Properties

Label 8-1200e4-1.1-c1e4-0-4
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  + 6·9-s − 16·13-s + 16·37-s + 4·49-s − 40·61-s + 8·73-s + 27·81-s − 40·97-s + 40·109-s − 96·117-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 108·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 2·9-s − 4.43·13-s + 2.63·37-s + 4/7·49-s − 5.12·61-s + 0.936·73-s + 3·81-s − 4.06·97-s + 3.83·109-s − 8.87·117-s − 1.81·121-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 + 8.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-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: Trivial
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

\(L(1)\) \(\approx\) \(0.4281722287\)
\(L(\frac12)\) \(\approx\) \(0.4281722287\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
11$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - p T^{2} )^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
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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

−7.05813651340714620350130199679, −6.78820023125593017989067828946, −6.71866854605241423728366863602, −6.46808189778991609006335920607, −5.85155468172535609992734301066, −5.84430080541105412271687470428, −5.84149255384127535926568052636, −5.29820230688059406651349201561, −5.06278832377350930361764924146, −4.75222893608096037942303067435, −4.68128785864173441191328753785, −4.51559807789055526208734608411, −4.30818213461729069166455355256, −4.27926499727315289089739939820, −3.70721354746632601359932475889, −3.42830021632819486182643434622, −3.04854332335338325993964436251, −2.84981593718342395936760095328, −2.51901527819013492998931905679, −2.34015288593240148834715663035, −2.05035438238152100739931983970, −1.74616850029122072339844568918, −1.24949974391807442269895498300, −0.927406560044210912211595088998, −0.14495271801349587772582375727, 0.14495271801349587772582375727, 0.927406560044210912211595088998, 1.24949974391807442269895498300, 1.74616850029122072339844568918, 2.05035438238152100739931983970, 2.34015288593240148834715663035, 2.51901527819013492998931905679, 2.84981593718342395936760095328, 3.04854332335338325993964436251, 3.42830021632819486182643434622, 3.70721354746632601359932475889, 4.27926499727315289089739939820, 4.30818213461729069166455355256, 4.51559807789055526208734608411, 4.68128785864173441191328753785, 4.75222893608096037942303067435, 5.06278832377350930361764924146, 5.29820230688059406651349201561, 5.84149255384127535926568052636, 5.84430080541105412271687470428, 5.85155468172535609992734301066, 6.46808189778991609006335920607, 6.71866854605241423728366863602, 6.78820023125593017989067828946, 7.05813651340714620350130199679

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