Properties

Label 4-4800e2-1.1-c1e2-0-52
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 $2$

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

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

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: \(2\)
Selberg data: \((4,\ 23040000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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

−8.148999417763595037242873834441, −7.71032510569072262086488957415, −7.20954408718917121186469512149, −7.17783413321688312870870099919, −6.36133176608047715519177047249, −6.35615423258339646253996866730, −5.98817181650402084242120908660, −5.93574274819714052762172486210, −4.87662642103056694520551797063, −4.83210746621997969870832160177, −4.16413160757843991694599718276, −4.09788697308073685171324925749, −3.59876710541155682093588445546, −3.29011294053351677908585571686, −2.51633575047117498215008641202, −2.10101473529025980216811474636, −1.49785612267003571358929907306, −1.45990141441098031357572591635, 0, 0, 1.45990141441098031357572591635, 1.49785612267003571358929907306, 2.10101473529025980216811474636, 2.51633575047117498215008641202, 3.29011294053351677908585571686, 3.59876710541155682093588445546, 4.09788697308073685171324925749, 4.16413160757843991694599718276, 4.83210746621997969870832160177, 4.87662642103056694520551797063, 5.93574274819714052762172486210, 5.98817181650402084242120908660, 6.35615423258339646253996866730, 6.36133176608047715519177047249, 7.17783413321688312870870099919, 7.20954408718917121186469512149, 7.71032510569072262086488957415, 8.148999417763595037242873834441

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