Properties

Label 8-608e4-1.1-c0e4-0-0
Degree $8$
Conductor $136651472896$
Sign $1$
Analytic cond. $0.00847701$
Root an. cond. $0.550846$
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  + 2·5-s − 9-s − 2·13-s − 2·17-s + 3·25-s − 2·29-s − 2·41-s − 2·45-s + 4·49-s + 2·53-s + 2·61-s − 4·65-s + 2·73-s + 81-s − 4·85-s − 2·89-s − 2·97-s − 2·101-s − 2·109-s + 2·117-s + 4·121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + ⋯
L(s)  = 1  + 2·5-s − 9-s − 2·13-s − 2·17-s + 3·25-s − 2·29-s − 2·41-s − 2·45-s + 4·49-s + 2·53-s + 2·61-s − 4·65-s + 2·73-s + 81-s − 4·85-s − 2·89-s − 2·97-s − 2·101-s − 2·109-s + 2·117-s + 4·121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(0.00847701\)
Root analytic conductor: \(0.550846\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 19^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7031794759\)
\(L(\frac12)\) \(\approx\) \(0.7031794759\)
\(L(1)\) 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 \( 1 \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
5$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
13$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
23$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
29$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_2$ \( ( 1 + T^{2} )^{4} \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
43$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
47$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
53$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
59$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
71$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
79$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
83$C_2$ \( ( 1 + T^{2} )^{4} \)
89$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
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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

−7.968856433519597690605333638530, −7.60040728919371533297932533832, −7.17375748317411581951368264135, −7.15947266291594548559820984459, −6.91479889423352608118592797952, −6.74703446302938217338749664756, −6.65820829769792790057472118000, −6.15609306423635503235058732235, −5.96510884259150119521028902577, −5.49755724781655768682368337989, −5.44132731449757688771682563388, −5.32969246559001744155013287283, −5.30889111847725754477391362922, −4.84145422907222959210007397475, −4.27624416230100452575007135126, −4.16908640299428815677374098549, −4.14650064633458554788531923730, −3.46141081493330175932817934696, −3.18827094450451246200727851099, −2.68355143182558775553460465950, −2.54060602965108863061103159857, −2.18537959018974700229037322651, −2.12173342786610617706684343655, −1.78396047555416462650800385223, −0.921356769282575708589389190251, 0.921356769282575708589389190251, 1.78396047555416462650800385223, 2.12173342786610617706684343655, 2.18537959018974700229037322651, 2.54060602965108863061103159857, 2.68355143182558775553460465950, 3.18827094450451246200727851099, 3.46141081493330175932817934696, 4.14650064633458554788531923730, 4.16908640299428815677374098549, 4.27624416230100452575007135126, 4.84145422907222959210007397475, 5.30889111847725754477391362922, 5.32969246559001744155013287283, 5.44132731449757688771682563388, 5.49755724781655768682368337989, 5.96510884259150119521028902577, 6.15609306423635503235058732235, 6.65820829769792790057472118000, 6.74703446302938217338749664756, 6.91479889423352608118592797952, 7.15947266291594548559820984459, 7.17375748317411581951368264135, 7.60040728919371533297932533832, 7.968856433519597690605333638530

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