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Properties

Label 1-108-108.31-r1-0-0
Degree $1$
Conductor $108$
Sign $0.549 - 0.835i$
Analytic cond. $11.6062$
Root an. cond. $11.6062$
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  + (−0.939 − 0.342i)5-s + (−0.173 + 0.984i)7-s + (0.939 − 0.342i)11-s + (0.766 − 0.642i)13-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.173 − 0.984i)23-s + (0.766 + 0.642i)25-s + (0.766 + 0.642i)29-s + (−0.173 − 0.984i)31-s + (0.5 − 0.866i)35-s + (−0.5 − 0.866i)37-s + (0.766 − 0.642i)41-s + (0.939 − 0.342i)43-s + (−0.173 + 0.984i)47-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)5-s + (−0.173 + 0.984i)7-s + (0.939 − 0.342i)11-s + (0.766 − 0.642i)13-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.173 − 0.984i)23-s + (0.766 + 0.642i)25-s + (0.766 + 0.642i)29-s + (−0.173 − 0.984i)31-s + (0.5 − 0.866i)35-s + (−0.5 − 0.866i)37-s + (0.766 − 0.642i)41-s + (0.939 − 0.342i)43-s + (−0.173 + 0.984i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.549 - 0.835i$
Analytic conductor: \(11.6062\)
Root analytic conductor: \(11.6062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 108,\ (1:\ ),\ 0.549 - 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.189251303 - 0.6412386004i\)
\(L(\frac12)\) \(\approx\) \(1.189251303 - 0.6412386004i\)
\(L(1)\) \(\approx\) \(0.9731570991 - 0.1504558005i\)
\(L(1)\) \(\approx\) \(0.9731570991 - 0.1504558005i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (-0.173 + 0.984i)T \)
11 \( 1 + (0.939 - 0.342i)T \)
13 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.766 + 0.642i)T \)
31 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 + T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.766 - 0.642i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−29.63325443394774628337953948060, −28.373245199265762073756971733438, −27.3158780573286748230264881823, −26.55304569682112847742593119980, −25.545417332170209265273039539117, −24.13303543178377285971294863215, −23.26856844292176330873936385898, −22.52524799990521403359446433382, −21.13822676709917034785733770910, −19.85444568946098821154671448272, −19.35218337027048608295789240435, −17.947844336316793439773152901547, −16.76144764959333878907932169190, −15.80201539706577086120141189415, −14.585788631825454802247407106918, −13.59495579786468517595228008656, −12.1328507614643398053371922105, −11.18322745720348536720148692563, −10.038873688854424447360882151414, −8.54934617177710374277179483399, −7.32735391125371678154693504264, −6.34965155622785947510178563416, −4.28026828751413952289730592408, −3.555400103466167621950082681, −1.31094625666797136036849037441, 0.680687088231189284268108572670, 2.80403041872457297747875547643, 4.19097742746192047116465086475, 5.61509364926853673818797126178, 7.004850430268229635814406512419, 8.48188260269982472038780689999, 9.18820944222679990475902429333, 11.02818836083804877697234710222, 11.899840527012818289094531271876, 12.91521577864121623051154712944, 14.36628646427153827923281127646, 15.63144347208411982109909046146, 16.16919237258197468261990485639, 17.72025006754034693955602524146, 18.79752865817135803171201603197, 19.77156441117551496363833014901, 20.71910470457140240515967910564, 22.15815571703681785970861664500, 22.81262139186725579876826336344, 24.23780323570191780682048220840, 24.8498872558269702108868822698, 26.1199189165170325909820233388, 27.387872795489671676112725100682, 27.96937271878471295562191865227, 29.02102434545339495037812956689

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