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Properties

Label 1-108-108.79-r1-0-0
Degree $1$
Conductor $108$
Sign $-0.998 + 0.0581i$
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  + (0.173 − 0.984i)5-s + (−0.766 + 0.642i)7-s + (−0.173 − 0.984i)11-s + (−0.939 + 0.342i)13-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.766 − 0.642i)23-s + (−0.939 − 0.342i)25-s + (−0.939 − 0.342i)29-s + (−0.766 − 0.642i)31-s + (0.5 + 0.866i)35-s + (−0.5 + 0.866i)37-s + (−0.939 + 0.342i)41-s + (−0.173 − 0.984i)43-s + (−0.766 + 0.642i)47-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)5-s + (−0.766 + 0.642i)7-s + (−0.173 − 0.984i)11-s + (−0.939 + 0.342i)13-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.766 − 0.642i)23-s + (−0.939 − 0.342i)25-s + (−0.939 − 0.342i)29-s + (−0.766 − 0.642i)31-s + (0.5 + 0.866i)35-s + (−0.5 + 0.866i)37-s + (−0.939 + 0.342i)41-s + (−0.173 − 0.984i)43-s + (−0.766 + 0.642i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $-0.998 + 0.0581i$
Analytic conductor: \(11.6062\)
Root analytic conductor: \(11.6062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 108,\ (1:\ ),\ -0.998 + 0.0581i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005637261771 - 0.1937401210i\)
\(L(\frac12)\) \(\approx\) \(0.005637261771 - 0.1937401210i\)
\(L(1)\) \(\approx\) \(0.7235793239 - 0.1140811864i\)
\(L(1)\) \(\approx\) \(0.7235793239 - 0.1140811864i\)

Euler product

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

−29.74635925789040377345773643821, −29.08969203951719946323477258317, −27.73414798980968646765029455077, −26.54255184788653495265750707981, −25.973198980899487806354745978560, −24.86177976216975136165683510171, −23.4633033767647995644838045612, −22.557367521925882157751744576078, −21.89486190629464450455717004637, −20.25027518748961521058972650563, −19.5844136742578715443194516898, −18.20349815780853281037549901037, −17.45363671726147112630189319954, −16.034921298204657974628586993550, −14.99650547919758668887117301046, −13.89556849194969727723303484267, −12.82352993759014476544418250017, −11.44117890713726345717632001524, −10.19506859669165869852358685388, −9.47740731051016530505740752021, −7.37710053254659487260793320781, −6.89423451303940983228257303829, −5.199022084705281823265089352696, −3.57586754371068264674591413487, −2.29939118186261670554638935834, 0.076214352714448403648577156021, 2.03275442250904354127102411597, 3.72868144414915311031690910159, 5.2927764476879315039427495817, 6.28268131890169960425328345994, 8.05817056872673716679972459811, 9.07203235062176204844666447760, 10.106225012741983650443688449067, 11.773866415050771834396547040716, 12.67032383515752773580531483536, 13.66727118081698792288103965372, 15.10285090121829470917661717770, 16.32214356563862837394017824006, 16.926197704315416200575329276271, 18.462397685567159037813237448500, 19.41510049183937675054721244310, 20.46622259454951333705270137679, 21.65635766369202777938886961685, 22.382113103082186573242603597199, 24.02101826281214550130592242513, 24.50926255573429368384020091912, 25.69214365309112308338930389662, 26.74841584224425304909986131414, 27.94662714904441256955095044664, 28.894541038141320034943665737529

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