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Properties

Label 1-117-117.58-r1-0-0
Degree $1$
Conductor $117$
Sign $0.612 - 0.790i$
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  − i·2-s − 4-s + (−0.866 + 0.5i)5-s + (0.866 − 0.5i)7-s + i·8-s + (0.5 + 0.866i)10-s + i·11-s + (−0.5 − 0.866i)14-s + 16-s + (0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (0.866 − 0.5i)20-s + 22-s + (0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (−0.866 + 0.5i)5-s + (0.866 − 0.5i)7-s + i·8-s + (0.5 + 0.866i)10-s + i·11-s + (−0.5 − 0.866i)14-s + 16-s + (0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (0.866 − 0.5i)20-s + 22-s + (0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.612 - 0.790i$
Analytic conductor: \(12.5733\)
Root analytic conductor: \(12.5733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 117,\ (1:\ ),\ 0.612 - 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.311803686 - 0.6428220937i\)
\(L(\frac12)\) \(\approx\) \(1.311803686 - 0.6428220937i\)
\(L(1)\) \(\approx\) \(0.9314498290 - 0.3803529071i\)
\(L(1)\) \(\approx\) \(0.9314498290 - 0.3803529071i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 - iT \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + iT \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + T \)
59 \( 1 + iT \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 + iT \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−28.81400197641957781854246347197, −27.61941872141063775711720928176, −27.2245161437101083429114698316, −26.04311816265569544952592134814, −24.795552879529797500932037532372, −24.07633881158172082412073429721, −23.43192205716674987066113579823, −22.05840928442179127624886159028, −21.12962900748328555669921798049, −19.58425919178591940429805006369, −18.650029538389801836510296415574, −17.50987415190272064896131944017, −16.45704476880794750489051411152, −15.554013176400746595633555310367, −14.65337022947350284861904700825, −13.47327891805675339023062121206, −12.207109986289325431817870035637, −11.061315216511918413909558470209, −9.22144651895436646823859214819, −8.296066334047723479203034361054, −7.507527441945038553555398288616, −5.84243119801184902973170819380, −4.86344352684280158747853970870, −3.53560117983593107183470174613, −0.919373630672993858600976197177, 0.98518963186528257293237542908, 2.668814303577126809469830376259, 4.04203215054343660449581065851, 5.013693258012522825347839068950, 7.22380071786761325630796967246, 8.17621104566678328392764276865, 9.69978318801586591120644352939, 10.78850308055680167564306396990, 11.670198798783784526723373830978, 12.58991995980910545705319533634, 14.14083696589439770440349898192, 14.78114424408115925203210005997, 16.3847823165433068802740208875, 17.86402098589956761803669644717, 18.45144635414476343519806317040, 19.801329420765340002325717704199, 20.41334596101702709982693832419, 21.47174129817429488051104503907, 22.99273074215333986244553253440, 23.08191066774540758412489832859, 24.60524282141720557368965283908, 26.197985969550915833648913329, 27.12989301608611061072362153081, 27.65911054323044118544402341733, 28.85198792771478494189819051147

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