Properties

Label 8-700e4-1.1-c1e4-0-18
Degree $8$
Conductor $240100000000$
Sign $1$
Analytic cond. $976.114$
Root an. cond. $2.36421$
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  + 2·2-s + 2·4-s + 8·7-s + 4·8-s + 6·9-s + 12·13-s + 16·14-s + 8·16-s + 12·17-s + 12·18-s − 24·19-s + 8·23-s + 24·26-s + 16·28-s − 4·29-s − 24·31-s + 8·32-s + 24·34-s + 12·36-s − 48·38-s − 8·43-s + 16·46-s + 34·49-s + 24·52-s + 32·56-s − 8·58-s − 48·62-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 3.02·7-s + 1.41·8-s + 2·9-s + 3.32·13-s + 4.27·14-s + 2·16-s + 2.91·17-s + 2.82·18-s − 5.50·19-s + 1.66·23-s + 4.70·26-s + 3.02·28-s − 0.742·29-s − 4.31·31-s + 1.41·32-s + 4.11·34-s + 2·36-s − 7.78·38-s − 1.21·43-s + 2.35·46-s + 34/7·49-s + 3.32·52-s + 4.27·56-s − 1.05·58-s − 6.09·62-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(976.114\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{700} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(18.62759719\)
\(L(\frac12)\) \(\approx\) \(18.62759719\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
5 \( 1 \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \)
11$D_4\times C_2$ \( 1 - 30 T^{2} + 419 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 - 6 T + 31 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
23$D_{4}$ \( ( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 + 2 T + 11 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
37$D_4\times C_2$ \( 1 - 52 T^{2} + 1686 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 - 91 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 - 148 T^{2} + 11190 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2$ \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 228 T^{2} + 22310 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \)
73$D_{4}$ \( ( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 94 T^{2} + 3891 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 20 T^{2} + 6966 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 260 T^{2} + 31014 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 - 6 T + 95 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−7.56968409200590920535286813631, −7.20355228998278615978580797545, −7.17835730567238871697323978817, −6.59950314775382354753394256828, −6.58772497602824339999792469557, −6.46296307562815356265840056595, −5.81167879997851008378160833835, −5.75360519527220709299336193616, −5.69016982181752223593134501351, −5.19221689701828347802642710032, −5.09337940252152245630653265129, −4.87725290516887990004804222771, −4.43313475396283319680502768366, −4.40573596692281607346223255677, −4.09079785462358892302223593350, −3.80686171244561784277982207743, −3.66872300217759907557538966672, −3.60080138354343885813528493040, −3.22403684253494160205145639730, −2.28883845418861232094950655946, −1.97529746044495163547024368569, −1.90623059456436271549450019071, −1.46790204204344805951313824957, −1.43826092197498725155777993635, −1.00877070449489424581020370471, 1.00877070449489424581020370471, 1.43826092197498725155777993635, 1.46790204204344805951313824957, 1.90623059456436271549450019071, 1.97529746044495163547024368569, 2.28883845418861232094950655946, 3.22403684253494160205145639730, 3.60080138354343885813528493040, 3.66872300217759907557538966672, 3.80686171244561784277982207743, 4.09079785462358892302223593350, 4.40573596692281607346223255677, 4.43313475396283319680502768366, 4.87725290516887990004804222771, 5.09337940252152245630653265129, 5.19221689701828347802642710032, 5.69016982181752223593134501351, 5.75360519527220709299336193616, 5.81167879997851008378160833835, 6.46296307562815356265840056595, 6.58772497602824339999792469557, 6.59950314775382354753394256828, 7.17835730567238871697323978817, 7.20355228998278615978580797545, 7.56968409200590920535286813631

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