Properties

Label 2-72e2-1.1-c1-0-82
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 + 1.44·7-s − 3.44·11-s + 3.89·13-s − 4.89·17-s + 4·19-s + 0.550·23-s − 4·25-s − 9.89·29-s − 7.44·31-s + 1.44·35-s − 8.89·37-s − 2.10·41-s + 12.3·43-s − 8.34·47-s − 4.89·49-s + 0.898·53-s − 3.44·55-s − 0.348·59-s − 1.89·61-s + 3.89·65-s − 2.34·67-s + 11.7·71-s + 4.89·73-s − 5·77-s − 8.55·79-s − 5.44·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.547·7-s − 1.04·11-s + 1.08·13-s − 1.18·17-s + 0.917·19-s + 0.114·23-s − 0.800·25-s − 1.83·29-s − 1.33·31-s + 0.245·35-s − 1.46·37-s − 0.328·41-s + 1.88·43-s − 1.21·47-s − 0.699·49-s + 0.123·53-s − 0.465·55-s − 0.0453·59-s − 0.243·61-s + 0.483·65-s − 0.286·67-s + 1.40·71-s + 0.573·73-s − 0.569·77-s − 0.962·79-s − 0.598·83-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: $\chi_{5184} (1, \cdot )$
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 \( 1 - 1.44T + 7T^{2} \)
11 \( 1 + 3.44T + 11T^{2} \)
13 \( 1 - 3.89T + 13T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 0.550T + 23T^{2} \)
29 \( 1 + 9.89T + 29T^{2} \)
31 \( 1 + 7.44T + 31T^{2} \)
37 \( 1 + 8.89T + 37T^{2} \)
41 \( 1 + 2.10T + 41T^{2} \)
43 \( 1 - 12.3T + 43T^{2} \)
47 \( 1 + 8.34T + 47T^{2} \)
53 \( 1 - 0.898T + 53T^{2} \)
59 \( 1 + 0.348T + 59T^{2} \)
61 \( 1 + 1.89T + 61T^{2} \)
67 \( 1 + 2.34T + 67T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 - 4.89T + 73T^{2} \)
79 \( 1 + 8.55T + 79T^{2} \)
83 \( 1 + 5.44T + 83T^{2} \)
89 \( 1 + 3.10T + 89T^{2} \)
97 \( 1 - 5.89T + 97T^{2} \)
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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

−7.78436193517376983201459774363, −7.27923038079325371447400723094, −6.33912391178795998387378496816, −5.53697001282244069863384846668, −5.15662977841655354746233760525, −4.06851775590045600191449366931, −3.34283937151822409720561206128, −2.20163995669978844907221252175, −1.55713001825501504061890467504, 0, 1.55713001825501504061890467504, 2.20163995669978844907221252175, 3.34283937151822409720561206128, 4.06851775590045600191449366931, 5.15662977841655354746233760525, 5.53697001282244069863384846668, 6.33912391178795998387378496816, 7.27923038079325371447400723094, 7.78436193517376983201459774363

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