Properties

Label 8-1078e4-1.1-c1e4-0-3
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 4-s − 4·5-s + 4·6-s + 2·8-s + 5·9-s + 8·10-s − 2·11-s − 2·12-s + 4·13-s + 8·15-s − 4·16-s − 4·17-s − 10·18-s − 4·19-s − 4·20-s + 4·22-s + 4·23-s − 4·24-s + 12·25-s − 8·26-s − 10·27-s − 12·29-s − 16·30-s − 8·31-s + 2·32-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 1/2·4-s − 1.78·5-s + 1.63·6-s + 0.707·8-s + 5/3·9-s + 2.52·10-s − 0.603·11-s − 0.577·12-s + 1.10·13-s + 2.06·15-s − 16-s − 0.970·17-s − 2.35·18-s − 0.917·19-s − 0.894·20-s + 0.852·22-s + 0.834·23-s − 0.816·24-s + 12/5·25-s − 1.56·26-s − 1.92·27-s − 2.22·29-s − 2.92·30-s − 1.43·31-s + 0.353·32-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

\(L(1)\) \(\approx\) \(0.1296647064\)
\(L(\frac12)\) \(\approx\) \(0.1296647064\)
\(L(\frac{3}{2})\) 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$D_4\times C_2$ \( 1 + 2 T - T^{2} - 2 T^{3} + 4 T^{4} - 2 p T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
5$D_4\times C_2$ \( 1 + 4 T + 4 T^{2} + 8 T^{3} + 39 T^{4} + 8 p T^{5} + 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
13$D_{4}$ \( ( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
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$D_4\times C_2$ \( 1 + 4 T - 24 T^{2} + 8 T^{3} + 935 T^{4} + 8 p T^{5} - 24 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 4 T - 16 T^{2} + 56 T^{3} + 127 T^{4} + 56 p T^{5} - 16 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \)
37$D_4\times C_2$ \( 1 - 16 T + 120 T^{2} - 992 T^{3} + 7655 T^{4} - 992 p T^{5} + 120 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 8 T + 96 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 4 T - 10 T^{2} - 272 T^{3} - 2285 T^{4} - 272 p T^{5} - 10 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 4 T + 4 T^{2} + 376 T^{3} - 3513 T^{4} + 376 p T^{5} + 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 14 T + 31 T^{2} + 658 T^{3} + 12180 T^{4} + 658 p T^{5} + 31 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 18 T + 129 T^{2} - 1314 T^{3} + 14540 T^{4} - 1314 p T^{5} + 129 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 + 14 T + 31 T^{2} + 434 T^{3} + 9604 T^{4} + 434 p T^{5} + 31 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 8 T + 108 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 16 T + 48 T^{2} + 992 T^{3} + 19247 T^{4} + 992 p T^{5} + 48 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 18 T + 103 T^{2} - 1134 T^{3} + 17004 T^{4} - 1134 p T^{5} + 103 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 + 4 T - 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 8 T - 58 T^{2} + 448 T^{3} + 1267 T^{4} + 448 p T^{5} - 58 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 - 2 T + 187 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−7.20918272536583867141327357672, −7.00992106102348367850459635149, −6.59433945383640406499528540087, −6.58503505583875957714547337521, −6.21584839128916204335007498033, −6.03924451627818579422295235550, −5.67361227770173730523194917779, −5.54654420051932405104961532181, −5.44564494912569839209184887590, −4.79049274183026592033559907792, −4.75686180566901043375997522135, −4.45774813667395885563101136753, −4.34023566181138521046815865278, −4.11045201834538240984786495522, −3.87862954999686439472186788093, −3.56103390192155205347672395577, −3.40756788457424491678596021631, −2.91865026607353780100327605550, −2.42339818101168106066961177737, −2.38261911480719988017602144868, −1.78676092649032180415758147830, −1.40449394789152773808825172811, −1.13214929686437944892638840308, −0.63712503260384284201440131060, −0.19556023110908663204122553119, 0.19556023110908663204122553119, 0.63712503260384284201440131060, 1.13214929686437944892638840308, 1.40449394789152773808825172811, 1.78676092649032180415758147830, 2.38261911480719988017602144868, 2.42339818101168106066961177737, 2.91865026607353780100327605550, 3.40756788457424491678596021631, 3.56103390192155205347672395577, 3.87862954999686439472186788093, 4.11045201834538240984786495522, 4.34023566181138521046815865278, 4.45774813667395885563101136753, 4.75686180566901043375997522135, 4.79049274183026592033559907792, 5.44564494912569839209184887590, 5.54654420051932405104961532181, 5.67361227770173730523194917779, 6.03924451627818579422295235550, 6.21584839128916204335007498033, 6.58503505583875957714547337521, 6.59433945383640406499528540087, 7.00992106102348367850459635149, 7.20918272536583867141327357672

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