Properties

Label 2-22848-1.1-c1-0-33
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 − 6·13-s + 2·15-s − 17-s − 2·19-s − 21-s − 25-s − 27-s − 8·29-s − 2·35-s − 2·37-s + 6·39-s + 2·41-s + 8·43-s − 2·45-s + 8·47-s + 49-s + 51-s + 2·53-s + 2·57-s + 12·59-s + 4·61-s + 63-s + 12·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.66·13-s + 0.516·15-s − 0.242·17-s − 0.458·19-s − 0.218·21-s − 1/5·25-s − 0.192·27-s − 1.48·29-s − 0.338·35-s − 0.328·37-s + 0.960·39-s + 0.312·41-s + 1.21·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.140·51-s + 0.274·53-s + 0.264·57-s + 1.56·59-s + 0.512·61-s + 0.125·63-s + 1.48·65-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 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−15.72057236116047, −15.24926287841839, −14.72434340561615, −14.34263629628976, −13.55127591118594, −12.84576381121543, −12.42966012201536, −11.92526971202685, −11.45441173948055, −10.93086552464108, −10.40612193654477, −9.677512172017061, −9.217772274889937, −8.440557677876152, −7.739927519320670, −7.400628014243907, −6.874359198677980, −6.064868444241493, −5.277255648013163, −4.954198550766992, −3.990286467246042, −3.839504897391624, −2.515994038434976, −2.077022901432816, −0.8009982056744012, 0, 0.8009982056744012, 2.077022901432816, 2.515994038434976, 3.839504897391624, 3.990286467246042, 4.954198550766992, 5.277255648013163, 6.064868444241493, 6.874359198677980, 7.400628014243907, 7.739927519320670, 8.440557677876152, 9.217772274889937, 9.677512172017061, 10.40612193654477, 10.93086552464108, 11.45441173948055, 11.92526971202685, 12.42966012201536, 12.84576381121543, 13.55127591118594, 14.34263629628976, 14.72434340561615, 15.24926287841839, 15.72057236116047

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