Properties

Label 8-1960e4-1.1-c1e4-0-4
Degree $8$
Conductor $1.476\times 10^{13}$
Sign $1$
Analytic cond. $59997.4$
Root an. cond. $3.95609$
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-s + 2·5-s − 2·9-s − 7·11-s + 6·13-s + 2·15-s − 5·17-s − 2·19-s − 2·23-s + 25-s − 13·27-s − 6·29-s + 16·31-s − 7·33-s + 4·37-s + 6·39-s + 4·41-s − 12·43-s − 4·45-s − 3·47-s − 5·51-s − 10·53-s − 14·55-s − 2·57-s − 16·59-s − 6·61-s + 12·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 2/3·9-s − 2.11·11-s + 1.66·13-s + 0.516·15-s − 1.21·17-s − 0.458·19-s − 0.417·23-s + 1/5·25-s − 2.50·27-s − 1.11·29-s + 2.87·31-s − 1.21·33-s + 0.657·37-s + 0.960·39-s + 0.624·41-s − 1.82·43-s − 0.596·45-s − 0.437·47-s − 0.700·51-s − 1.37·53-s − 1.88·55-s − 0.264·57-s − 2.08·59-s − 0.768·61-s + 1.48·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(59997.4\)
Root analytic conductor: \(3.95609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1960} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.345533696\)
\(L(\frac12)\) \(\approx\) \(3.345533696\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( ( 1 - T + T^{2} )^{2} \)
7 \( 1 \)
good3$C_2$$\times$$C_2^2$ \( ( 1 - T + p T^{2} )^{2}( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} ) \)
11$D_4\times C_2$ \( 1 + 7 T + 23 T^{2} + 28 T^{3} + 16 T^{4} + 28 p T^{5} + 23 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
13$D_{4}$ \( ( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 5 T - 7 T^{2} - 10 T^{3} + 310 T^{4} - 10 p T^{5} - 7 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 + 2 T - 2 T^{2} - 64 T^{3} - 401 T^{4} - 64 p T^{5} - 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 + 2 T - 10 T^{2} - 64 T^{3} - 425 T^{4} - 64 p T^{5} - 10 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 3 T - 13 T^{2} - 216 T^{3} - 2148 T^{4} - 216 p T^{5} - 13 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 + 10 T + 2 T^{2} - 80 T^{3} + 1495 T^{4} - 80 p T^{5} + 2 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 + 6 T - 62 T^{2} - 144 T^{3} + 3687 T^{4} - 144 p T^{5} - 62 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2^2$ \( ( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 + 13 T + 43 T^{2} - 416 T^{3} - 3716 T^{4} - 416 p T^{5} + 43 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 18 T + 98 T^{2} + 864 T^{3} + 14319 T^{4} + 864 p T^{5} + 98 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.42635390320469833423670983345, −6.23373760007879460888811108947, −6.17393457721935811324938319983, −6.10169129274440323748266669649, −5.62490633333546792270366503254, −5.59270882168321379392135761614, −5.19468840123823791003764299604, −5.11558015129126344475580157147, −4.68558004244876747260462772387, −4.64263065434443275130608608210, −4.63304864222192474172565493344, −4.02303340418776986821905333479, −3.78812735484311601744112288211, −3.63270243102510229865068218828, −3.48009759212399328790557296931, −2.98464112768167879233934969126, −2.97350550328332118497727542090, −2.62585239714198096983181825339, −2.36906392741702027741345667879, −2.05388001953233852715683069534, −1.97050123391877081624312103863, −1.65599864528202014112322764702, −1.27071357407278623969741918551, −0.48095394240061577358413906220, −0.47005717657123338054296365084, 0.47005717657123338054296365084, 0.48095394240061577358413906220, 1.27071357407278623969741918551, 1.65599864528202014112322764702, 1.97050123391877081624312103863, 2.05388001953233852715683069534, 2.36906392741702027741345667879, 2.62585239714198096983181825339, 2.97350550328332118497727542090, 2.98464112768167879233934969126, 3.48009759212399328790557296931, 3.63270243102510229865068218828, 3.78812735484311601744112288211, 4.02303340418776986821905333479, 4.63304864222192474172565493344, 4.64263065434443275130608608210, 4.68558004244876747260462772387, 5.11558015129126344475580157147, 5.19468840123823791003764299604, 5.59270882168321379392135761614, 5.62490633333546792270366503254, 6.10169129274440323748266669649, 6.17393457721935811324938319983, 6.23373760007879460888811108947, 6.42635390320469833423670983345

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