Properties

Label 2-22848-1.1-c1-0-53
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 + 2·11-s − 2·15-s − 17-s − 6·19-s − 21-s − 25-s − 27-s − 8·29-s − 8·31-s − 2·33-s + 2·35-s + 4·37-s − 6·41-s + 8·43-s + 2·45-s + 8·47-s + 49-s + 51-s − 2·53-s + 4·55-s + 6·57-s + 6·59-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 − 0.516·15-s − 0.242·17-s − 1.37·19-s − 0.218·21-s − 1/5·25-s − 0.192·27-s − 1.48·29-s − 1.43·31-s − 0.348·33-s + 0.338·35-s + 0.657·37-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 − 0.274·53-s + 0.539·55-s + 0.794·57-s + 0.781·59-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: \(1\)
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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
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

−15.70425092542605, −15.19630206771761, −14.60864356496173, −14.22967584934542, −13.53365925910904, −12.97247432527589, −12.63872120646390, −11.92812532652993, −11.26143385328528, −10.89744115799665, −10.39350522771997, −9.582954064089482, −9.278122625555277, −8.641163830509154, −7.898620343403542, −7.213776725719034, −6.614664941476271, −6.050241142079351, −5.521783927499377, −4.995610425654978, −4.016404866236248, −3.776246908087453, −2.336907143563695, −2.021803564754962, −1.135598312462055, 0, 1.135598312462055, 2.021803564754962, 2.336907143563695, 3.776246908087453, 4.016404866236248, 4.995610425654978, 5.521783927499377, 6.050241142079351, 6.614664941476271, 7.213776725719034, 7.898620343403542, 8.641163830509154, 9.278122625555277, 9.582954064089482, 10.39350522771997, 10.89744115799665, 11.26143385328528, 11.92812532652993, 12.63872120646390, 12.97247432527589, 13.53365925910904, 14.22967584934542, 14.60864356496173, 15.19630206771761, 15.70425092542605

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