Properties

Label 2-6e3-1.1-c3-0-4
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  + 13.2·5-s − 5.23·7-s − 11-s + 84.9·13-s − 40.9·17-s + 57.5·19-s − 114.·23-s + 50.1·25-s + 202.·29-s + 274.·31-s − 69.2·35-s + 242.·37-s + 328.·41-s + 281.·43-s − 23.8·47-s − 315.·49-s − 300.·53-s − 13.2·55-s − 753.·59-s + 495.·61-s + 1.12e3·65-s − 409.·67-s − 1.11e3·71-s − 287·73-s + 5.23·77-s − 1.23e3·79-s + 942.·83-s + ⋯
L(s)  = 1  + 1.18·5-s − 0.282·7-s − 0.0274·11-s + 1.81·13-s − 0.584·17-s + 0.694·19-s − 1.04·23-s + 0.401·25-s + 1.29·29-s + 1.58·31-s − 0.334·35-s + 1.07·37-s + 1.25·41-s + 0.997·43-s − 0.0740·47-s − 0.920·49-s − 0.777·53-s − 0.0324·55-s − 1.66·59-s + 1.03·61-s + 2.14·65-s − 0.747·67-s − 1.86·71-s − 0.460·73-s + 0.00774·77-s − 1.75·79-s + 1.24·83-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

\(L(2)\) \(\approx\) \(2.234462073\)
\(L(\frac12)\) \(\approx\) \(2.234462073\)
\(L(\frac{5}{2})\) 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 \( 1 - 13.2T + 125T^{2} \)
7 \( 1 + 5.23T + 343T^{2} \)
11 \( 1 + T + 1.33e3T^{2} \)
13 \( 1 - 84.9T + 2.19e3T^{2} \)
17 \( 1 + 40.9T + 4.91e3T^{2} \)
19 \( 1 - 57.5T + 6.85e3T^{2} \)
23 \( 1 + 114.T + 1.21e4T^{2} \)
29 \( 1 - 202.T + 2.43e4T^{2} \)
31 \( 1 - 274.T + 2.97e4T^{2} \)
37 \( 1 - 242.T + 5.06e4T^{2} \)
41 \( 1 - 328.T + 6.89e4T^{2} \)
43 \( 1 - 281.T + 7.95e4T^{2} \)
47 \( 1 + 23.8T + 1.03e5T^{2} \)
53 \( 1 + 300.T + 1.48e5T^{2} \)
59 \( 1 + 753.T + 2.05e5T^{2} \)
61 \( 1 - 495.T + 2.26e5T^{2} \)
67 \( 1 + 409.T + 3.00e5T^{2} \)
71 \( 1 + 1.11e3T + 3.57e5T^{2} \)
73 \( 1 + 287T + 3.89e5T^{2} \)
79 \( 1 + 1.23e3T + 4.93e5T^{2} \)
83 \( 1 - 942.T + 5.71e5T^{2} \)
89 \( 1 - 190.T + 7.04e5T^{2} \)
97 \( 1 + 306.T + 9.12e5T^{2} \)
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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

−11.83835515331070642536891061116, −10.78393469598445035558312701337, −9.905074771040248132470443821665, −9.031588197240277705946068079454, −7.982197569423095370460507509881, −6.34620304303927934957474450570, −5.95219174120883752919284721674, −4.35880564921701000208727633144, −2.80275563037754282454914468864, −1.27077781699008545401896784942, 1.27077781699008545401896784942, 2.80275563037754282454914468864, 4.35880564921701000208727633144, 5.95219174120883752919284721674, 6.34620304303927934957474450570, 7.982197569423095370460507509881, 9.031588197240277705946068079454, 9.905074771040248132470443821665, 10.78393469598445035558312701337, 11.83835515331070642536891061116

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