Properties

Label 2-22848-1.1-c1-0-30
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 + 2·11-s − 4·13-s + 2·15-s + 17-s + 2·19-s + 21-s − 4·23-s − 25-s + 27-s + 2·33-s + 2·35-s + 8·37-s − 4·39-s − 2·41-s + 4·43-s + 2·45-s + 49-s + 51-s + 6·53-s + 4·55-s + 2·57-s + 10·59-s + 6·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s + 0.516·15-s + 0.242·17-s + 0.458·19-s + 0.218·21-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 0.348·33-s + 0.338·35-s + 1.31·37-s − 0.640·39-s − 0.312·41-s + 0.609·43-s + 0.298·45-s + 1/7·49-s + 0.140·51-s + 0.824·53-s + 0.539·55-s + 0.264·57-s + 1.30·59-s + 0.768·61-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

\(L(1)\) \(\approx\) \(3.795700317\)
\(L(\frac12)\) \(\approx\) \(3.795700317\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.41257107133620, −14.67365339175757, −14.38390936573270, −14.10033751048642, −13.23044917045455, −13.06165860233018, −12.14714897215137, −11.77476775530275, −11.20469129487580, −10.15066709916453, −10.03963924097071, −9.495071955834997, −8.845116585901425, −8.339895305756977, −7.414531047113745, −7.344025492265821, −6.303277995446811, −5.833061664066512, −5.156904956626469, −4.434566848845625, −3.844713202750398, −2.930438446975555, −2.269374143114563, −1.728543206797101, −0.7624464658848059, 0.7624464658848059, 1.728543206797101, 2.269374143114563, 2.930438446975555, 3.844713202750398, 4.434566848845625, 5.156904956626469, 5.833061664066512, 6.303277995446811, 7.344025492265821, 7.414531047113745, 8.339895305756977, 8.845116585901425, 9.495071955834997, 10.03963924097071, 10.15066709916453, 11.20469129487580, 11.77476775530275, 12.14714897215137, 13.06165860233018, 13.23044917045455, 14.10033751048642, 14.38390936573270, 14.67365339175757, 15.41257107133620

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