Properties

Label 4-7920e2-1.1-c1e2-0-5
Degree $4$
Conductor $62726400$
Sign $1$
Analytic cond. $3999.48$
Root an. cond. $7.95245$
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  + 2·5-s + 4·7-s − 2·11-s + 8·17-s + 8·19-s − 8·23-s + 3·25-s + 4·29-s + 8·35-s + 12·37-s − 4·41-s + 12·43-s − 8·47-s + 6·49-s + 4·53-s − 4·55-s − 8·59-s − 12·61-s + 16·71-s − 8·77-s − 20·83-s + 16·85-s + 4·89-s + 16·95-s + 12·97-s − 4·101-s − 16·103-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.51·7-s − 0.603·11-s + 1.94·17-s + 1.83·19-s − 1.66·23-s + 3/5·25-s + 0.742·29-s + 1.35·35-s + 1.97·37-s − 0.624·41-s + 1.82·43-s − 1.16·47-s + 6/7·49-s + 0.549·53-s − 0.539·55-s − 1.04·59-s − 1.53·61-s + 1.89·71-s − 0.911·77-s − 2.19·83-s + 1.73·85-s + 0.423·89-s + 1.64·95-s + 1.21·97-s − 0.398·101-s − 1.57·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(62726400\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(3999.48\)
Root analytic conductor: \(7.95245\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 62726400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.362078992\)
\(L(\frac12)\) \(\approx\) \(6.362078992\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
good7$C_4$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−7.965381801156168283272471023746, −7.64370032542845224832630795707, −7.36149128963944271334102410866, −7.36095641566394068722333768797, −6.34615805811010462160664146287, −6.31981750987947601465700547473, −5.83607366290488814726543059911, −5.55527703368403334649323090761, −5.28952618023231337302848626887, −4.95400421429265281698702619064, −4.53368275081722657082823354252, −4.26923387341740387707431209998, −3.63495828976766875045445310362, −3.28085278881156929025595108900, −2.75780331969216714585310815898, −2.56074077660263243698300210894, −1.84624113797027591371481666839, −1.57212968246214759875529195810, −1.08216848792811576668245872007, −0.65584218080414115825377466433, 0.65584218080414115825377466433, 1.08216848792811576668245872007, 1.57212968246214759875529195810, 1.84624113797027591371481666839, 2.56074077660263243698300210894, 2.75780331969216714585310815898, 3.28085278881156929025595108900, 3.63495828976766875045445310362, 4.26923387341740387707431209998, 4.53368275081722657082823354252, 4.95400421429265281698702619064, 5.28952618023231337302848626887, 5.55527703368403334649323090761, 5.83607366290488814726543059911, 6.31981750987947601465700547473, 6.34615805811010462160664146287, 7.36095641566394068722333768797, 7.36149128963944271334102410866, 7.64370032542845224832630795707, 7.965381801156168283272471023746

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