Properties

Label 8-165e4-1.1-c3e4-0-0
Degree $8$
Conductor $741200625$
Sign $1$
Analytic cond. $8982.55$
Root an. cond. $3.12014$
Motivic weight $3$
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  + 4·2-s + 12·3-s + 5·4-s + 20·5-s + 48·6-s + 34·7-s + 4·8-s + 90·9-s + 80·10-s − 44·11-s + 60·12-s + 2·13-s + 136·14-s + 240·15-s − 23·16-s + 74·17-s + 360·18-s + 136·19-s + 100·20-s + 408·21-s − 176·22-s − 64·23-s + 48·24-s + 250·25-s + 8·26-s + 540·27-s + 170·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.30·3-s + 5/8·4-s + 1.78·5-s + 3.26·6-s + 1.83·7-s + 0.176·8-s + 10/3·9-s + 2.52·10-s − 1.20·11-s + 1.44·12-s + 0.0426·13-s + 2.59·14-s + 4.13·15-s − 0.359·16-s + 1.05·17-s + 4.71·18-s + 1.64·19-s + 1.11·20-s + 4.23·21-s − 1.70·22-s − 0.580·23-s + 0.408·24-s + 2·25-s + 0.0603·26-s + 3.84·27-s + 1.14·28-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(8982.55\)
Root analytic conductor: \(3.12014\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{4} \cdot 11^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(48.89519774\)
\(L(\frac12)\) \(\approx\) \(48.89519774\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{4} \)
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

−9.089922618354779870118949989772, −8.329606273408107556272876338666, −8.277180068951443661438294092648, −8.258936056653913063605766391868, −7.76663070042066493240854393144, −7.70820782069187426116160127553, −7.50139287119856281710070420224, −6.81258652239268613973793372121, −6.79862063125740981593087971302, −6.29839544205149230210849197289, −5.69102629869008333045958502837, −5.61632918539960980033110634356, −5.58365213359824708809163172614, −4.61749048961360917943542706486, −4.60083519576693327838925871031, −4.57919032015402283640228629244, −4.55622592142425963746678797951, −3.38921215744865041769093438211, −3.21853732966367196469296438572, −3.12791487994090885759162751116, −2.47907653558667954946509723646, −2.27655276201348294101160903521, −1.86464315129914068725919318431, −1.20747782620850776850538122683, −1.15355991645704823257212866536, 1.15355991645704823257212866536, 1.20747782620850776850538122683, 1.86464315129914068725919318431, 2.27655276201348294101160903521, 2.47907653558667954946509723646, 3.12791487994090885759162751116, 3.21853732966367196469296438572, 3.38921215744865041769093438211, 4.55622592142425963746678797951, 4.57919032015402283640228629244, 4.60083519576693327838925871031, 4.61749048961360917943542706486, 5.58365213359824708809163172614, 5.61632918539960980033110634356, 5.69102629869008333045958502837, 6.29839544205149230210849197289, 6.79862063125740981593087971302, 6.81258652239268613973793372121, 7.50139287119856281710070420224, 7.70820782069187426116160127553, 7.76663070042066493240854393144, 8.258936056653913063605766391868, 8.277180068951443661438294092648, 8.329606273408107556272876338666, 9.089922618354779870118949989772

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