Properties

Label 8-1456e4-1.1-c1e4-0-4
Degree $8$
Conductor $4.494\times 10^{12}$
Sign $1$
Analytic cond. $18270.6$
Root an. cond. $3.40972$
Motivic weight $1$
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  − 3·3-s + 6·5-s + 2·7-s + 7·9-s − 3·11-s − 10·13-s − 18·15-s − 6·17-s + 3·19-s − 6·21-s + 5·25-s − 18·27-s − 3·29-s − 8·31-s + 9·33-s + 12·35-s − 8·37-s + 30·39-s − 6·41-s + 5·43-s + 42·45-s + 49-s + 18·51-s + 24·53-s − 18·55-s − 9·57-s + 12·61-s + ⋯
L(s)  = 1  − 1.73·3-s + 2.68·5-s + 0.755·7-s + 7/3·9-s − 0.904·11-s − 2.77·13-s − 4.64·15-s − 1.45·17-s + 0.688·19-s − 1.30·21-s + 25-s − 3.46·27-s − 0.557·29-s − 1.43·31-s + 1.56·33-s + 2.02·35-s − 1.31·37-s + 4.80·39-s − 0.937·41-s + 0.762·43-s + 6.26·45-s + 1/7·49-s + 2.52·51-s + 3.29·53-s − 2.42·55-s − 1.19·57-s + 1.53·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 7^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(18270.6\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.232918853\)
\(L(\frac12)\) \(\approx\) \(1.232918853\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( ( 1 - T + T^{2} )^{2} \)
13$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
good3$D_4\times C_2$ \( 1 + p T + 2 T^{2} + p T^{3} + 13 T^{4} + p^{2} T^{5} + 2 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
5$D_{4}$ \( ( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 + 3 T - 4 T^{2} - 27 T^{3} - 51 T^{4} - 27 p T^{5} - 4 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 + 6 T + 13 T^{2} - 66 T^{3} - 372 T^{4} - 66 p T^{5} + 13 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 - 3 T - 20 T^{2} + 27 T^{3} + 309 T^{4} + 27 p T^{5} - 20 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^3$ \( 1 - 26 T^{2} + 147 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 + 3 T - 20 T^{2} - 3 p T^{3} - 9 p T^{4} - 3 p^{2} T^{5} - 20 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
31$D_{4}$ \( ( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 + 6 T - 50 T^{2} + 24 T^{3} + 4239 T^{4} + 24 p T^{5} - 50 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 5 T + 34 T^{2} + 475 T^{3} - 2843 T^{4} + 475 p T^{5} + 34 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 + 89 T^{2} + p^{2} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 - 113 T^{2} + 9288 T^{4} - 113 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + 12 T + 19 T^{2} - 108 T^{3} + 1488 T^{4} - 108 p T^{5} + 19 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
71$C_4\times C_2$ \( 1 + 6 T + 10 T^{2} - 696 T^{3} - 6921 T^{4} - 696 p T^{5} + 10 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 121 T^{2} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 21 T + 184 T^{2} + 1659 T^{3} + 18879 T^{4} + 1659 p T^{5} + 184 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 + 31 T + 538 T^{2} + 7099 T^{3} + 76303 T^{4} + 7099 p T^{5} + 538 p^{2} T^{6} + 31 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.79600763467352023239541919308, −6.72899878726561293060031101010, −6.17312679735306930656083857962, −5.88355651269250119452238942075, −5.82127914794909148561939263234, −5.71860605490438164010844193840, −5.42490829108155258210438819864, −5.42477486779351927903654911770, −5.17553296970179087253123296717, −4.91990889834404620433462475149, −4.90541504820762970113688005184, −4.21317027481963991441209650743, −4.14715661628870675886569742713, −4.14385511469237586900135048246, −3.90459372055797932037366676000, −3.12677367326224512025377922540, −2.92693113168608544465786712280, −2.52600830685656802271889325451, −2.49150510071916518720464521296, −1.91467115775109604064111656407, −1.79674521092784992695523399081, −1.73710827097335748374525934714, −1.62271613302544210424979023036, −0.54377516870769913214941928173, −0.33316712018401380513084832639, 0.33316712018401380513084832639, 0.54377516870769913214941928173, 1.62271613302544210424979023036, 1.73710827097335748374525934714, 1.79674521092784992695523399081, 1.91467115775109604064111656407, 2.49150510071916518720464521296, 2.52600830685656802271889325451, 2.92693113168608544465786712280, 3.12677367326224512025377922540, 3.90459372055797932037366676000, 4.14385511469237586900135048246, 4.14715661628870675886569742713, 4.21317027481963991441209650743, 4.90541504820762970113688005184, 4.91990889834404620433462475149, 5.17553296970179087253123296717, 5.42477486779351927903654911770, 5.42490829108155258210438819864, 5.71860605490438164010844193840, 5.82127914794909148561939263234, 5.88355651269250119452238942075, 6.17312679735306930656083857962, 6.72899878726561293060031101010, 6.79600763467352023239541919308

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