Properties

Label 2-22848-1.1-c1-0-21
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 + 6·11-s − 2·15-s − 17-s + 2·19-s − 21-s − 25-s + 27-s + 4·29-s + 6·33-s + 2·35-s − 8·37-s + 2·41-s + 4·43-s − 2·45-s − 8·47-s + 49-s − 51-s + 14·53-s − 12·55-s + 2·57-s + 6·59-s + 10·61-s − 63-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.80·11-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 + 0.742·29-s + 1.04·33-s + 0.338·35-s − 1.31·37-s + 0.312·41-s + 0.609·43-s − 0.298·45-s − 1.16·47-s + 1/7·49-s − 0.140·51-s + 1.92·53-s − 1.61·55-s + 0.264·57-s + 0.781·59-s + 1.28·61-s − 0.125·63-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

\(L(1)\) \(\approx\) \(2.493228496\)
\(L(\frac12)\) \(\approx\) \(2.493228496\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−15.53896427000276, −14.83553681105035, −14.52771055620314, −13.88209126733963, −13.47633593436112, −12.72576027205494, −12.00726792811339, −11.90858479286709, −11.23428817291643, −10.55583922422763, −9.769416075026305, −9.455821922943372, −8.630641068672632, −8.469091615692384, −7.615561079366689, −6.945495229082729, −6.701715368141973, −5.856407411793535, −5.033826995321101, −4.137320961987834, −3.860408191679798, −3.275502846733683, −2.396350448950240, −1.480390359108618, −0.6490834347432710, 0.6490834347432710, 1.480390359108618, 2.396350448950240, 3.275502846733683, 3.860408191679798, 4.137320961987834, 5.033826995321101, 5.856407411793535, 6.701715368141973, 6.945495229082729, 7.615561079366689, 8.469091615692384, 8.630641068672632, 9.455821922943372, 9.769416075026305, 10.55583922422763, 11.23428817291643, 11.90858479286709, 12.00726792811339, 12.72576027205494, 13.47633593436112, 13.88209126733963, 14.52771055620314, 14.83553681105035, 15.53896427000276

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