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Properties

Label 1-117-117.29-r1-0-0
Degree $1$
Conductor $117$
Sign $-0.578 + 0.815i$
Analytic cond. $12.5733$
Root an. cond. $12.5733$
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  − 2-s + 4-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s − 8-s + (−0.5 + 0.866i)10-s − 11-s + (0.5 − 0.866i)14-s + 16-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)20-s + 22-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  − 2-s + 4-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s − 8-s + (−0.5 + 0.866i)10-s − 11-s + (0.5 − 0.866i)14-s + 16-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)20-s + 22-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.578 + 0.815i$
Analytic conductor: \(12.5733\)
Root analytic conductor: \(12.5733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 117,\ (1:\ ),\ -0.578 + 0.815i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1951861729 + 0.3779022266i\)
\(L(\frac12)\) \(\approx\) \(0.1951861729 + 0.3779022266i\)
\(L(1)\) \(\approx\) \(0.5783337552 + 0.05433636154i\)
\(L(1)\) \(\approx\) \(0.5783337552 + 0.05433636154i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 - T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 - T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 - T \)
59 \( 1 - T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
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

−28.731586030574547342421447337586, −27.378760057914226441737270387012, −26.466695573065503402642625909869, −25.895981811338542000853208174496, −24.91170130524122668603411507515, −23.53325422342196508977843000526, −22.54894888024505310548896530741, −21.07667947997709611878553765606, −20.371446160245824240393372015407, −18.91777847856022021657202041046, −18.45652693304130748221808095908, −17.19824289915600597854853840254, −16.35683318321468073072354614178, −15.12953763745539797974716221746, −13.94438041016320977568287072937, −12.60317557594386651748959395668, −10.964367022897621166553647065384, −10.35262501873311727525770432879, −9.37619505638000013360358024556, −7.75089295481548137946320386245, −6.950459175789567384098542296672, −5.711849489236199189170860036852, −3.42445614460836614053338370217, −2.15900658889593221789719353227, −0.23154837220032149671351451455, 1.579024919052309978694223653101, 2.93944476720687290781792833959, 5.23056127598410788579730485809, 6.264366392263211243214864230240, 7.85591733031606308661372220461, 8.90167577036602780919817546741, 9.70910105397165985582725129654, 10.94391059481437796553337637041, 12.35451469077695574494656830971, 13.14148430931116422600554494968, 15.064513094804255953324045223930, 15.93621166226642286485914306754, 16.93360078527273806124489540182, 17.90209423674845748875851626888, 18.93249462280383310353233841939, 19.88286851121048718592077068264, 21.09026481377946409482969925509, 21.68595969391509867749345457165, 23.56599517799745689463056478720, 24.467192967164144538749441616982, 25.560126831159240710123520519884, 26.00319067717235564540562232707, 27.52386142920867788290713595015, 28.36550625815816314227632340061, 28.89609637118756700838598818409

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