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Properties

Label 2-6e3-216.211-c0-0-0
Degree $2$
Conductor $216$
Sign $0.230 + 0.973i$
Analytic cond. $0.107798$
Root an. cond. $0.328326$
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)2-s + (−0.939 − 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.939 + 0.342i)6-s + (−0.500 − 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.326 − 0.118i)11-s + (−0.499 + 0.866i)12-s + (−0.939 − 0.342i)16-s + (−0.766 + 1.32i)17-s + 18-s + (0.939 + 1.62i)19-s + (−0.326 + 0.118i)22-s + (0.173 + 0.984i)24-s + (0.766 − 0.642i)25-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (−0.939 − 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.939 + 0.342i)6-s + (−0.500 − 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.326 − 0.118i)11-s + (−0.499 + 0.866i)12-s + (−0.939 − 0.342i)16-s + (−0.766 + 1.32i)17-s + 18-s + (0.939 + 1.62i)19-s + (−0.326 + 0.118i)22-s + (0.173 + 0.984i)24-s + (0.766 − 0.642i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.230 + 0.973i$
Analytic conductor: \(0.107798\)
Root analytic conductor: \(0.328326\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :0),\ 0.230 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7913401073\)
\(L(\frac12)\) \(\approx\) \(0.7913401073\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
3 \( 1 + (0.939 + 0.342i)T \)
good5 \( 1 + (-0.766 + 0.642i)T^{2} \)
7 \( 1 + (0.939 - 0.342i)T^{2} \)
11 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
13 \( 1 + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.939 + 0.342i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
43 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 - 0.342i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.32953517194279755701245092231, −11.57148343112639087956361182369, −10.56101681617682719310884209506, −10.02325748065572413581706343970, −8.337953635355256334377047693001, −6.87212787298639068654634342695, −5.94026093401960317177308463747, −5.00456087207015534385987498116, −3.69748497958035591559088146679, −1.78266404823769550531317073504, 3.07553742771147273664702880928, 4.72744834922822389289818147415, 5.21108410862649946687208184609, 6.65256265235872878193443521054, 7.23380947614325617451175746356, 8.782026577647267199560345712244, 9.867880909656380328133404837866, 11.43014107959455302546820777587, 11.53446718819248208947848412418, 12.97332564740030393015892615720

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