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Properties

Label 1-129-129.59-r1-0-0
Degree $1$
Conductor $129$
Sign $-0.189 - 0.981i$
Analytic cond. $13.8629$
Root an. cond. $13.8629$
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  + (0.900 − 0.433i)2-s + (0.623 − 0.781i)4-s + (0.222 − 0.974i)5-s + 7-s + (0.222 − 0.974i)8-s + (−0.222 − 0.974i)10-s + (−0.623 − 0.781i)11-s + (−0.222 + 0.974i)13-s + (0.900 − 0.433i)14-s + (−0.222 − 0.974i)16-s + (0.222 + 0.974i)17-s + (0.623 − 0.781i)19-s + (−0.623 − 0.781i)20-s + (−0.900 − 0.433i)22-s + (−0.623 − 0.781i)23-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)2-s + (0.623 − 0.781i)4-s + (0.222 − 0.974i)5-s + 7-s + (0.222 − 0.974i)8-s + (−0.222 − 0.974i)10-s + (−0.623 − 0.781i)11-s + (−0.222 + 0.974i)13-s + (0.900 − 0.433i)14-s + (−0.222 − 0.974i)16-s + (0.222 + 0.974i)17-s + (0.623 − 0.781i)19-s + (−0.623 − 0.781i)20-s + (−0.900 − 0.433i)22-s + (−0.623 − 0.781i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $-0.189 - 0.981i$
Analytic conductor: \(13.8629\)
Root analytic conductor: \(13.8629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 129,\ (1:\ ),\ -0.189 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.113796740 - 2.561966540i\)
\(L(\frac12)\) \(\approx\) \(2.113796740 - 2.561966540i\)
\(L(1)\) \(\approx\) \(1.721633883 - 1.021940663i\)
\(L(1)\) \(\approx\) \(1.721633883 - 1.021940663i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.900 - 0.433i)T \)
5 \( 1 + (0.222 - 0.974i)T \)
7 \( 1 + T \)
11 \( 1 + (-0.623 - 0.781i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 + (0.222 + 0.974i)T \)
19 \( 1 + (0.623 - 0.781i)T \)
23 \( 1 + (-0.623 - 0.781i)T \)
29 \( 1 + (0.900 - 0.433i)T \)
31 \( 1 + (-0.900 + 0.433i)T \)
37 \( 1 + T \)
41 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (0.222 + 0.974i)T \)
59 \( 1 + (0.222 + 0.974i)T \)
61 \( 1 + (-0.900 - 0.433i)T \)
67 \( 1 + (0.623 - 0.781i)T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 + T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (0.900 + 0.433i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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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

−29.22795111291217952248512659297, −27.5722939429326018506349135550, −26.648916157433042051874343507265, −25.51342193135585933269662257494, −24.8682727088595217500267684033, −23.5762391380506866841468130497, −22.84281218696032420015966155840, −21.91263790928665222688254809009, −20.907243539278450201776195922688, −20.04442428087432292910544995494, −18.13035859546211639032498661397, −17.75647662392822926812941559328, −16.18434635412521687587276235882, −15.07010144348501396576081250176, −14.44305377844456595228968559920, −13.42414219710417570588241685550, −12.11617365727199467841874216357, −11.12329456454616528678213312949, −9.96170499024211028184700430370, −7.88059648331419233844539963452, −7.350217373141528237637676981862, −5.81435662007521385345244337005, −4.87036973520268272171001600742, −3.31274841637385459497270162622, −2.11872618032725080037840825450, 1.060923281662657720335502926602, 2.348049872723634097666494786792, 4.156147115145855187476458714878, 5.04650140989337065279173087679, 6.12126516980477855881494782231, 7.84950007071385691497528917991, 9.12010917888080413705758630114, 10.5662350446761823954761552350, 11.593005062346316569596179388169, 12.56748286556636455643945268637, 13.66236855728114256378602144898, 14.45238462678235409919946548750, 15.79425413795140068263785024758, 16.72407521132864113927201332420, 18.102986983886232077217643012075, 19.40068861806402437042803538369, 20.38188921604248166354183060150, 21.35741496259107076728790961432, 21.75287060290344457434426440293, 23.4943671165981474273107088429, 24.10353638414998725104080981705, 24.692841386589433762496216116396, 26.191710151966587584213621180064, 27.54212772318265243118689717267, 28.54909886820976888887253211359

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