Properties

Label 8-140e4-1.1-c1e4-0-4
Degree $8$
Conductor $384160000$
Sign $1$
Analytic cond. $1.56178$
Root an. cond. $1.05731$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s − 2·5-s + 3·7-s + 15·9-s + 3·11-s − 12·15-s − 9·17-s − 19-s + 18·21-s + 12·23-s − 7·25-s + 18·27-s − 2·29-s + 31-s + 18·33-s − 6·35-s − 27·37-s − 30·41-s − 30·45-s + 15·47-s + 7·49-s − 54·51-s + 3·53-s − 6·55-s − 6·57-s + 59-s + 12·61-s + ⋯
L(s)  = 1  + 3.46·3-s − 0.894·5-s + 1.13·7-s + 5·9-s + 0.904·11-s − 3.09·15-s − 2.18·17-s − 0.229·19-s + 3.92·21-s + 2.50·23-s − 7/5·25-s + 3.46·27-s − 0.371·29-s + 0.179·31-s + 3.13·33-s − 1.01·35-s − 4.43·37-s − 4.68·41-s − 4.47·45-s + 2.18·47-s + 49-s − 7.56·51-s + 0.412·53-s − 0.809·55-s − 0.794·57-s + 0.130·59-s + 1.53·61-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.56178\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{140} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.336135032\)
\(L(\frac12)\) \(\approx\) \(3.336135032\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )^{2}( 1 + p T^{2} )^{2} \)
11$D_4\times C_2$ \( 1 - 3 T - T^{2} + 36 T^{3} - 120 T^{4} + 36 p T^{5} - p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 - 8 T^{2} + 126 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 + 9 T + 63 T^{2} + 324 T^{3} + 1466 T^{4} + 324 p T^{5} + 63 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 + T - 23 T^{2} - 14 T^{3} + 196 T^{4} - 14 p T^{5} - 23 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
23$C_2$$\times$$C_2^2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} ) \)
29$D_{4}$ \( ( 1 + T + 44 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - T - 47 T^{2} + 14 T^{3} + 1312 T^{4} + 14 p T^{5} - 47 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 + 27 T + 373 T^{2} + 3510 T^{3} + 24522 T^{4} + 3510 p T^{5} + 373 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 15 T + 124 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 125 T^{2} + 7248 T^{4} - 125 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 15 T + 183 T^{2} - 1620 T^{3} + 12980 T^{4} - 1620 p T^{5} + 183 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 3 T + 67 T^{2} - 192 T^{3} + 1446 T^{4} - 192 p T^{5} + 67 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - T - 103 T^{2} + 14 T^{3} + 7276 T^{4} + 14 p T^{5} - 103 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 12 T + 43 T^{2} + 252 T^{3} - 2304 T^{4} + 252 p T^{5} + 43 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 18 T + 193 T^{2} - 1530 T^{3} + 9972 T^{4} - 1530 p T^{5} + 193 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 15 T + 235 T^{2} - 2400 T^{3} + 25746 T^{4} - 2400 p T^{5} + 235 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 7 T - 107 T^{2} + 14 T^{3} + 14224 T^{4} + 14 p T^{5} - 107 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 245 T^{2} + 28656 T^{4} - 245 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 + 7 T - 40 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \)
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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

−9.486777281139055447548499103507, −9.016022891953113342712542735998, −9.006782490216052697291878239909, −8.891019769530485354095453845206, −8.352237345153777076379897922362, −8.320085466032543741524198492635, −8.297116144780486952166476848643, −8.200790875209265107201527491740, −7.56724516418755785646199133188, −6.97272893606860122951055659982, −6.91060570836426987869113010442, −6.85207858241255026999163430946, −6.78512934521630292500291983834, −5.67768649321110556795598673553, −5.42045278587186085757282105169, −4.94560266360483320586645260055, −4.87039846142984731138562535907, −4.21641432278640671822632457388, −3.71715217461558743430330436490, −3.55780134359169229725739791947, −3.53518049086950142524561968582, −2.92974880914809004783737323761, −2.25357071367651651651678217180, −2.15059284060156860198596146756, −1.70187967642289420603536752132, 1.70187967642289420603536752132, 2.15059284060156860198596146756, 2.25357071367651651651678217180, 2.92974880914809004783737323761, 3.53518049086950142524561968582, 3.55780134359169229725739791947, 3.71715217461558743430330436490, 4.21641432278640671822632457388, 4.87039846142984731138562535907, 4.94560266360483320586645260055, 5.42045278587186085757282105169, 5.67768649321110556795598673553, 6.78512934521630292500291983834, 6.85207858241255026999163430946, 6.91060570836426987869113010442, 6.97272893606860122951055659982, 7.56724516418755785646199133188, 8.200790875209265107201527491740, 8.297116144780486952166476848643, 8.320085466032543741524198492635, 8.352237345153777076379897922362, 8.891019769530485354095453845206, 9.006782490216052697291878239909, 9.016022891953113342712542735998, 9.486777281139055447548499103507

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