Properties

Degree $2$
Conductor $6720$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 7-s + 9-s − 4·11-s + 2·13-s − 15-s + 2·17-s + 4·19-s + 21-s − 8·23-s + 25-s + 27-s − 6·29-s − 8·31-s − 4·33-s − 35-s + 2·37-s + 2·39-s + 2·41-s + 12·43-s − 45-s − 8·47-s + 49-s + 2·51-s − 6·53-s + 4·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s + 0.917·19-s + 0.218·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.696·33-s − 0.169·35-s + 0.328·37-s + 0.320·39-s + 0.312·41-s + 1.82·43-s − 0.149·45-s − 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6720\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{6720} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6720,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−17.82710431960323, −16.82224584367180, −16.06290450989836, −15.93397721046913, −15.24076386863519, −14.38001433305821, −14.24827917497797, −13.31435494780140, −12.89604416330735, −12.22277976212394, −11.46639824525728, −10.93923699148887, −10.25941733153313, −9.594042346412457, −8.932283085896344, −8.165482597585322, −7.560513326646139, −7.433065467507025, −6.055571681101853, −5.573606633093107, −4.689747871670962, −3.871848280441057, −3.242435135868658, −2.324406566694845, −1.423315021105893, 0, 1.423315021105893, 2.324406566694845, 3.242435135868658, 3.871848280441057, 4.689747871670962, 5.573606633093107, 6.055571681101853, 7.433065467507025, 7.560513326646139, 8.165482597585322, 8.932283085896344, 9.594042346412457, 10.25941733153313, 10.93923699148887, 11.46639824525728, 12.22277976212394, 12.89604416330735, 13.31435494780140, 14.24827917497797, 14.38001433305821, 15.24076386863519, 15.93397721046913, 16.06290450989836, 16.82224584367180, 17.82710431960323

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