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Properties

Label 8-608e4-1.1-c1e4-0-0
Degree $8$
Conductor $136651472896$
Sign $1$
Analytic cond. $555.549$
Root an. cond. $2.20338$
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 + 5-s + 7-s − 9-s − 7·11-s + 10·13-s − 2·15-s + 5·17-s − 4·19-s − 2·21-s + 8·23-s − 25-s + 2·27-s + 10·29-s + 6·31-s + 14·33-s + 35-s + 14·37-s − 20·39-s − 2·41-s − 3·43-s − 45-s + 3·47-s − 10·49-s − 10·51-s + 4·53-s − 7·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s + 0.377·7-s − 1/3·9-s − 2.11·11-s + 2.77·13-s − 0.516·15-s + 1.21·17-s − 0.917·19-s − 0.436·21-s + 1.66·23-s − 1/5·25-s + 0.384·27-s + 1.85·29-s + 1.07·31-s + 2.43·33-s + 0.169·35-s + 2.30·37-s − 3.20·39-s − 0.312·41-s − 0.457·43-s − 0.149·45-s + 0.437·47-s − 1.42·49-s − 1.40·51-s + 0.549·53-s − 0.943·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(555.549\)
Root analytic conductor: \(2.20338\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.503115960\)
\(L(\frac12)\) \(\approx\) \(1.503115960\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 + T )^{4} \)
good3$D_{4}$ \( ( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
5$C_2 \wr C_2\wr C_2$ \( 1 - T + 2 T^{2} + 9 T^{3} + 2 T^{4} + 9 p T^{5} + 2 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - T + 11 T^{2} - 40 T^{3} + 60 T^{4} - 40 p T^{5} + 11 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 4 p T^{2} + 203 T^{3} + 742 T^{4} + 203 p T^{5} + 4 p^{3} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 59 T^{2} - 232 T^{3} + 852 T^{4} - 232 p T^{5} + 59 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + p T^{2} - 114 T^{3} + 698 T^{4} - 114 p T^{5} + p^{3} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 75 T^{2} - 420 T^{3} + 2456 T^{4} - 420 p T^{5} + 75 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 123 T^{2} - 712 T^{3} + 5108 T^{4} - 712 p T^{5} + 123 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 68 T^{2} - 142 T^{3} + 1782 T^{4} - 142 p T^{5} + 68 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 152 T^{2} - 1018 T^{3} + 7006 T^{4} - 1018 p T^{5} + 152 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 92 T^{2} + 54 T^{3} + 4694 T^{4} + 54 p T^{5} + 92 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 140 T^{2} + 291 T^{3} + 8278 T^{4} + 291 p T^{5} + 140 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 80 T^{2} - 247 T^{3} + 3038 T^{4} - 247 p T^{5} + 80 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 143 T^{2} - 252 T^{3} + 9032 T^{4} - 252 p T^{5} + 143 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 321 T^{2} + 3390 T^{3} + 30764 T^{4} + 3390 p T^{5} + 321 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 178 T^{2} + 521 T^{3} + 14618 T^{4} + 521 p T^{5} + 178 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 193 T^{2} + 1006 T^{3} + 16060 T^{4} + 1006 p T^{5} + 193 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 30 T + 552 T^{2} - 6814 T^{3} + 65870 T^{4} - 6814 p T^{5} + 552 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 245 T^{2} - 1662 T^{3} + 25114 T^{4} - 1662 p T^{5} + 245 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 216 T^{2} + 1082 T^{3} + 18510 T^{4} + 1082 p T^{5} + 216 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 216 T^{2} + 212 T^{3} + 21054 T^{4} + 212 p T^{5} + 216 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 152 T^{2} + 224 T^{3} - 4082 T^{4} + 224 p T^{5} + 152 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 328 T^{2} + 1658 T^{3} + 45422 T^{4} + 1658 p T^{5} + 328 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.77454623900532973025907131074, −7.71628544232690523294263363375, −6.93444878712010645181800671551, −6.82406195250834855024347141590, −6.77389117415901392003830991004, −6.29362316127869186284936383111, −6.19435138676468426777749806087, −5.95773628257800292327958979909, −5.86534732208057167535562191637, −5.43531100704852863368288043836, −5.28290744138970174470089862419, −5.02926061034421063496546199502, −5.02454952987412360949421269722, −4.41908281348470975355167675132, −4.24903991070791365842302556112, −3.95305371397522074169362689592, −3.66468006284354467491265928533, −3.05558974670843458495821775117, −2.81979441333074951592183675639, −2.78560795102084779332930551786, −2.59239824471435804869879595117, −1.59563453040170557506753101078, −1.50168463122502933759317682040, −1.04227641104780986626024553948, −0.45769094431582547630532906869, 0.45769094431582547630532906869, 1.04227641104780986626024553948, 1.50168463122502933759317682040, 1.59563453040170557506753101078, 2.59239824471435804869879595117, 2.78560795102084779332930551786, 2.81979441333074951592183675639, 3.05558974670843458495821775117, 3.66468006284354467491265928533, 3.95305371397522074169362689592, 4.24903991070791365842302556112, 4.41908281348470975355167675132, 5.02454952987412360949421269722, 5.02926061034421063496546199502, 5.28290744138970174470089862419, 5.43531100704852863368288043836, 5.86534732208057167535562191637, 5.95773628257800292327958979909, 6.19435138676468426777749806087, 6.29362316127869186284936383111, 6.77389117415901392003830991004, 6.82406195250834855024347141590, 6.93444878712010645181800671551, 7.71628544232690523294263363375, 7.77454623900532973025907131074

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