Properties

Degree $2$
Conductor $23184$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 7-s − 4·11-s − 2·13-s + 6·17-s − 4·19-s − 23-s − 25-s + 2·29-s + 8·31-s − 2·35-s + 6·37-s + 6·41-s + 4·43-s − 8·47-s + 49-s − 6·53-s − 8·55-s + 4·59-s − 10·61-s − 4·65-s − 4·67-s − 8·71-s − 6·73-s + 4·77-s − 12·83-s + 12·85-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s − 1.20·11-s − 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.208·23-s − 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.338·35-s + 0.986·37-s + 0.937·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 1.07·55-s + 0.520·59-s − 1.28·61-s − 0.496·65-s − 0.488·67-s − 0.949·71-s − 0.702·73-s + 0.455·77-s − 1.31·83-s + 1.30·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23184\)    =    \(2^{4} \cdot 3^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{23184} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 23184,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
23 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
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

−15.91658664733396, −15.11226139481915, −14.62762381468798, −14.12078146997591, −13.52742367719048, −13.06308825501840, −12.56900023186403, −12.08952558142323, −11.36633387872881, −10.62312305393256, −10.12846425482690, −9.845325155099745, −9.286960892857077, −8.431252697907433, −7.896594011520097, −7.450678521553555, −6.585363244393579, −5.902529326320718, −5.707431616789349, −4.773927418288151, −4.329869337714126, −3.164715575430224, −2.743019194853641, −2.043872746206005, −1.093940138478493, 0, 1.093940138478493, 2.043872746206005, 2.743019194853641, 3.164715575430224, 4.329869337714126, 4.773927418288151, 5.707431616789349, 5.902529326320718, 6.585363244393579, 7.450678521553555, 7.896594011520097, 8.431252697907433, 9.286960892857077, 9.845325155099745, 10.12846425482690, 10.62312305393256, 11.36633387872881, 12.08952558142323, 12.56900023186403, 13.06308825501840, 13.52742367719048, 14.12078146997591, 14.62762381468798, 15.11226139481915, 15.91658664733396

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