Properties

Label 4-168e2-1.1-c0e2-0-1
Degree $4$
Conductor $28224$
Sign $1$
Analytic cond. $0.00702963$
Root an. cond. $0.289556$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s − 5-s + 6-s − 7-s − 8-s − 10-s − 11-s − 14-s − 15-s − 16-s − 21-s − 22-s − 24-s + 25-s − 27-s + 2·29-s − 30-s + 31-s − 33-s + 35-s + 40-s − 42-s − 48-s + 50-s − 53-s − 54-s + ⋯
L(s)  = 1  + 2-s + 3-s − 5-s + 6-s − 7-s − 8-s − 10-s − 11-s − 14-s − 15-s − 16-s − 21-s − 22-s − 24-s + 25-s − 27-s + 2·29-s − 30-s + 31-s − 33-s + 35-s + 40-s − 42-s − 48-s + 50-s − 53-s − 54-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(28224\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.00702963\)
Root analytic conductor: \(0.289556\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{168} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 28224,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6320305171\)
\(L(\frac12)\) \(\approx\) \(0.6320305171\)
\(L(1)\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + T + T^{2} \)
good5$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
11$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
29$C_2$ \( ( 1 - T + T^{2} )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
59$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 + T + T^{2} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
83$C_2$ \( ( 1 - T + T^{2} )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
97$C_2$ \( ( 1 + T + T^{2} )^{2} \)
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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

−13.16302320037969944480696072567, −13.00038542653060252709731992770, −12.45124170982219093015462280986, −11.83714033401652427485785498122, −11.74342602875900181326235820481, −10.67912812258416350923297324413, −10.44604020380389069179961353883, −9.606095076322175874873032937294, −9.240445997654200465171850406238, −8.642314643704530090554462014690, −7.983189307617049929617508331280, −7.943921129814027549837558082507, −6.85306887210919170952066197257, −6.45840513587427204111383799514, −5.72205468344983056887725200940, −4.90885335948892201490429824040, −4.41569849154479348210501520321, −3.63772716762778935296786994848, −2.94693018909009501599165498144, −2.75321242507591566263979590084, 2.75321242507591566263979590084, 2.94693018909009501599165498144, 3.63772716762778935296786994848, 4.41569849154479348210501520321, 4.90885335948892201490429824040, 5.72205468344983056887725200940, 6.45840513587427204111383799514, 6.85306887210919170952066197257, 7.943921129814027549837558082507, 7.983189307617049929617508331280, 8.642314643704530090554462014690, 9.240445997654200465171850406238, 9.606095076322175874873032937294, 10.44604020380389069179961353883, 10.67912812258416350923297324413, 11.74342602875900181326235820481, 11.83714033401652427485785498122, 12.45124170982219093015462280986, 13.00038542653060252709731992770, 13.16302320037969944480696072567

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