Properties

Label 8-50e8-1.1-c0e4-0-0
Degree $8$
Conductor $3.906\times 10^{13}$
Sign $1$
Analytic cond. $2.42319$
Root an. cond. $1.11698$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 2·6-s − 2·7-s + 9-s + 2·14-s − 18-s + 4·21-s − 2·23-s − 2·29-s + 32-s + 3·41-s − 4·42-s − 2·43-s + 2·46-s − 2·47-s + 49-s + 2·58-s − 2·61-s − 2·63-s − 64-s − 2·67-s + 4·69-s − 3·82-s + 3·83-s + 2·86-s + 4·87-s + ⋯
L(s)  = 1  − 2-s − 2·3-s + 2·6-s − 2·7-s + 9-s + 2·14-s − 18-s + 4·21-s − 2·23-s − 2·29-s + 32-s + 3·41-s − 4·42-s − 2·43-s + 2·46-s − 2·47-s + 49-s + 2·58-s − 2·61-s − 2·63-s − 64-s − 2·67-s + 4·69-s − 3·82-s + 3·83-s + 2·86-s + 4·87-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(2.42319\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{16} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.007228967133\)
\(L(\frac12)\) \(\approx\) \(0.007228967133\)
\(L(1)\) 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$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
5 \( 1 \)
good3$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
7$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
11$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
13$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
17$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
19$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
23$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
29$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
41$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
43$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
47$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
53$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
59$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
61$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
67$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
71$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
73$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
79$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
83$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
89$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
97$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
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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

−6.42759653189872476815767693084, −6.31542385665047452944571800113, −6.08077878203059521257669762957, −5.95594482080390037506365610999, −5.91519100068082864209217899615, −5.53751144888620573910112577479, −5.49351499201191168127081265547, −5.18435912036940852342520134532, −4.93527975255815716123480287763, −4.76434976190238893648256653769, −4.37639939261935441017678161702, −4.21342182419586493320455145483, −4.13660115052624248464680896495, −3.57870559537665670100833798410, −3.51816609724517987376546169782, −3.45798275297208945867002549464, −3.03417081999184028604173930513, −2.69945112830998560793613442524, −2.57085981066576402223042236850, −2.27050737583388664364618984720, −1.76415357208800149807363057969, −1.51541286343495239524025177349, −1.31329506776727121976079557250, −0.51602764847268626932077886377, −0.10226218936003877317716632035, 0.10226218936003877317716632035, 0.51602764847268626932077886377, 1.31329506776727121976079557250, 1.51541286343495239524025177349, 1.76415357208800149807363057969, 2.27050737583388664364618984720, 2.57085981066576402223042236850, 2.69945112830998560793613442524, 3.03417081999184028604173930513, 3.45798275297208945867002549464, 3.51816609724517987376546169782, 3.57870559537665670100833798410, 4.13660115052624248464680896495, 4.21342182419586493320455145483, 4.37639939261935441017678161702, 4.76434976190238893648256653769, 4.93527975255815716123480287763, 5.18435912036940852342520134532, 5.49351499201191168127081265547, 5.53751144888620573910112577479, 5.91519100068082864209217899615, 5.95594482080390037506365610999, 6.08077878203059521257669762957, 6.31542385665047452944571800113, 6.42759653189872476815767693084

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