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Properties

Label 1-139-139.24-r0-0-0
Degree $1$
Conductor $139$
Sign $-0.508 + 0.861i$
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  + (−0.419 + 0.907i)2-s + (0.898 + 0.439i)3-s + (−0.648 − 0.761i)4-s + (−0.877 + 0.480i)5-s + (−0.775 + 0.631i)6-s + (0.934 − 0.356i)7-s + (0.962 − 0.269i)8-s + (0.613 + 0.789i)9-s + (−0.0682 − 0.997i)10-s + (0.113 + 0.993i)11-s + (−0.247 − 0.968i)12-s + (−0.829 + 0.557i)13-s + (−0.0682 + 0.997i)14-s + (−0.998 + 0.0455i)15-s + (−0.158 + 0.987i)16-s + (0.746 + 0.665i)17-s + ⋯
L(s)  = 1  + (−0.419 + 0.907i)2-s + (0.898 + 0.439i)3-s + (−0.648 − 0.761i)4-s + (−0.877 + 0.480i)5-s + (−0.775 + 0.631i)6-s + (0.934 − 0.356i)7-s + (0.962 − 0.269i)8-s + (0.613 + 0.789i)9-s + (−0.0682 − 0.997i)10-s + (0.113 + 0.993i)11-s + (−0.247 − 0.968i)12-s + (−0.829 + 0.557i)13-s + (−0.0682 + 0.997i)14-s + (−0.998 + 0.0455i)15-s + (−0.158 + 0.987i)16-s + (0.746 + 0.665i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.508 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.508 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(139\)
Sign: $-0.508 + 0.861i$
Analytic conductor: \(0.645513\)
Root analytic conductor: \(0.645513\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{139} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 139,\ (0:\ ),\ -0.508 + 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5018540550 + 0.8790835745i\)
\(L(\frac12)\) \(\approx\) \(0.5018540550 + 0.8790835745i\)
\(L(1)\) \(\approx\) \(0.7736299647 + 0.6303533516i\)
\(L(1)\) \(\approx\) \(0.7736299647 + 0.6303533516i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad139 \( 1 \)
good2 \( 1 + (-0.419 + 0.907i)T \)
3 \( 1 + (0.898 + 0.439i)T \)
5 \( 1 + (-0.877 + 0.480i)T \)
7 \( 1 + (0.934 - 0.356i)T \)
11 \( 1 + (0.113 + 0.993i)T \)
13 \( 1 + (-0.829 + 0.557i)T \)
17 \( 1 + (0.746 + 0.665i)T \)
19 \( 1 + (-0.949 + 0.313i)T \)
23 \( 1 + (-0.775 - 0.631i)T \)
29 \( 1 + (0.983 - 0.181i)T \)
31 \( 1 + (0.613 - 0.789i)T \)
37 \( 1 + (-0.998 - 0.0455i)T \)
41 \( 1 + (0.291 + 0.956i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.0227 + 0.999i)T \)
53 \( 1 + (0.995 - 0.0909i)T \)
59 \( 1 + (-0.334 - 0.942i)T \)
61 \( 1 + (0.291 - 0.956i)T \)
67 \( 1 + (0.803 + 0.595i)T \)
71 \( 1 + (0.0227 - 0.999i)T \)
73 \( 1 + (-0.715 - 0.699i)T \)
79 \( 1 + (0.682 + 0.730i)T \)
83 \( 1 + (0.803 - 0.595i)T \)
89 \( 1 + (0.538 + 0.842i)T \)
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

−27.756481774629246392409046144961, −27.35031803498133468668735657640, −26.43081875070859939542106087391, −25.1129525979712559214764994826, −24.26154272005191710988277041608, −23.21000048193507929637187647546, −21.613545571545513051281160992, −20.95159581333243379172650673653, −19.84928420229527405947366680671, −19.32140109471000637651461164581, −18.32880226413528975778956967914, −17.26879477058058763582923074127, −15.8596809214274654738750024528, −14.5733033497063479082714868129, −13.56995100325156067008814779585, −12.28164851029059779700083565262, −11.755807129317781260660520953077, −10.32485652760442525975637212381, −8.84752811513027135762132200176, −8.29032924224813489479596087968, −7.39593540017279006231548326701, −4.999521179707806537662067536986, −3.699809312505134789843327336904, −2.55203359360003141821768053789, −1.048091075882382199862743628536, 1.99970638342237409257611548605, 4.06993739326882999336658040203, 4.71897736177788683630872900250, 6.756945956442918794537771211622, 7.803960122700937106766038211620, 8.34748695230698556616090060919, 9.84033366628409559496737357886, 10.62674307603727307723101728109, 12.28119948095439118215001903158, 14.018789430884465529453748118218, 14.724098957559486396434601715695, 15.2384950694922225838350055467, 16.48333033439629530547827601962, 17.51856982107690444874971279892, 18.82073669728666655180644083049, 19.525306834865441374725282668657, 20.547639383671382078742205917490, 21.864356350522486259862599265575, 23.09446708785357168832937560925, 23.9552585880197197324335197801, 24.9126789752784545223888517129, 26.02652995981463212915130820559, 26.61774920986264963659956314777, 27.539762390418492108765176316769, 28.063895404220153975943984828119

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