Properties

Label 8-825e4-1.1-c3e4-0-2
Degree $8$
Conductor $463250390625$
Sign $1$
Analytic cond. $5.61409\times 10^{6}$
Root an. cond. $6.97686$
Motivic weight $3$
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  − 4·2-s − 12·3-s + 5·4-s + 48·6-s − 34·7-s − 4·8-s + 90·9-s − 44·11-s − 60·12-s − 2·13-s + 136·14-s − 23·16-s − 74·17-s − 360·18-s + 136·19-s + 408·21-s + 176·22-s + 64·23-s + 48·24-s + 8·26-s − 540·27-s − 170·28-s + 52·29-s + 492·31-s + 156·32-s + 528·33-s + 296·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 5/8·4-s + 3.26·6-s − 1.83·7-s − 0.176·8-s + 10/3·9-s − 1.20·11-s − 1.44·12-s − 0.0426·13-s + 2.59·14-s − 0.359·16-s − 1.05·17-s − 4.71·18-s + 1.64·19-s + 4.23·21-s + 1.70·22-s + 0.580·23-s + 0.408·24-s + 0.0603·26-s − 3.84·27-s − 1.14·28-s + 0.332·29-s + 2.85·31-s + 0.861·32-s + 2.78·33-s + 1.49·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(5.61409\times 10^{6}\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.2876775679\)
\(L(\frac12)\) \(\approx\) \(0.2876775679\)
\(L(\frac{5}{2})\) 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)$
bad3$C_1$ \( ( 1 + p T )^{4} \)
5 \( 1 \)
11$C_1$ \( ( 1 + p T )^{4} \)
good2$C_2 \wr S_4$ \( 1 + p^{2} T + 11 T^{2} + 7 p^{2} T^{3} + 3 p^{5} T^{4} + 7 p^{5} T^{5} + 11 p^{6} T^{6} + p^{11} T^{7} + p^{12} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 34 T + 216 p T^{2} + 30650 T^{3} + 769870 T^{4} + 30650 p^{3} T^{5} + 216 p^{7} T^{6} + 34 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 2 T + 2204 T^{2} - 131802 T^{3} + 98902 T^{4} - 131802 p^{3} T^{5} + 2204 p^{6} T^{6} + 2 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 74 T + 9268 T^{2} + 839286 T^{3} + 66563030 T^{4} + 839286 p^{3} T^{5} + 9268 p^{6} T^{6} + 74 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 136 T + 29216 T^{2} - 2775000 T^{3} + 306693902 T^{4} - 2775000 p^{3} T^{5} + 29216 p^{6} T^{6} - 136 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 64 T + 9180 T^{2} - 736576 T^{3} + 174843814 T^{4} - 736576 p^{3} T^{5} + 9180 p^{6} T^{6} - 64 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 52 T + 40192 T^{2} - 1067100 T^{3} + 1086313358 T^{4} - 1067100 p^{3} T^{5} + 40192 p^{6} T^{6} - 52 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 492 T + 159660 T^{2} - 35934012 T^{3} + 6834585382 T^{4} - 35934012 p^{3} T^{5} + 159660 p^{6} T^{6} - 492 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 4 T + 71204 T^{2} - 2105612 T^{3} + 3341948742 T^{4} - 2105612 p^{3} T^{5} + 71204 p^{6} T^{6} - 4 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 268 T + 87440 T^{2} + 18299532 T^{3} - 3675373698 T^{4} + 18299532 p^{3} T^{5} + 87440 p^{6} T^{6} - 268 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 546 T + 360936 T^{2} + 117370434 T^{3} + 43550599390 T^{4} + 117370434 p^{3} T^{5} + 360936 p^{6} T^{6} + 546 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 276 T + 225516 T^{2} - 70999812 T^{3} + 30897856486 T^{4} - 70999812 p^{3} T^{5} + 225516 p^{6} T^{6} - 276 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 184 T + 277116 T^{2} - 74769448 T^{3} + 51214149142 T^{4} - 74769448 p^{3} T^{5} + 277116 p^{6} T^{6} - 184 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 1032 T + 677724 T^{2} + 369637448 T^{3} + 191407046422 T^{4} + 369637448 p^{3} T^{5} + 677724 p^{6} T^{6} + 1032 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 116 T + 356564 T^{2} - 212884668 T^{3} + 50278980358 T^{4} - 212884668 p^{3} T^{5} + 356564 p^{6} T^{6} - 116 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 552 T + 1238636 T^{2} - 483921960 T^{3} + 563245413654 T^{4} - 483921960 p^{3} T^{5} + 1238636 p^{6} T^{6} - 552 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 920 T + 545004 T^{2} + 500512760 T^{3} + 425381878342 T^{4} + 500512760 p^{3} T^{5} + 545004 p^{6} T^{6} + 920 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 926 T + 356132 T^{2} - 167124414 T^{3} - 159336444362 T^{4} - 167124414 p^{3} T^{5} + 356132 p^{6} T^{6} + 926 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 1152 T + 1552592 T^{2} - 1451664720 T^{3} + 1092950826078 T^{4} - 1451664720 p^{3} T^{5} + 1552592 p^{6} T^{6} - 1152 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 134 T + 1310120 T^{2} + 31395930 T^{3} + 826949758590 T^{4} + 31395930 p^{3} T^{5} + 1310120 p^{6} T^{6} - 134 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 1064 T + 103244 T^{2} - 552303208 T^{3} - 369353813754 T^{4} - 552303208 p^{3} T^{5} + 103244 p^{6} T^{6} + 1064 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 1648 T + 4080476 T^{2} - 4443876752 T^{3} + 5783943978438 T^{4} - 4443876752 p^{3} T^{5} + 4080476 p^{6} T^{6} - 1648 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.81113123966579368842072365721, −6.64793796779989864567040771196, −6.55752222548203851150900402441, −6.19547146977596008063366292741, −6.19297851827383244175495823273, −5.89254115554206835270875342554, −5.56684693384781573414045891298, −5.39129010626140626323981043959, −5.17249283676517564982794944095, −4.72557065167434015558778232018, −4.52933749780731299755788595156, −4.48202384216771394280160542389, −4.42384484925535726599683533098, −3.67608447760003216773888503742, −3.34813854355423216941823578883, −3.12679995554369658483584952101, −2.88293246248269445896109067128, −2.79072913182173734402393813232, −2.15414247791255037928271882696, −1.77231352563960757477423881588, −1.52498954813841366137012010758, −0.855915469735104951197775373955, −0.75058665251066263969345500238, −0.40288930299521032155519409383, −0.28473393326003029133472420205, 0.28473393326003029133472420205, 0.40288930299521032155519409383, 0.75058665251066263969345500238, 0.855915469735104951197775373955, 1.52498954813841366137012010758, 1.77231352563960757477423881588, 2.15414247791255037928271882696, 2.79072913182173734402393813232, 2.88293246248269445896109067128, 3.12679995554369658483584952101, 3.34813854355423216941823578883, 3.67608447760003216773888503742, 4.42384484925535726599683533098, 4.48202384216771394280160542389, 4.52933749780731299755788595156, 4.72557065167434015558778232018, 5.17249283676517564982794944095, 5.39129010626140626323981043959, 5.56684693384781573414045891298, 5.89254115554206835270875342554, 6.19297851827383244175495823273, 6.19547146977596008063366292741, 6.55752222548203851150900402441, 6.64793796779989864567040771196, 6.81113123966579368842072365721

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