Properties

Label 2-288-1.1-c5-0-15
Degree $2$
Conductor $288$
Sign $-1$
Analytic cond. $46.1905$
Root an. cond. $6.79636$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 14·5-s − 208·7-s + 536·11-s + 694·13-s + 1.27e3·17-s + 1.11e3·19-s − 3.21e3·23-s − 2.92e3·25-s − 2.91e3·29-s − 2.62e3·31-s + 2.91e3·35-s − 9.45e3·37-s − 170·41-s − 1.99e4·43-s − 32·47-s + 2.64e4·49-s + 2.21e4·53-s − 7.50e3·55-s − 4.14e4·59-s + 1.54e4·61-s − 9.71e3·65-s − 2.07e4·67-s − 2.85e4·71-s − 5.36e4·73-s − 1.11e5·77-s − 6.91e4·79-s + 3.78e4·83-s + ⋯
L(s)  = 1  − 0.250·5-s − 1.60·7-s + 1.33·11-s + 1.13·13-s + 1.07·17-s + 0.706·19-s − 1.26·23-s − 0.937·25-s − 0.644·29-s − 0.490·31-s + 0.401·35-s − 1.13·37-s − 0.0157·41-s − 1.64·43-s − 0.00211·47-s + 1.57·49-s + 1.08·53-s − 0.334·55-s − 1.55·59-s + 0.532·61-s − 0.285·65-s − 0.564·67-s − 0.673·71-s − 1.17·73-s − 2.14·77-s − 1.24·79-s + 0.602·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(46.1905\)
Root analytic conductor: \(6.79636\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 288,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 14 T + p^{5} T^{2} \)
7 \( 1 + 208 T + p^{5} T^{2} \)
11 \( 1 - 536 T + p^{5} T^{2} \)
13 \( 1 - 694 T + p^{5} T^{2} \)
17 \( 1 - 1278 T + p^{5} T^{2} \)
19 \( 1 - 1112 T + p^{5} T^{2} \)
23 \( 1 + 3216 T + p^{5} T^{2} \)
29 \( 1 + 2918 T + p^{5} T^{2} \)
31 \( 1 + 2624 T + p^{5} T^{2} \)
37 \( 1 + 9458 T + p^{5} T^{2} \)
41 \( 1 + 170 T + p^{5} T^{2} \)
43 \( 1 + 19928 T + p^{5} T^{2} \)
47 \( 1 + 32 T + p^{5} T^{2} \)
53 \( 1 - 22178 T + p^{5} T^{2} \)
59 \( 1 + 41480 T + p^{5} T^{2} \)
61 \( 1 - 15462 T + p^{5} T^{2} \)
67 \( 1 + 20744 T + p^{5} T^{2} \)
71 \( 1 + 28592 T + p^{5} T^{2} \)
73 \( 1 + 53670 T + p^{5} T^{2} \)
79 \( 1 + 69152 T + p^{5} T^{2} \)
83 \( 1 - 37800 T + p^{5} T^{2} \)
89 \( 1 - 126806 T + p^{5} T^{2} \)
97 \( 1 - 62290 T + p^{5} T^{2} \)
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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

−10.29691457140724241923641197082, −9.588199326663946803387691909715, −8.736436700841341542076340691560, −7.47274002533097765388158355512, −6.42671536227933670425745713959, −5.73734274248781564415688595461, −3.83248816480442242060782819005, −3.38708595207640471215548489945, −1.46692674533019571594759359576, 0, 1.46692674533019571594759359576, 3.38708595207640471215548489945, 3.83248816480442242060782819005, 5.73734274248781564415688595461, 6.42671536227933670425745713959, 7.47274002533097765388158355512, 8.736436700841341542076340691560, 9.588199326663946803387691909715, 10.29691457140724241923641197082

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