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Properties

Label 8-2016e4-1.1-c0e4-0-2
Degree $8$
Conductor $1.652\times 10^{13}$
Sign $1$
Analytic cond. $1.02468$
Root an. cond. $1.00305$
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  − 4·2-s + 10·4-s − 20·8-s + 35·16-s − 56·32-s − 4·37-s + 84·64-s − 4·67-s + 4·71-s + 16·74-s + 8·113-s − 2·121-s + 127-s − 120·128-s + 131-s + 16·134-s + 137-s + 139-s − 16·142-s − 40·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + ⋯
L(s)  = 1  − 4·2-s + 10·4-s − 20·8-s + 35·16-s − 56·32-s − 4·37-s + 84·64-s − 4·67-s + 4·71-s + 16·74-s + 8·113-s − 2·121-s + 127-s − 120·128-s + 131-s + 16·134-s + 137-s + 139-s − 16·142-s − 40·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.02468\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1613207014\)
\(L(\frac12)\) \(\approx\) \(0.1613207014\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{4} \)
3 \( 1 \)
7$C_2^2$ \( 1 + T^{4} \)
good5$C_4\times C_2$ \( 1 + T^{8} \)
11$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
13$C_4\times C_2$ \( 1 + T^{8} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
19$C_4\times C_2$ \( 1 + T^{8} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
31$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
59$C_4\times C_2$ \( 1 + T^{8} \)
61$C_4\times C_2$ \( 1 + T^{8} \)
67$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
71$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + T^{4} )^{2} \)
83$C_4\times C_2$ \( 1 + T^{8} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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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

−6.89141021847263827147911690026, −6.57757259852417036526733768489, −6.55159505199474326552679683291, −6.31337914067575404348146795501, −6.15169297187923427910019536140, −5.76829299546126172531517036531, −5.61879126760612385310179043535, −5.45535340260555968904411124971, −5.27161458509778567789529720085, −4.89943604328929353183270299990, −4.72904336615261449074694070346, −4.06922833846949171414156592637, −4.05194059186810699415429646301, −3.56070618477746718440297683794, −3.26683398549657741636414371551, −3.23249212489918434301172589328, −3.02960876366254441446318137034, −2.78735109666548780952418628151, −2.17722260769148271210566573378, −2.08544524374370391571382113657, −1.93449400373786570754117932742, −1.63816856164113404029374405661, −1.40997760053730715258220385359, −0.813611574727140676023112931511, −0.51000993293398624026820823225, 0.51000993293398624026820823225, 0.813611574727140676023112931511, 1.40997760053730715258220385359, 1.63816856164113404029374405661, 1.93449400373786570754117932742, 2.08544524374370391571382113657, 2.17722260769148271210566573378, 2.78735109666548780952418628151, 3.02960876366254441446318137034, 3.23249212489918434301172589328, 3.26683398549657741636414371551, 3.56070618477746718440297683794, 4.05194059186810699415429646301, 4.06922833846949171414156592637, 4.72904336615261449074694070346, 4.89943604328929353183270299990, 5.27161458509778567789529720085, 5.45535340260555968904411124971, 5.61879126760612385310179043535, 5.76829299546126172531517036531, 6.15169297187923427910019536140, 6.31337914067575404348146795501, 6.55159505199474326552679683291, 6.57757259852417036526733768489, 6.89141021847263827147911690026

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