Properties

Label 4-225792-1.1-c1e2-0-25
Degree $4$
Conductor $225792$
Sign $1$
Analytic cond. $14.3966$
Root an. cond. $1.94789$
Motivic weight $1$
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  + 4·5-s − 3·9-s + 16·19-s + 2·25-s + 12·29-s − 8·43-s − 12·45-s − 16·47-s + 49-s + 12·53-s − 8·67-s − 16·71-s + 20·73-s + 9·81-s + 64·95-s − 12·97-s + 4·101-s − 6·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 48·145-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 1.78·5-s − 9-s + 3.67·19-s + 2/5·25-s + 2.22·29-s − 1.21·43-s − 1.78·45-s − 2.33·47-s + 1/7·49-s + 1.64·53-s − 0.977·67-s − 1.89·71-s + 2.34·73-s + 81-s + 6.56·95-s − 1.21·97-s + 0.398·101-s − 0.545·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.98·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225792 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225792 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(225792\)    =    \(2^{9} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(14.3966\)
Root analytic conductor: \(1.94789\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 225792,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.497939845\)
\(L(\frac12)\) \(\approx\) \(2.497939845\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−9.280504664575016660349009750311, −8.509354664621903192772158046209, −8.188420417684287628984093773695, −7.58750440596713449607840635902, −7.07204727903132066232126574118, −6.42262084813563967264199691637, −6.11340833663381996666748371567, −5.48790807221177914788155934026, −5.22316409435338364257345412913, −4.86758911716003878330800803137, −3.73233091133641602778161609628, −2.99737772195369908741226640050, −2.79183800612725741835263061431, −1.79239147115953487472752121748, −1.07526755386223660246135627512, 1.07526755386223660246135627512, 1.79239147115953487472752121748, 2.79183800612725741835263061431, 2.99737772195369908741226640050, 3.73233091133641602778161609628, 4.86758911716003878330800803137, 5.22316409435338364257345412913, 5.48790807221177914788155934026, 6.11340833663381996666748371567, 6.42262084813563967264199691637, 7.07204727903132066232126574118, 7.58750440596713449607840635902, 8.188420417684287628984093773695, 8.509354664621903192772158046209, 9.280504664575016660349009750311

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