Properties

Label 8-560e4-1.1-c1e4-0-6
Degree $8$
Conductor $98344960000$
Sign $1$
Analytic cond. $399.816$
Root an. cond. $2.11462$
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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(399.816\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{560} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.460462104\)
\(L(\frac12)\) \(\approx\) \(1.460462104\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \)
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

−7.65913339828836832565532654024, −7.65429634246921221814901801122, −7.42671116477326197263980006442, −7.20226220740905371285949491999, −6.89858435197585086374222599199, −6.31983640096746890680335091522, −6.20514548278450681386118297719, −6.00678235773738969096745108412, −5.84083781942698524420119909191, −5.77296498686735187792720309104, −5.21591744697219998737670193448, −5.12877884704281348068737638064, −4.79871513576928201258031087100, −4.66578558696794469330022086297, −3.93604656073639836598130336332, −3.88671544927220471547075621835, −3.80036155467176529376893035210, −3.51280139105421303898109979248, −3.47148343844777682612497715294, −2.52688002110971297911429488291, −2.03853526150052686716080141763, −1.86449883122570750352496448683, −1.77476745804642224710503056426, −0.75983506415339821548267893998, −0.58943952282039319654075048682, 0.58943952282039319654075048682, 0.75983506415339821548267893998, 1.77476745804642224710503056426, 1.86449883122570750352496448683, 2.03853526150052686716080141763, 2.52688002110971297911429488291, 3.47148343844777682612497715294, 3.51280139105421303898109979248, 3.80036155467176529376893035210, 3.88671544927220471547075621835, 3.93604656073639836598130336332, 4.66578558696794469330022086297, 4.79871513576928201258031087100, 5.12877884704281348068737638064, 5.21591744697219998737670193448, 5.77296498686735187792720309104, 5.84083781942698524420119909191, 6.00678235773738969096745108412, 6.20514548278450681386118297719, 6.31983640096746890680335091522, 6.89858435197585086374222599199, 7.20226220740905371285949491999, 7.42671116477326197263980006442, 7.65429634246921221814901801122, 7.65913339828836832565532654024

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