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Properties

Label 1-6e3-216.11-r0-0-0
Degree $1$
Conductor $216$
Sign $0.686 + 0.727i$
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  + (0.766 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.686 + 0.727i$
Analytic conductor: \(1.00309\)
Root analytic conductor: \(1.00309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 216,\ (0:\ ),\ 0.686 + 0.727i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.268122606 + 0.5470146968i\)
\(L(\frac12)\) \(\approx\) \(1.268122606 + 0.5470146968i\)
\(L(1)\) \(\approx\) \(1.197619251 + 0.2601451590i\)
\(L(1)\) \(\approx\) \(1.197619251 + 0.2601451590i\)

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

−26.45933387410720418989663114180, −25.47442318319188969966824416328, −24.51445005261646408346636795590, −23.90026463463450681529274536164, −22.78949119341890716930199013382, −21.44978457075182898250025685724, −20.99623789137178033832630597396, −20.099843584035680250103106426539, −18.81689777394834663085278787174, −17.74905765630233018729229407398, −17.12092594618797500777593488919, −16.08593718377447783812022796530, −14.85407372895063785326643116987, −13.89441259981947991492683634111, −13.00822292991836650137012124533, −12.010409869493434611441406404, −10.583616708332280422061452308508, −10.017188839940767777521412191804, −8.32780571566543582021803378251, −8.00651875428176034555795542012, −6.11424828563414607703560429453, −5.335718804329431230215902052919, −4.145646890779517865769388121773, −2.48280313857338781008850559355, −1.15549354846413981429751721465, 1.827614631225820624040073901087, 2.70532858503934653811895744192, 4.518365378385205997059154925031, 5.477341428453428116395716871409, 6.76222969882995597077275349586, 7.742959640228015209399548932001, 9.0885521324710452839907790588, 10.04894145133898482394281894, 11.10465434426823546669530539772, 12.04093945566273377424368150862, 13.38303234693537958872459010639, 14.27204887695825969909181790962, 15.029987861382077316566655952751, 16.21099666453937053240402845417, 17.52305013004852141166198197601, 18.07950207999695490551922553763, 18.93620666743768793622325056980, 20.29286139160743169520319558321, 21.31079668556209882176154145867, 21.765641357764172846327867537495, 23.02513075948709162144239620681, 23.92458176689139710012376916993, 24.917591597621569843191616678, 25.86315855490148361191312004911, 26.52796874653015166384905566017

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