Properties

Label 2-22848-1.1-c1-0-29
Degree $2$
Conductor $22848$
Sign $-1$
Analytic cond. $182.442$
Root an. cond. $13.5071$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s − 7-s + 9-s − 4·11-s + 2·13-s + 2·15-s + 17-s − 8·19-s + 21-s − 25-s − 27-s − 2·29-s + 8·31-s + 4·33-s + 2·35-s − 2·37-s − 2·39-s − 6·41-s + 8·43-s − 2·45-s + 8·47-s + 49-s − 51-s − 6·53-s + 8·55-s + 8·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.516·15-s + 0.242·17-s − 1.83·19-s + 0.218·21-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.696·33-s + 0.338·35-s − 0.328·37-s − 0.320·39-s − 0.937·41-s + 1.21·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s − 0.140·51-s − 0.824·53-s + 1.07·55-s + 1.05·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22848\)    =    \(2^{6} \cdot 3 \cdot 7 \cdot 17\)
Sign: $-1$
Analytic conductor: \(182.442\)
Root analytic conductor: \(13.5071\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{22848} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 22848,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 + T \)
17 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
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

−15.75314502744150, −15.39087484037667, −14.93806424850301, −14.03699949659456, −13.52432523447432, −12.93248900438829, −12.42574955252444, −12.10561881536747, −11.22170399330313, −10.94843484587712, −10.37773203153820, −9.886347104810937, −9.066268652965478, −8.288837320722456, −8.073042778478100, −7.328934518934888, −6.672292131440796, −6.118998368022908, −5.509651959307806, −4.752626002901718, −4.166131371479899, −3.594448959729256, −2.722003787195920, −1.988843375876162, −0.7580253744357592, 0, 0.7580253744357592, 1.988843375876162, 2.722003787195920, 3.594448959729256, 4.166131371479899, 4.752626002901718, 5.509651959307806, 6.118998368022908, 6.672292131440796, 7.328934518934888, 8.073042778478100, 8.288837320722456, 9.066268652965478, 9.886347104810937, 10.37773203153820, 10.94843484587712, 11.22170399330313, 12.10561881536747, 12.42574955252444, 12.93248900438829, 13.52432523447432, 14.03699949659456, 14.93806424850301, 15.39087484037667, 15.75314502744150

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