Properties

Label 8-1078e4-1.1-c1e4-0-4
Degree $8$
Conductor $1.350\times 10^{12}$
Sign $1$
Analytic cond. $5490.14$
Root an. cond. $2.93391$
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 + 2·3-s + 4-s − 2·5-s − 4·6-s + 2·8-s + 2·9-s + 4·10-s − 2·11-s + 2·12-s − 4·13-s − 4·15-s − 4·16-s + 4·17-s − 4·18-s + 10·19-s − 2·20-s + 4·22-s − 8·23-s + 4·24-s + 6·25-s + 8·26-s − 8·27-s + 8·30-s − 4·31-s + 2·32-s − 4·33-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s − 1.63·6-s + 0.707·8-s + 2/3·9-s + 1.26·10-s − 0.603·11-s + 0.577·12-s − 1.10·13-s − 1.03·15-s − 16-s + 0.970·17-s − 0.942·18-s + 2.29·19-s − 0.447·20-s + 0.852·22-s − 1.66·23-s + 0.816·24-s + 6/5·25-s + 1.56·26-s − 1.53·27-s + 1.46·30-s − 0.718·31-s + 0.353·32-s − 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 7^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(5490.14\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1078} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 7^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.9288603294\)
\(L(\frac12)\) \(\approx\) \(0.9288603294\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T + T^{2} )^{2} \)
7 \( 1 \)
11$C_2$ \( ( 1 + T + T^{2} )^{2} \)
good3$C_2^3$ \( 1 - 2 T + 2 T^{2} + 8 T^{3} - 17 T^{4} + 8 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
5$D_4\times C_2$ \( 1 + 2 T - 2 T^{2} - 8 T^{3} - 9 T^{4} - 8 p T^{5} - 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$D_{4}$ \( ( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 4 T - 2 T^{2} + 64 T^{3} - 237 T^{4} + 64 p T^{5} - 2 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 - 10 T + 42 T^{2} - 200 T^{3} + 1103 T^{4} - 200 p T^{5} + 42 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 4 T + 18 T^{2} + 304 T^{3} - 1957 T^{4} + 304 p T^{5} + 18 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 + 8 T - 38 T^{2} - 32 T^{3} + 4203 T^{4} - 32 p T^{5} - 38 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 10 T - 38 T^{2} + 200 T^{3} + 9663 T^{4} + 200 p T^{5} - 38 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 6 T - 90 T^{2} - 24 T^{3} + 9959 T^{4} - 24 p T^{5} - 90 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 4 T + 58 T^{2} + 704 T^{3} - 3797 T^{4} + 704 p T^{5} + 58 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
71$C_4$ \( ( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 8 T - 18 T^{2} - 512 T^{3} - 3277 T^{4} - 512 p T^{5} - 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
83$D_{4}$ \( ( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−7.36816339226097863728342080878, −7.02810085546022361708441714654, −6.93021602216735072285287830156, −6.44618252827781767675097793514, −6.13561696880837170244503892386, −6.11971204866869489111216787781, −5.79223650833497379561752628318, −5.29389836332510262909658975736, −5.27911606995012011257866548434, −4.83929534306983776112955265874, −4.83826726405224177056753981407, −4.78982086568281023286513943456, −4.00611796174435292129632030225, −3.87936246501667235564082525803, −3.67350723487970640813544777321, −3.64336587891415512283406583339, −3.17674722836156686213198933326, −2.80902468521012605189956680420, −2.68056070979100287332154591508, −2.45247768376633488459939643326, −1.74263471571307998272752780359, −1.64250698663615982986929583990, −1.47440520033644363541171751711, −0.62274468191681132681374089533, −0.38203445109774266797855121817, 0.38203445109774266797855121817, 0.62274468191681132681374089533, 1.47440520033644363541171751711, 1.64250698663615982986929583990, 1.74263471571307998272752780359, 2.45247768376633488459939643326, 2.68056070979100287332154591508, 2.80902468521012605189956680420, 3.17674722836156686213198933326, 3.64336587891415512283406583339, 3.67350723487970640813544777321, 3.87936246501667235564082525803, 4.00611796174435292129632030225, 4.78982086568281023286513943456, 4.83826726405224177056753981407, 4.83929534306983776112955265874, 5.27911606995012011257866548434, 5.29389836332510262909658975736, 5.79223650833497379561752628318, 6.11971204866869489111216787781, 6.13561696880837170244503892386, 6.44618252827781767675097793514, 6.93021602216735072285287830156, 7.02810085546022361708441714654, 7.36816339226097863728342080878

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