Properties

Label 2-24-24.5-c2-0-3
Degree $2$
Conductor $24$
Sign $1$
Analytic cond. $0.653952$
Root an. cond. $0.808673$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 4·4-s − 2·5-s − 6·6-s − 10·7-s + 8·8-s + 9·9-s − 4·10-s + 10·11-s − 12·12-s − 20·14-s + 6·15-s + 16·16-s + 18·18-s − 8·20-s + 30·21-s + 20·22-s − 24·24-s − 21·25-s − 27·27-s − 40·28-s − 50·29-s + 12·30-s + 38·31-s + 32·32-s − 30·33-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 2/5·5-s − 6-s − 1.42·7-s + 8-s + 9-s − 2/5·10-s + 0.909·11-s − 12-s − 1.42·14-s + 2/5·15-s + 16-s + 18-s − 2/5·20-s + 10/7·21-s + 0.909·22-s − 24-s − 0.839·25-s − 27-s − 1.42·28-s − 1.72·29-s + 2/5·30-s + 1.22·31-s + 32-s − 0.909·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $1$
Analytic conductor: \(0.653952\)
Root analytic conductor: \(0.808673\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{24} (5, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 24,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.109272074\)
\(L(\frac12)\) \(\approx\) \(1.109272074\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 + p T \)
good5 \( 1 + 2 T + p^{2} T^{2} \)
7 \( 1 + 10 T + p^{2} T^{2} \)
11 \( 1 - 10 T + p^{2} T^{2} \)
13 \( ( 1 - p T )( 1 + p T ) \)
17 \( ( 1 - p T )( 1 + p T ) \)
19 \( ( 1 - p T )( 1 + p T ) \)
23 \( ( 1 - p T )( 1 + p T ) \)
29 \( 1 + 50 T + p^{2} T^{2} \)
31 \( 1 - 38 T + p^{2} T^{2} \)
37 \( ( 1 - p T )( 1 + p T ) \)
41 \( ( 1 - p T )( 1 + p T ) \)
43 \( ( 1 - p T )( 1 + p T ) \)
47 \( ( 1 - p T )( 1 + p T ) \)
53 \( 1 - 94 T + p^{2} T^{2} \)
59 \( 1 - 10 T + p^{2} T^{2} \)
61 \( ( 1 - p T )( 1 + p T ) \)
67 \( ( 1 - p T )( 1 + p T ) \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( 1 - 50 T + p^{2} T^{2} \)
79 \( 1 + 58 T + p^{2} T^{2} \)
83 \( 1 + 134 T + p^{2} T^{2} \)
89 \( ( 1 - p T )( 1 + p T ) \)
97 \( 1 + 190 T + p^{2} T^{2} \)
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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

−17.05900125367894776227961880775, −16.19870908313102715155501224887, −15.21918399726442928042470941707, −13.44962302934968010035656911083, −12.38348407530844087816136359301, −11.41784072404285207042171459035, −9.892069267421358887642098215380, −7.04559610571669524325508573917, −5.91633467858724488813659514310, −3.91830577714857009768874565814, 3.91830577714857009768874565814, 5.91633467858724488813659514310, 7.04559610571669524325508573917, 9.892069267421358887642098215380, 11.41784072404285207042171459035, 12.38348407530844087816136359301, 13.44962302934968010035656911083, 15.21918399726442928042470941707, 16.19870908313102715155501224887, 17.05900125367894776227961880775

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