Properties

Label 2-72e2-1.1-c1-0-57
Degree $2$
Conductor $5184$
Sign $-1$
Analytic cond. $41.3944$
Root an. cond. $6.43385$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 1.73·7-s + 1.73·11-s + 3·13-s − 4·17-s − 6.92·19-s + 8.66·23-s − 4·25-s + 29-s + 5.19·31-s + 1.73·35-s + 8·37-s − 5·41-s + 8.66·43-s − 12.1·47-s − 4·49-s − 8·53-s − 1.73·55-s + 1.73·59-s + 7·61-s − 3·65-s + 8.66·67-s − 3.46·71-s − 12·73-s − 2.99·77-s − 5.19·79-s − 8.66·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.654·7-s + 0.522·11-s + 0.832·13-s − 0.970·17-s − 1.58·19-s + 1.80·23-s − 0.800·25-s + 0.185·29-s + 0.933·31-s + 0.292·35-s + 1.31·37-s − 0.780·41-s + 1.32·43-s − 1.76·47-s − 0.571·49-s − 1.09·53-s − 0.233·55-s + 0.225·59-s + 0.896·61-s − 0.372·65-s + 1.05·67-s − 0.411·71-s − 1.40·73-s − 0.341·77-s − 0.584·79-s − 0.950·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $-1$
Analytic conductor: \(41.3944\)
Root analytic conductor: \(6.43385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5184,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T + 5T^{2} \)
7 \( 1 + 1.73T + 7T^{2} \)
11 \( 1 - 1.73T + 11T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 + 6.92T + 19T^{2} \)
23 \( 1 - 8.66T + 23T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 - 5.19T + 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 - 8.66T + 43T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 + 8T + 53T^{2} \)
59 \( 1 - 1.73T + 59T^{2} \)
61 \( 1 - 7T + 61T^{2} \)
67 \( 1 - 8.66T + 67T^{2} \)
71 \( 1 + 3.46T + 71T^{2} \)
73 \( 1 + 12T + 73T^{2} \)
79 \( 1 + 5.19T + 79T^{2} \)
83 \( 1 + 8.66T + 83T^{2} \)
89 \( 1 - 4T + 89T^{2} \)
97 \( 1 + 3T + 97T^{2} \)
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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

−8.017079140951883488070335008157, −6.87439164003596768971370692637, −6.55697203242958446216014355797, −5.88096134532344983890968030900, −4.69051408731632134830622147168, −4.17060095760202355627080048597, −3.32985615785838910909568164509, −2.48022858347745485693222102074, −1.27928161018926854390990233596, 0, 1.27928161018926854390990233596, 2.48022858347745485693222102074, 3.32985615785838910909568164509, 4.17060095760202355627080048597, 4.69051408731632134830622147168, 5.88096134532344983890968030900, 6.55697203242958446216014355797, 6.87439164003596768971370692637, 8.017079140951883488070335008157

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