Properties

Label 4-168e2-1.1-c3e2-0-3
Degree $4$
Conductor $28224$
Sign $1$
Analytic cond. $98.2541$
Root an. cond. $3.14838$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 2·5-s + 35·7-s + 18·11-s + 66·13-s − 6·15-s + 68·17-s − 25·19-s + 105·21-s − 92·23-s + 125·25-s − 27·27-s + 184·29-s − 25·31-s + 54·33-s − 70·35-s + 213·37-s + 198·39-s + 188·41-s − 134·43-s − 278·47-s + 882·49-s + 204·51-s + 400·53-s − 36·55-s − 75·57-s − 744·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.178·5-s + 1.88·7-s + 0.493·11-s + 1.40·13-s − 0.103·15-s + 0.970·17-s − 0.301·19-s + 1.09·21-s − 0.834·23-s + 25-s − 0.192·27-s + 1.17·29-s − 0.144·31-s + 0.284·33-s − 0.338·35-s + 0.946·37-s + 0.812·39-s + 0.716·41-s − 0.475·43-s − 0.862·47-s + 18/7·49-s + 0.560·51-s + 1.03·53-s − 0.0882·55-s − 0.174·57-s − 1.64·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(28224\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(98.2541\)
Root analytic conductor: \(3.14838\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 28224,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.177718767\)
\(L(\frac12)\) \(\approx\) \(4.177718767\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 - p T + p^{2} T^{2} \)
7$C_2$ \( 1 - 5 p T + p^{3} T^{2} \)
good5$C_2^2$ \( 1 + 2 T - 121 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 18 T - 1007 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 33 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 4 p T - p^{2} T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 25 T - 6234 T^{2} + 25 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 4 p T - 7 p^{2} T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 92 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 25 T - 29166 T^{2} + 25 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 213 T - 5284 T^{2} - 213 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 94 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 67 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 278 T - 26539 T^{2} + 278 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 400 T + 11123 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 744 T + 348157 T^{2} + 744 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 734 T + 311775 T^{2} - 734 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 555 T + 7262 T^{2} + 555 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 642 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 973 T + 557712 T^{2} + 973 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 785 T + 123186 T^{2} - 785 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 822 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 424 T - 525193 T^{2} + 424 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 734 T + p^{3} T^{2} )^{2} \)
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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

−12.43640495540356873268671480603, −12.06333618662214552821539035762, −11.43758551122152971292876806008, −11.27052678409937702414855955124, −10.58183272967781755286407685014, −10.22197667793205238802035459152, −9.458119420793879734094666438344, −8.757686137049928052091729091563, −8.340010853066541709275023503331, −8.256394316350968245010379053760, −7.47500484261877181607105383178, −6.98140600752645564681399916220, −5.95390611052833781534537451694, −5.74669477323666047790154147731, −4.61615489084740071455955144058, −4.39768393450666439888051214931, −3.52227183615706175947489380356, −2.70876883768963688033426113023, −1.61647319045703788701912693309, −1.07902011614553030221086211987, 1.07902011614553030221086211987, 1.61647319045703788701912693309, 2.70876883768963688033426113023, 3.52227183615706175947489380356, 4.39768393450666439888051214931, 4.61615489084740071455955144058, 5.74669477323666047790154147731, 5.95390611052833781534537451694, 6.98140600752645564681399916220, 7.47500484261877181607105383178, 8.256394316350968245010379053760, 8.340010853066541709275023503331, 8.757686137049928052091729091563, 9.458119420793879734094666438344, 10.22197667793205238802035459152, 10.58183272967781755286407685014, 11.27052678409937702414855955124, 11.43758551122152971292876806008, 12.06333618662214552821539035762, 12.43640495540356873268671480603

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