Properties

Degree $4$
Conductor $28224$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s + 5·7-s + 6·11-s − 6·13-s − 2·15-s − 4·17-s + 5·19-s + 5·21-s + 4·23-s + 5·25-s − 27-s − 8·29-s − 7·31-s + 6·33-s − 10·35-s + 9·37-s − 6·39-s − 4·41-s − 2·43-s − 2·47-s + 18·49-s − 4·51-s − 8·53-s − 12·55-s + 5·57-s − 10·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1.88·7-s + 1.80·11-s − 1.66·13-s − 0.516·15-s − 0.970·17-s + 1.14·19-s + 1.09·21-s + 0.834·23-s + 25-s − 0.192·27-s − 1.48·29-s − 1.25·31-s + 1.04·33-s − 1.69·35-s + 1.47·37-s − 0.960·39-s − 0.624·41-s − 0.304·43-s − 0.291·47-s + 18/7·49-s − 0.560·51-s − 1.09·53-s − 1.61·55-s + 0.662·57-s − 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(28224\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{168} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 28224,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56176\)
\(L(\frac12)\) \(\approx\) \(1.56176\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 8 T + 11 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 15 T + 158 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 8 T - 25 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.93998168163479237060150960111, −12.49647059581523935845655071106, −11.80810672278820619933441987182, −11.63213957703065132972287836513, −11.10923750331065684197624462419, −10.93984127357960550447846175547, −9.865810713128648151345970312920, −9.267857396078675292804220194486, −9.066908169917119266659675428788, −8.455168944411838828307103830164, −7.74939842826450915271496415580, −7.46661102487355113936002959526, −7.06202613067146614143981319815, −6.22004389258745725172411486743, −4.99206537175416333524925458485, −4.98644188984549650667574953911, −4.10700387459099160093599550272, −3.51752340962738967894359338796, −2.39332483157576102990205156124, −1.45509825848733654655920391659, 1.45509825848733654655920391659, 2.39332483157576102990205156124, 3.51752340962738967894359338796, 4.10700387459099160093599550272, 4.98644188984549650667574953911, 4.99206537175416333524925458485, 6.22004389258745725172411486743, 7.06202613067146614143981319815, 7.46661102487355113936002959526, 7.74939842826450915271496415580, 8.455168944411838828307103830164, 9.066908169917119266659675428788, 9.267857396078675292804220194486, 9.865810713128648151345970312920, 10.93984127357960550447846175547, 11.10923750331065684197624462419, 11.63213957703065132972287836513, 11.80810672278820619933441987182, 12.49647059581523935845655071106, 12.93998168163479237060150960111

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