Properties

Label 8-1200e4-1.1-c1e4-0-7
Degree $8$
Conductor $2.074\times 10^{12}$
Sign $1$
Analytic cond. $8430.11$
Root an. cond. $3.09548$
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  + 4·3-s + 6·9-s − 24·23-s − 4·27-s + 24·47-s + 28·49-s − 8·61-s − 96·69-s − 37·81-s − 24·83-s − 72·107-s + 8·109-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 96·141-s + 112·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 2.30·3-s + 2·9-s − 5.00·23-s − 0.769·27-s + 3.50·47-s + 4·49-s − 1.02·61-s − 11.5·69-s − 4.11·81-s − 2.63·83-s − 6.96·107-s + 0.766·109-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 8.08·141-s + 9.23·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.307·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(8430.11\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.641535451\)
\(L(\frac12)\) \(\approx\) \(1.641535451\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - p T^{2} )^{4} \)
11$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
29$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \)
71$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + p T^{2} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.11577066413084289315137422344, −7.08322716753657859101295343288, −6.39168010752464140424389693861, −6.35277375173126196220610173641, −5.98556502377901711600667024005, −5.83272815448742066084512827065, −5.65237751979499147345805771279, −5.64704405021879485678770775430, −5.16861511076408231112749362784, −5.03309726173317248910347170342, −4.23247609880593210699259738993, −4.14946578398659871203693389002, −4.06295508142864887722798714083, −4.02941466392208582516434478564, −3.94536854376273629142937536651, −3.47303397879361899317441194527, −2.94307584961044825849348005237, −2.83003784890277061352165408781, −2.70018377210525305784776554293, −2.23837138895444839726466662636, −2.05428805373599319694578494794, −2.05313929833207383003595654285, −1.46059831993586108228041325740, −1.04285687428598778156383408146, −0.20404618580031993280575827241, 0.20404618580031993280575827241, 1.04285687428598778156383408146, 1.46059831993586108228041325740, 2.05313929833207383003595654285, 2.05428805373599319694578494794, 2.23837138895444839726466662636, 2.70018377210525305784776554293, 2.83003784890277061352165408781, 2.94307584961044825849348005237, 3.47303397879361899317441194527, 3.94536854376273629142937536651, 4.02941466392208582516434478564, 4.06295508142864887722798714083, 4.14946578398659871203693389002, 4.23247609880593210699259738993, 5.03309726173317248910347170342, 5.16861511076408231112749362784, 5.64704405021879485678770775430, 5.65237751979499147345805771279, 5.83272815448742066084512827065, 5.98556502377901711600667024005, 6.35277375173126196220610173641, 6.39168010752464140424389693861, 7.08322716753657859101295343288, 7.11577066413084289315137422344

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