Properties

Degree $2$
Conductor $24$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s + 9-s + 4·11-s − 2·13-s + 2·15-s + 2·17-s − 4·19-s − 8·23-s − 25-s − 27-s + 6·29-s + 8·31-s − 4·33-s + 6·37-s + 2·39-s − 6·41-s + 4·43-s − 2·45-s − 7·49-s − 2·51-s − 2·53-s − 8·55-s + 4·57-s + 4·59-s − 2·61-s + 4·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s + 0.485·17-s − 0.917·19-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 + 0.986·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.298·45-s − 49-s − 0.280·51-s − 0.274·53-s − 1.07·55-s + 0.529·57-s + 0.520·59-s − 0.256·61-s + 0.496·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{24} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 24,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5391289118\)
\(L(\frac12)\) \(\approx\) \(0.5391289118\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
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 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.89343166445288, −19.15515379485895, −17.67075195000178, −16.62726100086659, −15.47325164750051, −14.18198826155077, −12.31725686074734, −11.58204746144152, −9.964671915727173, −8.098990694093692, −6.428971070727447, −4.253030286927965, 4.253030286927965, 6.428971070727447, 8.098990694093692, 9.964671915727173, 11.58204746144152, 12.31725686074734, 14.18198826155077, 15.47325164750051, 16.62726100086659, 17.67075195000178, 19.15515379485895, 19.89343166445288

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