Properties

Label 4-2400e2-1.1-c1e2-0-12
Degree $4$
Conductor $5760000$
Sign $1$
Analytic cond. $367.262$
Root an. cond. $4.37768$
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  − 9-s + 8·11-s − 16·19-s + 12·29-s − 12·41-s + 14·49-s − 24·59-s + 28·61-s + 16·79-s + 81-s − 4·89-s − 8·99-s − 28·101-s − 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 16·171-s + 173-s + ⋯
L(s)  = 1  − 1/3·9-s + 2.41·11-s − 3.67·19-s + 2.22·29-s − 1.87·41-s + 2·49-s − 3.12·59-s + 3.58·61-s + 1.80·79-s + 1/9·81-s − 0.423·89-s − 0.804·99-s − 2.78·101-s − 2.68·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 1.22·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5760000\)    =    \(2^{10} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(367.262\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5760000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.022128074\)
\(L(\frac12)\) \(\approx\) \(2.022128074\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−9.121279321727361796187495882666, −8.653946391450262269850011610284, −8.566199376355159919024556049375, −8.132350852563471362107334593035, −7.80110789286529074190981222881, −6.86163984448130820996516660373, −6.66034590661104915103867788837, −6.59253762829136573373045722505, −6.33179208332781865686470221518, −5.72318174511009985932257059866, −5.17977672209596164305187515504, −4.68200779026234768406191851070, −4.22025835593867010029583925639, −3.85858199368142659427749753835, −3.80609050352547795382138013611, −2.74778045478718415433800835631, −2.52612844324430918158692015706, −1.76875518447980779067063967005, −1.37731319719758585072401552200, −0.49200684404884067164167360943, 0.49200684404884067164167360943, 1.37731319719758585072401552200, 1.76875518447980779067063967005, 2.52612844324430918158692015706, 2.74778045478718415433800835631, 3.80609050352547795382138013611, 3.85858199368142659427749753835, 4.22025835593867010029583925639, 4.68200779026234768406191851070, 5.17977672209596164305187515504, 5.72318174511009985932257059866, 6.33179208332781865686470221518, 6.59253762829136573373045722505, 6.66034590661104915103867788837, 6.86163984448130820996516660373, 7.80110789286529074190981222881, 8.132350852563471362107334593035, 8.566199376355159919024556049375, 8.653946391450262269850011610284, 9.121279321727361796187495882666

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