354516039, 2.61461183233135575195986290192, 3.13056377536281939006265302866, 4.56254303929839450131792823044, 6.27359449770045391368018602871, 7.26579538781708956244093482095, 7.99622247066305552295468473943, 9.81620402739185762458989406639, 10.81794237735755808628298188103, 12.53382654149642199526221436350, 13.36311852705991750655461724650, 14.17935780440580850779283668586, 15.131430754020531992761789055379, 15.90806976491255614758339867250, 17.34095077629724954023130500920, 18.63284109376743328117885694068, 19.68113417706597686189039814189, 20.710148361327456068050444354371, 21.5909644584735972795080165862, 22.87786138113885028335125873318, 23.33908478279692407336480369912, 24.86620870909473098146555077843, 25.395483560675100455230043193602, 26.551907185767240740860342532350, 26.79011640738624079182830485667, 29.17843798760508025170562000246

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