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Properties

Label 1-6e3-216.211-r1-0-0
Degree $1$
Conductor $216$
Sign $0.727 + 0.686i$
Analytic cond. $23.2124$
Root an. cond. $23.2124$
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

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

Invariants

Degree: \(1\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.727 + 0.686i$
Analytic conductor: \(23.2124\)
Root analytic conductor: \(23.2124\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 216,\ (1:\ ),\ 0.727 + 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.660714579 + 0.6597596871i\)
\(L(\frac12)\) \(\approx\) \(1.660714579 + 0.6597596871i\)
\(L(1)\) \(\approx\) \(1.109807176 + 0.1602385115i\)
\(L(1)\) \(\approx\) \(1.109807176 + 0.1602385115i\)

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

−26.52901283286057048113404882551, −25.08894792890982654379832195406, −24.15792721497486815197231671337, −23.85286614444625725143426636791, −22.47134285618835466146452088500, −21.42977375024282914622454545709, −20.73528172142340475753379742126, −19.41811662375969589498974722530, −19.03328857240505585161692651553, −17.44732589452591486540297497305, −16.86602637020643519779423873378, −15.64291013823818750779992459729, −14.83463515198730093028981327278, −13.75448391841100398104774279308, −12.53004420525508975196236553555, −11.56782422959004749064932249529, −10.95255770548146096689399465840, −9.08010409701764847902111064730, −8.61073971844785608470792289028, −7.40874761098979858606476364478, −6.1137415332297385374679283193, −4.698563549091719734726264380299, −3.99055145005761725836292442810, −2.177490773341472543284244633512, −0.75311994642897371661002012710, 1.098175829357173356078595940677, 2.73892936431059408688312843515, 4.03514373915112041608805478629, 5.034704510142316072294222800736, 6.66388303488027109868707712489, 7.54377311436314958174630471449, 8.46562654509946618002347840692, 9.920583543011664656661675432763, 10.98132646429128852052376158017, 11.71858739320161373118154431794, 12.83185639713039271999610137433, 14.28232290123929751990126805362, 14.84301971818051583389769514708, 15.79915395124465766983287625699, 17.13286444103056482104631060744, 17.90452595835015337040346827981, 18.904425338359912389799087757571, 19.99792115723206811001455469805, 20.6172902734845487043504207472, 21.9504002050570559173691534414, 22.86117919579912400642118719929, 23.477646355410294690969058569374, 24.70546257896656806563644165460, 25.40914526121826128349415688343, 26.830253043589087226150196846467

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