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Properties

Label 1-6e3-216.5-r1-0-0
Degree $1$
Conductor $216$
Sign $0.686 - 0.727i$
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

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

Invariants

Degree: \(1\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.686 - 0.727i$
Analytic conductor: \(23.2124\)
Root analytic conductor: \(23.2124\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 216,\ (1:\ ),\ 0.686 - 0.727i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.834222606 - 0.7912064003i\)
\(L(\frac12)\) \(\approx\) \(1.834222606 - 0.7912064003i\)
\(L(1)\) \(\approx\) \(1.207872582 - 0.1923071876i\)
\(L(1)\) \(\approx\) \(1.207872582 - 0.1923071876i\)

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

−26.602856010910219127135560941362, −25.430920112189142679438354481545, −24.926947908700402746494316308124, −23.54933883598797461260493419229, −22.623417661781779605349509638172, −21.90606512837551964216086290157, −20.99096704425457339564698309546, −19.759647197168316565972532680998, −18.87455424954052070226308550811, −18.11198436541260357130435406511, −16.63715110961780689543705599063, −16.41231279439144275621883474605, −14.653461479114922515695063879957, −14.06562468556871525600980242722, −13.09501072932311939331079160949, −11.85870215038422482874878741932, −10.76657169946681222392640474608, −9.69591192559409418162573509823, −9.022417308754101195237195656798, −7.223830253823847679147784515945, −6.536334460571880242786126917260, −5.44330424402104570075139817038, −3.77364855598044453752445745318, −2.78655786191821494962425210553, −1.19480017725069802211493235598, 0.779372974517978715159466357392, 2.260981958242089252785598355016, 3.61682136695616948492997046097, 5.11388979594427503187967898231, 6.018334100858587439959676058758, 7.15744594058662294160572812960, 8.6400944870100799352217973519, 9.5313037718519903559446830187, 10.27258942779747494468119174095, 11.8583820371071143043184853142, 12.81025187034163786529731998928, 13.42295884063764475620613250249, 14.82347596648283438265287782446, 15.681570572448776314198137060334, 16.957469663326197315189254579515, 17.42574849700697958118739839759, 18.70223614293459600233939718652, 19.78390666602725731969178385876, 20.47531710325868047308965083188, 21.726803180856186108539796874138, 22.33314739707069353533963636204, 23.40462094482128446883152278873, 24.62215248640747636196767543744, 25.33809438439529739804273584709, 25.92048907402230761421599178578

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