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Properties

Label 1-108-108.11-r0-0-0
Degree $1$
Conductor $108$
Sign $0.835 - 0.549i$
Analytic cond. $0.501549$
Root an. cond. $0.501549$
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  + (−0.766 − 0.642i)5-s + (0.939 + 0.342i)7-s + (0.766 − 0.642i)11-s + (0.173 − 0.984i)13-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.939 + 0.342i)23-s + (0.173 + 0.984i)25-s + (−0.173 − 0.984i)29-s + (0.939 − 0.342i)31-s + (−0.5 − 0.866i)35-s + (−0.5 + 0.866i)37-s + (−0.173 + 0.984i)41-s + (−0.766 + 0.642i)43-s + (−0.939 − 0.342i)47-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)5-s + (0.939 + 0.342i)7-s + (0.766 − 0.642i)11-s + (0.173 − 0.984i)13-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.939 + 0.342i)23-s + (0.173 + 0.984i)25-s + (−0.173 − 0.984i)29-s + (0.939 − 0.342i)31-s + (−0.5 − 0.866i)35-s + (−0.5 + 0.866i)37-s + (−0.173 + 0.984i)41-s + (−0.766 + 0.642i)43-s + (−0.939 − 0.342i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.835 - 0.549i$
Analytic conductor: \(0.501549\)
Root analytic conductor: \(0.501549\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 108,\ (0:\ ),\ 0.835 - 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9781863630 - 0.2928497729i\)
\(L(\frac12)\) \(\approx\) \(0.9781863630 - 0.2928497729i\)
\(L(1)\) \(\approx\) \(1.025005483 - 0.1592095066i\)
\(L(1)\) \(\approx\) \(1.025005483 - 0.1592095066i\)

Euler product

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

−30.12070512759085710523731113850, −28.415131632699486066197216579806, −27.63174101446545766676872881189, −26.62519287666737866555622432874, −25.803695909608090658834926606, −24.27357030743845203575702864751, −23.62312323181905468157736124069, −22.49213361177083335262626924130, −21.46220996185484522384344531136, −20.19975500986039200667140076880, −19.31826599738831847881742790100, −18.15631018912069960544404195038, −17.15676916838782564071850111305, −15.858913714850679535663833203952, −14.674179593788947058137493079247, −14.04034171315031165829410442265, −12.22231507751378645721011780220, −11.40700028095423065824732017378, −10.325580970182756982295893078262, −8.77007212577008655132246458754, −7.55111390298205282364817430699, −6.57761289210904084081951771237, −4.69647296061458142007018171589, −3.6576536444983275868107787277, −1.760824527843204233887955414541, 1.249769475689516605608505998837, 3.32819997943692241393208766460, 4.6996792813107099634099913695, 5.88605527699139420812814896228, 7.78401711334032457969058111793, 8.39231639576046258993232151842, 9.84310932826649050421255709548, 11.515222848116827289893963218868, 11.969715349260240286350523334217, 13.49981968148518016116211094578, 14.68693284193660419381135588300, 15.76805559360681256388212120855, 16.77975290495532004701541605328, 17.97395724159464233640015625744, 19.08504268629091930721769525902, 20.25459518484623184732394144378, 20.9904292860498972934327451993, 22.34511957349440518819194234530, 23.361891040704692026545303762200, 24.559973745404836896183413159476, 24.9704376304836701847227387807, 26.74479581567678296068338620819, 27.54292939421717311246029719458, 28.13628295138077508893888006133, 29.62279171888400572099173645639

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