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Properties

Label 8-1568e4-1.1-c0e4-0-1
Degree $8$
Conductor $6.045\times 10^{12}$
Sign $1$
Analytic cond. $0.374983$
Root an. cond. $0.884609$
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  − 16-s + 4·23-s + 4·43-s − 4·53-s + 4·67-s + 4·107-s − 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  − 16-s + 4·23-s + 4·43-s − 4·53-s + 4·67-s + 4·107-s − 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(0.374983\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 7^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.324724652\)
\(L(\frac12)\) \(\approx\) \(1.324724652\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
7 \( 1 \)
good3$C_4\times C_2$ \( 1 + T^{8} \)
5$C_4\times C_2$ \( 1 + T^{8} \)
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_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
29$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
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$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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$C_2$ \( ( 1 + T^{2} )^{4} \)
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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

−6.86572924659071774845739012619, −6.73119279021922069434551461936, −6.66640746776151022767153792532, −6.37757125908925135101429686566, −6.00187430441943463057250871193, −5.84485244555263839439400979653, −5.64530623472431618094629711453, −5.46892101243739222507119211348, −5.06437242676469845021734271573, −4.92963636392468015994646322892, −4.79903656229115455758747502400, −4.48005800797459833174797614894, −4.42820208402534793262029742496, −4.12536091010976957403868168768, −3.68892987091514333311053187165, −3.43879139777043829952931053248, −3.22093298491869310120391106001, −3.18410944080351939300328375512, −2.53256787234169924544659223634, −2.51712324361124875632430646967, −2.36621571555812480692928309701, −1.91227291388706616183797107536, −1.36601141306477706755349510952, −1.07220278636671478710628024423, −0.830273354711556467065703469392, 0.830273354711556467065703469392, 1.07220278636671478710628024423, 1.36601141306477706755349510952, 1.91227291388706616183797107536, 2.36621571555812480692928309701, 2.51712324361124875632430646967, 2.53256787234169924544659223634, 3.18410944080351939300328375512, 3.22093298491869310120391106001, 3.43879139777043829952931053248, 3.68892987091514333311053187165, 4.12536091010976957403868168768, 4.42820208402534793262029742496, 4.48005800797459833174797614894, 4.79903656229115455758747502400, 4.92963636392468015994646322892, 5.06437242676469845021734271573, 5.46892101243739222507119211348, 5.64530623472431618094629711453, 5.84485244555263839439400979653, 6.00187430441943463057250871193, 6.37757125908925135101429686566, 6.66640746776151022767153792532, 6.73119279021922069434551461936, 6.86572924659071774845739012619

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