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Properties

Label 1-139-139.83-r0-0-0
Degree $1$
Conductor $139$
Sign $0.997 + 0.0650i$
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  + (0.746 − 0.665i)2-s + (0.0227 + 0.999i)3-s + (0.113 − 0.993i)4-s + (0.983 + 0.181i)5-s + (0.682 + 0.730i)6-s + (0.291 + 0.956i)7-s + (−0.576 − 0.816i)8-s + (−0.998 + 0.0455i)9-s + (0.854 − 0.519i)10-s + (0.377 + 0.926i)11-s + (0.995 + 0.0909i)12-s + (−0.877 − 0.480i)13-s + (0.854 + 0.519i)14-s + (−0.158 + 0.987i)15-s + (−0.974 − 0.225i)16-s + (−0.829 − 0.557i)17-s + ⋯
L(s)  = 1  + (0.746 − 0.665i)2-s + (0.0227 + 0.999i)3-s + (0.113 − 0.993i)4-s + (0.983 + 0.181i)5-s + (0.682 + 0.730i)6-s + (0.291 + 0.956i)7-s + (−0.576 − 0.816i)8-s + (−0.998 + 0.0455i)9-s + (0.854 − 0.519i)10-s + (0.377 + 0.926i)11-s + (0.995 + 0.0909i)12-s + (−0.877 − 0.480i)13-s + (0.854 + 0.519i)14-s + (−0.158 + 0.987i)15-s + (−0.974 − 0.225i)16-s + (−0.829 − 0.557i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(139\)
Sign: $0.997 + 0.0650i$
Analytic conductor: \(0.645513\)
Root analytic conductor: \(0.645513\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{139} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 139,\ (0:\ ),\ 0.997 + 0.0650i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.764018122 + 0.05746979507i\)
\(L(\frac12)\) \(\approx\) \(1.764018122 + 0.05746979507i\)
\(L(1)\) \(\approx\) \(1.631693849 - 0.03539159831i\)
\(L(1)\) \(\approx\) \(1.631693849 - 0.03539159831i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad139 \( 1 \)
good2 \( 1 + (0.746 - 0.665i)T \)
3 \( 1 + (0.0227 + 0.999i)T \)
5 \( 1 + (0.983 + 0.181i)T \)
7 \( 1 + (0.291 + 0.956i)T \)
11 \( 1 + (0.377 + 0.926i)T \)
13 \( 1 + (-0.877 - 0.480i)T \)
17 \( 1 + (-0.829 - 0.557i)T \)
19 \( 1 + (0.898 - 0.439i)T \)
23 \( 1 + (0.682 - 0.730i)T \)
29 \( 1 + (0.803 + 0.595i)T \)
31 \( 1 + (-0.998 - 0.0455i)T \)
37 \( 1 + (-0.158 - 0.987i)T \)
41 \( 1 + (-0.247 + 0.968i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.648 - 0.761i)T \)
53 \( 1 + (-0.949 - 0.313i)T \)
59 \( 1 + (-0.917 - 0.398i)T \)
61 \( 1 + (-0.247 - 0.968i)T \)
67 \( 1 + (0.613 + 0.789i)T \)
71 \( 1 + (-0.648 + 0.761i)T \)
73 \( 1 + (-0.419 + 0.907i)T \)
79 \( 1 + (0.962 + 0.269i)T \)
83 \( 1 + (0.613 - 0.789i)T \)
89 \( 1 + (0.934 - 0.356i)T \)
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

−29.19763958924352603794781309064, −27.028802103088213053591950765194, −26.177151170852228785790182008701, −25.12329510268314525483173854055, −24.33250778407062752404803609836, −23.825586221831014847610136695967, −22.56181490118786199029825320207, −21.65900845445392291973615531759, −20.548370397717150128423839770887, −19.43603088323432299521397867471, −17.928753148549991080543725173, −17.19030656521620785402585227960, −16.52647927052569672218325736509, −14.71031181847466201404256658049, −13.79885486736005307891780873940, −13.42263482077659944855354178064, −12.18465292543091947642800451864, −11.042888402287862887512757668187, −9.21181370715086376552289738419, −7.97284024406675443021073342292, −6.90691291654650714367219172104, −6.04836376798159600778121965574, −4.83635701592190661835734566840, −3.16815388655212033954021693321, −1.62076926150603855346454728680, 2.12075272040848894053723211104, 3.02685546020306095013490995699, 4.825447883270329786949572985815, 5.276833528168657847193651965637, 6.700778852386188882461651514573, 9.0630470099719345402698016819, 9.67780254467778963571419327883, 10.73330244855448967216693686270, 11.825275079381929734177907681085, 12.92409129130793644115797027321, 14.32488012888497562678146423812, 14.86056220811992826998082923924, 15.87374332455747924214550948169, 17.45031160623636510325775785470, 18.328436323089442399957830970146, 19.93458206673036419302043050177, 20.55803635395504934820600776332, 21.789475919005097943288140440714, 22.03147568880109717231339926539, 22.96435364668094908556707801577, 24.69741856495843197722473766504, 25.17651179848906198207470193398, 26.65294579074243162104604146547, 27.705225430249558956277595273046, 28.589698502225437140434573373503

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