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Properties

Label 2-6e3-72.67-c0-0-0
Degree $2$
Conductor $216$
Sign $0.342 - 0.939i$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 0.999·8-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)16-s + 17-s − 19-s + (0.499 − 0.866i)22-s + (−0.5 − 0.866i)25-s + (0.499 − 0.866i)32-s + (0.5 + 0.866i)34-s + (−0.5 − 0.866i)38-s + (−0.5 + 0.866i)41-s + (0.5 + 0.866i)43-s + 0.999·44-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 0.999·8-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)16-s + 17-s − 19-s + (0.499 − 0.866i)22-s + (−0.5 − 0.866i)25-s + (0.499 − 0.866i)32-s + (0.5 + 0.866i)34-s + (−0.5 − 0.866i)38-s + (−0.5 + 0.866i)41-s + (0.5 + 0.866i)43-s + 0.999·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.342 - 0.939i$
Analytic conductor: \(0.107798\)
Root analytic conductor: \(0.328326\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :0),\ 0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8257943004\)
\(L(\frac12)\) \(\approx\) \(0.8257943004\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−12.89328439347955743801991004717, −12.05278273691497617937970632253, −10.89885766179764084292457815165, −9.666611710270148306101285950825, −8.440327092698017982364765255790, −7.78871758008494411283579834540, −6.45581861224275953103461660851, −5.60946620480892776908866980382, −4.34688403368470327698843044606, −2.99953893435201362555814620289, 2.02895605873748635540766396231, 3.54587282395174421740539371584, 4.80359899466792150821964388887, 5.85262563460310281494664278878, 7.27330252395249658553950690551, 8.628332674260032220341229115632, 9.785940083012428641225926651121, 10.45068527584334776278651051760, 11.50416066905037812463826442051, 12.43047215731515122245881153187

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