Properties

Label 2-6e3-1.1-c3-0-2
Degree $2$
Conductor $216$
Sign $1$
Analytic cond. $12.7444$
Root an. cond. $3.56993$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 21.2·5-s + 29.2·7-s − 11-s − 52.9·13-s + 96.9·17-s + 126.·19-s + 22.9·23-s + 325.·25-s + 133.·29-s + 101.·31-s − 620.·35-s + 105.·37-s − 16.3·41-s − 201.·43-s + 251.·47-s + 511.·49-s + 148.·53-s + 21.2·55-s + 73.6·59-s − 607.·61-s + 1.12e3·65-s + 761.·67-s − 701.·71-s − 287·73-s − 29.2·77-s − 128.·79-s − 160.·83-s + ⋯
L(s)  = 1  − 1.89·5-s + 1.57·7-s − 0.0274·11-s − 1.12·13-s + 1.38·17-s + 1.52·19-s + 0.207·23-s + 2.60·25-s + 0.855·29-s + 0.589·31-s − 2.99·35-s + 0.466·37-s − 0.0622·41-s − 0.713·43-s + 0.781·47-s + 1.49·49-s + 0.383·53-s + 0.0520·55-s + 0.162·59-s − 1.27·61-s + 2.14·65-s + 1.38·67-s − 1.17·71-s − 0.460·73-s − 0.0432·77-s − 0.183·79-s − 0.212·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(12.7444\)
Root analytic conductor: \(3.56993\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.461604606\)
\(L(\frac12)\) \(\approx\) \(1.461604606\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + 21.2T + 125T^{2} \)
7 \( 1 - 29.2T + 343T^{2} \)
11 \( 1 + T + 1.33e3T^{2} \)
13 \( 1 + 52.9T + 2.19e3T^{2} \)
17 \( 1 - 96.9T + 4.91e3T^{2} \)
19 \( 1 - 126.T + 6.85e3T^{2} \)
23 \( 1 - 22.9T + 1.21e4T^{2} \)
29 \( 1 - 133.T + 2.43e4T^{2} \)
31 \( 1 - 101.T + 2.97e4T^{2} \)
37 \( 1 - 105.T + 5.06e4T^{2} \)
41 \( 1 + 16.3T + 6.89e4T^{2} \)
43 \( 1 + 201.T + 7.95e4T^{2} \)
47 \( 1 - 251.T + 1.03e5T^{2} \)
53 \( 1 - 148.T + 1.48e5T^{2} \)
59 \( 1 - 73.6T + 2.05e5T^{2} \)
61 \( 1 + 607.T + 2.26e5T^{2} \)
67 \( 1 - 761.T + 3.00e5T^{2} \)
71 \( 1 + 701.T + 3.57e5T^{2} \)
73 \( 1 + 287T + 3.89e5T^{2} \)
79 \( 1 + 128.T + 4.93e5T^{2} \)
83 \( 1 + 160.T + 5.71e5T^{2} \)
89 \( 1 + 430.T + 7.04e5T^{2} \)
97 \( 1 + 31.1T + 9.12e5T^{2} \)
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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

−11.87155239491171279064759460021, −11.22681793610697157558217488445, −10.05377568913918971547139798921, −8.549719304513415609985442299902, −7.72568856430581117914485565018, −7.33661269288757715741243858205, −5.20682554496329626333244700609, −4.44422655077233422999691416673, −3.10120864927989580428109827646, −0.955214406059313780669523281826, 0.955214406059313780669523281826, 3.10120864927989580428109827646, 4.44422655077233422999691416673, 5.20682554496329626333244700609, 7.33661269288757715741243858205, 7.72568856430581117914485565018, 8.549719304513415609985442299902, 10.05377568913918971547139798921, 11.22681793610697157558217488445, 11.87155239491171279064759460021

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