Properties

Label 2-22848-1.1-c1-0-40
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 $0$

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 + 4·19-s + 21-s + 4·23-s − 25-s + 27-s + 2·29-s + 8·31-s + 4·33-s + 2·35-s + 2·37-s − 2·39-s − 2·41-s − 8·43-s + 2·45-s + 49-s − 51-s − 6·53-s + 8·55-s + 4·57-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 + 0.917·19-s + 0.218·21-s + 0.834·23-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 − 0.312·41-s − 1.21·43-s + 0.298·45-s + 1/7·49-s − 0.140·51-s − 0.824·53-s + 1.07·55-s + 0.529·57-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: \(0\)
Selberg data: \((2,\ 22848,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.504953289\)
\(L(\frac12)\) \(\approx\) \(4.504953289\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−15.52839597125005, −14.75639318760263, −14.32718632877217, −14.00221686826813, −13.38744575777874, −12.99982677676629, −12.11216547772852, −11.76650829471218, −11.20561851455658, −10.31908102375951, −9.922360098555288, −9.363826248803939, −8.973818779990695, −8.281689813114133, −7.683351624986457, −6.978688350892364, −6.447632326343756, −5.894398639655565, −4.906270416585534, −4.693954861042079, −3.669904529257684, −3.065881463217556, −2.273400452579268, −1.596565125396446, −0.8800869309078331, 0.8800869309078331, 1.596565125396446, 2.273400452579268, 3.065881463217556, 3.669904529257684, 4.693954861042079, 4.906270416585534, 5.894398639655565, 6.447632326343756, 6.978688350892364, 7.683351624986457, 8.281689813114133, 8.973818779990695, 9.363826248803939, 9.922360098555288, 10.31908102375951, 11.20561851455658, 11.76650829471218, 12.11216547772852, 12.99982677676629, 13.38744575777874, 14.00221686826813, 14.32718632877217, 14.75639318760263, 15.52839597125005

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