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Properties

Label 1-116-116.11-r0-0-0
Degree $1$
Conductor $116$
Sign $0.886 - 0.462i$
Analytic cond. $0.538701$
Root an. cond. $0.538701$
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  + (0.974 − 0.222i)3-s + (−0.623 − 0.781i)5-s + (0.222 + 0.974i)7-s + (0.900 − 0.433i)9-s + (0.433 − 0.900i)11-s + (0.900 + 0.433i)13-s + (−0.781 − 0.623i)15-s − i·17-s + (−0.974 − 0.222i)19-s + (0.433 + 0.900i)21-s + (−0.623 + 0.781i)23-s + (−0.222 + 0.974i)25-s + (0.781 − 0.623i)27-s + (−0.781 + 0.623i)31-s + (0.222 − 0.974i)33-s + ⋯
L(s)  = 1  + (0.974 − 0.222i)3-s + (−0.623 − 0.781i)5-s + (0.222 + 0.974i)7-s + (0.900 − 0.433i)9-s + (0.433 − 0.900i)11-s + (0.900 + 0.433i)13-s + (−0.781 − 0.623i)15-s − i·17-s + (−0.974 − 0.222i)19-s + (0.433 + 0.900i)21-s + (−0.623 + 0.781i)23-s + (−0.222 + 0.974i)25-s + (0.781 − 0.623i)27-s + (−0.781 + 0.623i)31-s + (0.222 − 0.974i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $0.886 - 0.462i$
Analytic conductor: \(0.538701\)
Root analytic conductor: \(0.538701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 116,\ (0:\ ),\ 0.886 - 0.462i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.307249510 - 0.3201338209i\)
\(L(\frac12)\) \(\approx\) \(1.307249510 - 0.3201338209i\)
\(L(1)\) \(\approx\) \(1.292248926 - 0.1943983457i\)
\(L(1)\) \(\approx\) \(1.292248926 - 0.1943983457i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 \)
good3 \( 1 + (0.974 - 0.222i)T \)
5 \( 1 + (-0.623 - 0.781i)T \)
7 \( 1 + (0.222 + 0.974i)T \)
11 \( 1 + (0.433 - 0.900i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.974 - 0.222i)T \)
23 \( 1 + (-0.623 + 0.781i)T \)
31 \( 1 + (-0.781 + 0.623i)T \)
37 \( 1 + (-0.433 - 0.900i)T \)
41 \( 1 + iT \)
43 \( 1 + (0.781 + 0.623i)T \)
47 \( 1 + (-0.433 + 0.900i)T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.974 + 0.222i)T \)
67 \( 1 + (-0.900 + 0.433i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (0.781 + 0.623i)T \)
79 \( 1 + (-0.433 - 0.900i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (0.781 - 0.623i)T \)
97 \( 1 + (-0.974 - 0.222i)T \)
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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

−29.76610964077596693772999083631, −27.87843210080783993620912160622, −27.29211430774194683729909447643, −26.01236713519890114743245363596, −25.7824803612755216186597122518, −24.19322505742120946253612411608, −23.22440902803753855654906400456, −22.22060215583718068634773986953, −20.84763841898262324600090163184, −20.07109837741547492898803812142, −19.20936829069354899837715795812, −18.101275611897587132946982280972, −16.74446571769234075197286024532, −15.36670563958490819922822509598, −14.72757205453436585587161001340, −13.71418599549701408024389473095, −12.48611525073086817233170148603, −10.80483492156841661410958753313, −10.18448357223475635166067385364, −8.55175741467920199115419576435, −7.631561790627214480696833297328, −6.557461705258427812991665074073, −4.249437637820645917830486053278, −3.62928956367148749374754758404, −1.93661263316342968295386265094, 1.55004184081524912688968710140, 3.167077396782535221799818549070, 4.41303074966667331474062109264, 6.0244267258889274748641788854, 7.63721617213023875853255641084, 8.75875904731175674616780258842, 9.16022330481072603968008660475, 11.22476169346061689949562230498, 12.24801785295124339876167755089, 13.34992757315985594437421154663, 14.41444844647345067074079425595, 15.618828579820764680077715169129, 16.310169926879952570351356800691, 18.04012002408494034738715137380, 19.03508318931086568748541288840, 19.78915259554796104891319003233, 20.966293152329827183273489566994, 21.64258535837937505374845606534, 23.3416933940071219497777293432, 24.31636567518075729263547120463, 24.99365281643528875261688938080, 26.00164576295474870252026279466, 27.28661478543945340310651756973, 27.91280911376469474333953711667, 29.22383129522989676864100269994

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