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Properties

Label 1-139-139.25-r0-0-0
Degree $1$
Conductor $139$
Sign $-0.865 + 0.500i$
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  + (0.0227 + 0.999i)2-s + (0.803 + 0.595i)3-s + (−0.998 + 0.0455i)4-s + (0.377 + 0.926i)5-s + (−0.576 + 0.816i)6-s + (−0.419 + 0.907i)7-s + (−0.0682 − 0.997i)8-s + (0.291 + 0.956i)9-s + (−0.917 + 0.398i)10-s + (−0.158 − 0.987i)11-s + (−0.829 − 0.557i)12-s + (0.113 − 0.993i)13-s + (−0.917 − 0.398i)14-s + (−0.247 + 0.968i)15-s + (0.995 − 0.0909i)16-s + (−0.648 + 0.761i)17-s + ⋯
L(s)  = 1  + (0.0227 + 0.999i)2-s + (0.803 + 0.595i)3-s + (−0.998 + 0.0455i)4-s + (0.377 + 0.926i)5-s + (−0.576 + 0.816i)6-s + (−0.419 + 0.907i)7-s + (−0.0682 − 0.997i)8-s + (0.291 + 0.956i)9-s + (−0.917 + 0.398i)10-s + (−0.158 − 0.987i)11-s + (−0.829 − 0.557i)12-s + (0.113 − 0.993i)13-s + (−0.917 − 0.398i)14-s + (−0.247 + 0.968i)15-s + (0.995 − 0.0909i)16-s + (−0.648 + 0.761i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(139\)
Sign: $-0.865 + 0.500i$
Analytic conductor: \(0.645513\)
Root analytic conductor: \(0.645513\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{139} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 139,\ (0:\ ),\ -0.865 + 0.500i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3255842979 + 1.214644221i\)
\(L(\frac12)\) \(\approx\) \(0.3255842979 + 1.214644221i\)
\(L(1)\) \(\approx\) \(0.7709036232 + 0.9340011301i\)
\(L(1)\) \(\approx\) \(0.7709036232 + 0.9340011301i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad139 \( 1 \)
good2 \( 1 + (0.0227 + 0.999i)T \)
3 \( 1 + (0.803 + 0.595i)T \)
5 \( 1 + (0.377 + 0.926i)T \)
7 \( 1 + (-0.419 + 0.907i)T \)
11 \( 1 + (-0.158 - 0.987i)T \)
13 \( 1 + (0.113 - 0.993i)T \)
17 \( 1 + (-0.648 + 0.761i)T \)
19 \( 1 + (0.983 + 0.181i)T \)
23 \( 1 + (-0.576 - 0.816i)T \)
29 \( 1 + (0.538 + 0.842i)T \)
31 \( 1 + (0.291 - 0.956i)T \)
37 \( 1 + (-0.247 - 0.968i)T \)
41 \( 1 + (0.746 - 0.665i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.613 + 0.789i)T \)
53 \( 1 + (-0.877 - 0.480i)T \)
59 \( 1 + (0.460 + 0.887i)T \)
61 \( 1 + (0.746 + 0.665i)T \)
67 \( 1 + (0.934 - 0.356i)T \)
71 \( 1 + (0.613 - 0.789i)T \)
73 \( 1 + (0.898 + 0.439i)T \)
79 \( 1 + (0.203 - 0.979i)T \)
83 \( 1 + (0.934 + 0.356i)T \)
89 \( 1 + (-0.715 - 0.699i)T \)
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

−28.4441065099881094639425784501, −26.95640131203408323416308232193, −26.18948446602859930206476376940, −25.10555627301966103890330946070, −23.89105288925731269603502766264, −23.16148836353464363982936981994, −21.73408317555604085449813750017, −20.53131876634764696638806798301, −20.21745457495416303201292982101, −19.30060203232730746955220509919, −18.047868393166922865410167443996, −17.262562286011444419182473475816, −15.76081460650825835122496774726, −13.9246951433783348714066590242, −13.66340906609769341889105829807, −12.58064891135502784398654539842, −11.673083816833109511932540286631, −9.84243869795657358056540908826, −9.40546309479097583116873347532, −8.124747081786128704595107281156, −6.82249004368378600873790158226, −4.86233835901110304259088646592, −3.79915678034112937808989613181, −2.28266370629248268221406952851, −1.14660303545294584314892015245, 2.68255658934143832036545546832, 3.68677255113145687047917703159, 5.41765936289300747723461715267, 6.29612630363524590879197252166, 7.80023547813942309629508690703, 8.71772124237238999151182652682, 9.76338810041721159250251427219, 10.78817633665990111027461845649, 12.803819470466534539903048957623, 13.838428409422006873968869629685, 14.67957803727054440199225159208, 15.55588421234804610280222101719, 16.246045781817396695326907331, 17.80716220639253065794429305759, 18.64623044642333413942401244453, 19.55643640414380168555639587171, 21.19250118336481204864723971740, 22.16558536458875627723093445798, 22.517746818886654874119380067947, 24.283019400927915402027382836493, 25.04364441822344594017316618189, 25.93788534754340385845048339263, 26.53371804248741680035150452629, 27.42892276615182570297158803731, 28.58450211122583129231047077772

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