Properties

Label 8-280e4-1.1-c1e4-0-7
Degree $8$
Conductor $6146560000$
Sign $1$
Analytic cond. $24.9885$
Root an. cond. $1.49526$
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  + 2·3-s − 2·5-s − 2·7-s + 5·9-s + 4·11-s + 8·13-s − 4·15-s + 4·17-s − 4·21-s + 14·23-s + 25-s + 10·27-s − 20·29-s + 4·31-s + 8·33-s + 4·35-s + 16·39-s + 12·41-s + 12·43-s − 10·45-s − 12·47-s + 7·49-s + 8·51-s − 8·55-s + 8·59-s − 2·61-s − 10·63-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s − 0.755·7-s + 5/3·9-s + 1.20·11-s + 2.21·13-s − 1.03·15-s + 0.970·17-s − 0.872·21-s + 2.91·23-s + 1/5·25-s + 1.92·27-s − 3.71·29-s + 0.718·31-s + 1.39·33-s + 0.676·35-s + 2.56·39-s + 1.87·41-s + 1.82·43-s − 1.49·45-s − 1.75·47-s + 49-s + 1.12·51-s − 1.07·55-s + 1.04·59-s − 0.256·61-s − 1.25·63-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(24.9885\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{280} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.367913427\)
\(L(\frac12)\) \(\approx\) \(3.367913427\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
good3$D_4\times C_2$ \( 1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 2 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 4 T - 2 T^{2} + 16 T^{3} + 27 T^{4} + 16 p T^{5} - 2 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
17$C_4\times C_2$ \( 1 - 4 T + 10 T^{2} + 112 T^{3} - 525 T^{4} + 112 p T^{5} + 10 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^3$ \( 1 - 6 T^{2} - 325 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 14 T + 103 T^{2} - 658 T^{3} + 3612 T^{4} - 658 p T^{5} + 103 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 10 T + 75 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 4 T - 42 T^{2} + 16 T^{3} + 1907 T^{4} + 16 p T^{5} - 42 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^3$ \( 1 - 42 T^{2} + 395 T^{4} - 42 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 - 6 T - 3 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 12 T + 46 T^{2} + 48 T^{3} + 627 T^{4} + 48 p T^{5} + 46 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 74 T^{2} + 2667 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 + 2 T - 87 T^{2} - 62 T^{3} + 4316 T^{4} - 62 p T^{5} - 87 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 6 T - 9 T^{2} + 534 T^{3} - 4876 T^{4} + 534 p T^{5} - 9 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
73$D_4\times C_2$ \( 1 - 4 T - 102 T^{2} + 112 T^{3} + 7427 T^{4} + 112 p T^{5} - 102 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \)
83$D_{4}$ \( ( 1 + 18 T + 229 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 22 T + 217 T^{2} - 22 p T^{3} + 228 p T^{4} - 22 p^{2} T^{5} + 217 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
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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

−8.752062120743038113528435166263, −8.673342836698844699501629770381, −7.990939318172307776076728691178, −7.77056097420570372136053467131, −7.71611618647286109197631231783, −7.33317776428191700037641836602, −7.20226861002066741457509607574, −6.92504814515629327873159953462, −6.69957839915771996295503766977, −6.24846153389133326059142244730, −5.93320887717007170126663228607, −5.88126113513778346760081005637, −5.50931223384563075237847348737, −5.06629752891740358246001047760, −4.56465708627764315012572501010, −4.38171337426815920232364466510, −3.96709219272168696580941660623, −3.79191535636515774799593132944, −3.52318138277711229973587567654, −3.34266005373798626897546140278, −2.82777380518921199031991292441, −2.61710245773607030521634850411, −1.75438787182776435296767405066, −1.24851763638738315928648747943, −1.08428940480459934603475646725, 1.08428940480459934603475646725, 1.24851763638738315928648747943, 1.75438787182776435296767405066, 2.61710245773607030521634850411, 2.82777380518921199031991292441, 3.34266005373798626897546140278, 3.52318138277711229973587567654, 3.79191535636515774799593132944, 3.96709219272168696580941660623, 4.38171337426815920232364466510, 4.56465708627764315012572501010, 5.06629752891740358246001047760, 5.50931223384563075237847348737, 5.88126113513778346760081005637, 5.93320887717007170126663228607, 6.24846153389133326059142244730, 6.69957839915771996295503766977, 6.92504814515629327873159953462, 7.20226861002066741457509607574, 7.33317776428191700037641836602, 7.71611618647286109197631231783, 7.77056097420570372136053467131, 7.990939318172307776076728691178, 8.673342836698844699501629770381, 8.752062120743038113528435166263

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