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Properties

Label 1-6e3-216.133-r0-0-0
Degree $1$
Conductor $216$
Sign $0.957 + 0.286i$
Analytic cond. $1.00309$
Root an. cond. $1.00309$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.957 + 0.286i$
Analytic conductor: \(1.00309\)
Root analytic conductor: \(1.00309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 216,\ (0:\ ),\ 0.957 + 0.286i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.257866916 + 0.1842503730i\)
\(L(\frac12)\) \(\approx\) \(1.257866916 + 0.1842503730i\)
\(L(1)\) \(\approx\) \(1.135363809 + 0.09577605295i\)
\(L(1)\) \(\approx\) \(1.135363809 + 0.09577605295i\)

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

−26.64544518685096762244152203042, −25.38212458959143275535756928205, −24.72539320865827943825883182628, −23.837587186255422669056248375581, −22.94590534617130625927546544517, −21.74771674130143027385851456529, −20.67940107446380854743517370848, −20.33015109574250703128468501411, −18.91794269145315822670708007075, −17.95040889521677191077228052588, −17.14587123432016922549978538219, −15.856792212076318351062470700798, −15.317149719914611086127217145263, −13.93976983923118826841834393541, −12.97559519831054976245828938300, −11.930254069834562469692060382908, −11.19297943365634251775486933950, −9.6116306914776663179135668210, −8.77022763049119369400888738970, −7.85560748970738410821159479808, −6.48863941473506986484563432000, −5.002317028958204641224530108689, −4.50203358369391784250635296555, −2.61755500908425245293553670510, −1.24994491370326922680253222886, 1.36992684820515898410561699280, 3.075750905591395117690663826881, 3.99633353539115871576593529286, 5.56570127435706559664697645682, 6.64266898929762676301546705797, 7.799062280184697706659853336845, 8.63679167029275999749472503554, 10.39082709952588641905667309733, 10.85196116421502373175456754375, 11.831813300231638549779951996856, 13.4162630020725314433630169463, 14.07052058418697392133617903678, 15.09421142517076935948597231673, 16.053586712022488101087773817742, 17.27530335273719602714179881277, 18.15190780081587066203873968946, 18.9942067186433439989229575748, 20.01230855770255415142473746397, 21.143064313891686317183484310679, 21.855998048196325018763399448974, 23.13809898026275230811989582707, 23.589177606125501275310849147469, 24.79578001885283717632607509163, 25.80685209660666627360951614566, 26.95188956029754060523053423184

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