Properties

Label 4-2880e2-1.1-c1e2-0-29
Degree $4$
Conductor $8294400$
Sign $1$
Analytic cond. $528.858$
Root an. cond. $4.79550$
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  − 8·7-s + 4·23-s − 25-s − 16·31-s + 4·41-s + 20·47-s + 34·49-s − 24·71-s − 28·73-s − 16·79-s − 36·89-s − 12·97-s − 32·103-s + 40·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 32·161-s + 163-s + 167-s + 22·169-s + 173-s + ⋯
L(s)  = 1  − 3.02·7-s + 0.834·23-s − 1/5·25-s − 2.87·31-s + 0.624·41-s + 2.91·47-s + 34/7·49-s − 2.84·71-s − 3.27·73-s − 1.80·79-s − 3.81·89-s − 1.21·97-s − 3.15·103-s + 3.76·113-s + 6/11·121-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 − 2.52·161-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8294400\)    =    \(2^{12} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(528.858\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 8294400,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_2$ \( 1 + T^{2} \)
good7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
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

−8.694075923882302512842694211987, −8.575959756909580328537814441038, −7.37442523734176563416851148320, −7.35281773056447823465597169468, −7.16896869400984052141077865702, −6.86849658158331012564599294703, −6.14327578343807227443782365367, −5.88721366877117713622398204739, −5.69786403664708467998158248543, −5.41454715346349893808092207943, −4.29962952503001469251322412214, −4.29157023170158336192993164055, −3.76477967976697913435000135753, −3.22174139329960197830469308146, −2.85782607181659317622961978511, −2.73327454524921032577164931679, −1.82615454885304414205704042256, −1.11359543999282860657323264902, 0, 0, 1.11359543999282860657323264902, 1.82615454885304414205704042256, 2.73327454524921032577164931679, 2.85782607181659317622961978511, 3.22174139329960197830469308146, 3.76477967976697913435000135753, 4.29157023170158336192993164055, 4.29962952503001469251322412214, 5.41454715346349893808092207943, 5.69786403664708467998158248543, 5.88721366877117713622398204739, 6.14327578343807227443782365367, 6.86849658158331012564599294703, 7.16896869400984052141077865702, 7.35281773056447823465597169468, 7.37442523734176563416851148320, 8.575959756909580328537814441038, 8.694075923882302512842694211987

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