Properties

Label 4-4800e2-1.1-c1e2-0-3
Degree $4$
Conductor $23040000$
Sign $1$
Analytic cond. $1469.05$
Root an. cond. $6.19097$
Motivic weight $1$
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  − 9-s − 4·11-s − 14·19-s − 4·29-s − 10·31-s + 24·41-s + 5·49-s − 12·59-s + 26·61-s − 8·71-s + 16·79-s + 81-s − 32·89-s + 4·99-s + 18·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 17·169-s + 14·171-s + ⋯
L(s)  = 1  − 1/3·9-s − 1.20·11-s − 3.21·19-s − 0.742·29-s − 1.79·31-s + 3.74·41-s + 5/7·49-s − 1.56·59-s + 3.32·61-s − 0.949·71-s + 1.80·79-s + 1/9·81-s − 3.39·89-s + 0.402·99-s + 1.72·109-s − 0.909·121-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 + 1.30·169-s + 1.07·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(23040000\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1469.05\)
Root analytic conductor: \(6.19097\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 23040000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5473711105\)
\(L(\frac12)\) \(\approx\) \(0.5473711105\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 77 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 145 T^{2} + p^{2} T^{4} \)
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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

−8.382180679225378054065618073833, −8.251711654258807428969878703982, −7.61960859333694318278678758499, −7.48864256819948257202563261604, −7.13719852191543241130875719535, −6.48954271962727643542959951163, −6.39676063706975165436882843172, −5.81477532772943756049781928814, −5.53724449978106674949920861592, −5.39102918851410175269321190964, −4.62142884779934403994223876580, −4.35569055707554968331559678215, −3.99056034914336343200142618524, −3.71503505122769703031839500594, −2.94736955497131800590406989821, −2.57417481469800850481440410396, −2.06766597255283751538373498990, −2.03726676943916773429991317158, −0.987627823977419018255840836011, −0.21998672096031077742442897395, 0.21998672096031077742442897395, 0.987627823977419018255840836011, 2.03726676943916773429991317158, 2.06766597255283751538373498990, 2.57417481469800850481440410396, 2.94736955497131800590406989821, 3.71503505122769703031839500594, 3.99056034914336343200142618524, 4.35569055707554968331559678215, 4.62142884779934403994223876580, 5.39102918851410175269321190964, 5.53724449978106674949920861592, 5.81477532772943756049781928814, 6.39676063706975165436882843172, 6.48954271962727643542959951163, 7.13719852191543241130875719535, 7.48864256819948257202563261604, 7.61960859333694318278678758499, 8.251711654258807428969878703982, 8.382180679225378054065618073833

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