Properties

Label 2-6e3-216.115-c0-0-0
Degree $2$
Conductor $216$
Sign $0.727 + 0.686i$
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  + (−0.939 + 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (0.173 + 0.984i)6-s + (−0.500 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.266 − 1.50i)11-s + (−0.5 − 0.866i)12-s + (0.173 − 0.984i)16-s + (0.939 + 1.62i)17-s + 0.999·18-s + (−0.173 + 0.300i)19-s + (0.266 + 1.50i)22-s + (0.766 + 0.642i)24-s + (−0.939 + 0.342i)25-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (0.173 + 0.984i)6-s + (−0.500 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.266 − 1.50i)11-s + (−0.5 − 0.866i)12-s + (0.173 − 0.984i)16-s + (0.939 + 1.62i)17-s + 0.999·18-s + (−0.173 + 0.300i)19-s + (0.266 + 1.50i)22-s + (0.766 + 0.642i)24-s + (−0.939 + 0.342i)25-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.727 + 0.686i$
Analytic conductor: \(0.107798\)
Root analytic conductor: \(0.328326\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :0),\ 0.727 + 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5069317621\)
\(L(\frac12)\) \(\approx\) \(0.5069317621\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 - 0.342i)T \)
3 \( 1 + (-0.173 + 0.984i)T \)
good5 \( 1 + (0.939 - 0.342i)T^{2} \)
7 \( 1 + (-0.173 - 0.984i)T^{2} \)
11 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
13 \( 1 + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (-0.173 + 0.984i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.173 - 0.984i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
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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

−12.29254082495383292127253192517, −11.39380713297973286218062862119, −10.51065905503647610025758177583, −9.220764964272969675686154178365, −8.263283939764319724125817982648, −7.73169680743094807641806986817, −6.28765311599362488319783499160, −5.82766409258450834246922328554, −3.26958848221300922261022230232, −1.50292278577570919072792984455, 2.40222102885017417011513209495, 3.83801659828596263560765615994, 5.19134054502797220968515953638, 6.89665279786183129393175293035, 7.87354725130302796243081668553, 9.087986225846525406667411900895, 9.765531946411498251344924338403, 10.39022260009512089582157799522, 11.63939972661834752743021834987, 12.16673406637422495019177931662

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