Properties

Label 8-768e4-1.1-c0e4-0-1
Degree $8$
Conductor $347892350976$
Sign $1$
Analytic cond. $0.0215810$
Root an. cond. $0.619097$
Motivic weight $0$
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  + 4·13-s − 4·37-s − 4·61-s − 81-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  + 4·13-s − 4·37-s − 4·61-s − 81-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(0.0215810\)
Root analytic conductor: \(0.619097\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 3^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9008900305\)
\(L(\frac12)\) \(\approx\) \(0.9008900305\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
good5$C_2^2$ \( ( 1 + T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_2$ \( ( 1 + T^{2} )^{4} \)
29$C_2^2$ \( ( 1 + T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + T^{4} )^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
79$C_2^2$ \( ( 1 + T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_2$ \( ( 1 + T^{2} )^{4} \)
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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

−7.84516857620233318507649007436, −7.23657877521383497767263340020, −7.14117594131401513683028128746, −6.91575487484178863634416985091, −6.81527305640456196656582093577, −6.31594235116094141169306955317, −6.25480779584041687786407899601, −6.00712842047157907449913149768, −5.97765466754364041880612449728, −5.43722649342321044303429795478, −5.43626017265676189353926315350, −5.05874528787727002927089794351, −4.87568833691894492966308118606, −4.40429449059889297546425732176, −4.06236654040940025278172943283, −4.03831900037673161004832773549, −3.70082047242043809456201066502, −3.36019090097599852242217091277, −3.31422842149176062958162581066, −2.81946286797672009368096705447, −2.76129092971241327325370869272, −1.73880738035066004212755957453, −1.66274092463359388474131334946, −1.63729939982513489239532469311, −1.02178174975497537173849244892, 1.02178174975497537173849244892, 1.63729939982513489239532469311, 1.66274092463359388474131334946, 1.73880738035066004212755957453, 2.76129092971241327325370869272, 2.81946286797672009368096705447, 3.31422842149176062958162581066, 3.36019090097599852242217091277, 3.70082047242043809456201066502, 4.03831900037673161004832773549, 4.06236654040940025278172943283, 4.40429449059889297546425732176, 4.87568833691894492966308118606, 5.05874528787727002927089794351, 5.43626017265676189353926315350, 5.43722649342321044303429795478, 5.97765466754364041880612449728, 6.00712842047157907449913149768, 6.25480779584041687786407899601, 6.31594235116094141169306955317, 6.81527305640456196656582093577, 6.91575487484178863634416985091, 7.14117594131401513683028128746, 7.23657877521383497767263340020, 7.84516857620233318507649007436

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