Properties

Label 8-55e4-1.1-c7e4-0-1
Degree $8$
Conductor $9150625$
Sign $1$
Analytic cond. $87138.8$
Root an. cond. $4.14501$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 21·2-s + 59·4-s − 500·5-s + 1.26e3·7-s + 1.95e3·8-s − 5.72e3·9-s + 1.05e4·10-s + 5.32e3·11-s + 2.69e3·13-s − 2.65e4·14-s − 2.08e4·16-s − 2.48e4·17-s + 1.20e5·18-s − 2.27e4·19-s − 2.95e4·20-s − 1.11e5·22-s − 160·23-s + 1.56e5·25-s − 5.66e4·26-s − 8.80e4·27-s + 7.44e4·28-s − 4.33e5·29-s − 5.17e5·31-s + 1.98e4·32-s + 5.21e5·34-s − 6.31e5·35-s − 3.37e5·36-s + ⋯
L(s)  = 1  − 1.85·2-s + 0.460·4-s − 1.78·5-s + 1.39·7-s + 1.34·8-s − 2.61·9-s + 3.32·10-s + 1.20·11-s + 0.340·13-s − 2.58·14-s − 1.27·16-s − 1.22·17-s + 4.86·18-s − 0.760·19-s − 0.824·20-s − 2.23·22-s − 0.00274·23-s + 2·25-s − 0.631·26-s − 0.860·27-s + 0.641·28-s − 3.30·29-s − 3.12·31-s + 0.107·32-s + 2.27·34-s − 2.48·35-s − 1.20·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(9150625\)    =    \(5^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(87138.8\)
Root analytic conductor: \(4.14501\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 9150625,\ (\ :7/2, 7/2, 7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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)$
bad5$C_1$ \( ( 1 + p^{3} T )^{4} \)
11$C_1$ \( ( 1 - p^{3} T )^{4} \)
good2$C_2 \wr S_4$ \( 1 + 21 T + 191 p T^{2} + 1207 p^{2} T^{3} + 7331 p^{3} T^{4} + 1207 p^{9} T^{5} + 191 p^{15} T^{6} + 21 p^{21} T^{7} + p^{28} T^{8} \)
3$C_2 \wr S_4$ \( 1 + 1909 p T^{2} + 3260 p^{3} T^{3} + 547384 p^{3} T^{4} + 3260 p^{10} T^{5} + 1909 p^{15} T^{6} + p^{28} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 1262 T + 331811 p T^{2} - 2690598426 T^{3} + 2496628718408 T^{4} - 2690598426 p^{7} T^{5} + 331811 p^{15} T^{6} - 1262 p^{21} T^{7} + p^{28} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 2696 T + 58180856 T^{2} - 600972371592 T^{3} + 506872398943678 T^{4} - 600972371592 p^{7} T^{5} + 58180856 p^{14} T^{6} - 2696 p^{21} T^{7} + p^{28} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 24824 T + 1205824543 T^{2} + 22979754624732 T^{3} + 713606671254624428 T^{4} + 22979754624732 p^{7} T^{5} + 1205824543 p^{14} T^{6} + 24824 p^{21} T^{7} + p^{28} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 22730 T + 3664393781 T^{2} + 60486317296410 T^{3} + 4952655568614982076 T^{4} + 60486317296410 p^{7} T^{5} + 3664393781 p^{14} T^{6} + 22730 p^{21} T^{7} + p^{28} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 160 T + 9847605952 T^{2} + 2089321487600 p T^{3} + 44809440919485094878 T^{4} + 2089321487600 p^{8} T^{5} + 9847605952 p^{14} T^{6} + 160 p^{21} T^{7} + p^{28} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 433924 T + 131007797503 T^{2} + 25678208682175608 T^{3} + \)\(39\!\cdots\!92\)\( T^{4} + 25678208682175608 p^{7} T^{5} + 131007797503 p^{14} T^{6} + 433924 p^{21} T^{7} + p^{28} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 16696 p T + 167567168363 T^{2} + 39927536639344008 T^{3} + \)\(76\!\cdots\!44\)\( T^{4} + 39927536639344008 p^{7} T^{5} + 167567168363 p^{14} T^{6} + 16696 p^{22} T^{7} + p^{28} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 523762 T + 389032704989 T^{2} - 139205255244220986 T^{3} + \)\(56\!\cdots\!68\)\( T^{4} - 139205255244220986 p^{7} T^{5} + 389032704989 p^{14} T^{6} - 523762 p^{21} T^{7} + p^{28} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 277004 T + 748891640052 T^{2} + 157066211495406420 T^{3} + \)\(21\!\cdots\!46\)\( T^{4} + 157066211495406420 p^{7} T^{5} + 748891640052 p^{14} T^{6} + 277004 p^{21} T^{7} + p^{28} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 251596 T + 868481895968 T^{2} + 191130979950156972 T^{3} + \)\(33\!\cdots\!98\)\( T^{4} + 191130979950156972 p^{7} T^{5} + 868481895968 p^{14} T^{6} + 251596 p^{21} T^{7} + p^{28} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 711180 T + 1799315585304 T^{2} + 929266313308814220 T^{3} + \)\(13\!\cdots\!98\)\( T^{4} + 929266313308814220 p^{7} T^{5} + 1799315585304 p^{14} T^{6} + 711180 p^{21} T^{7} + p^{28} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 940654 T + 3867641398525 T^{2} + 2963023233003223078 T^{3} + \)\(64\!\cdots\!28\)\( T^{4} + 2963023233003223078 p^{7} T^{5} + 3867641398525 p^{14} T^{6} + 940654 p^{21} T^{7} + p^{28} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 1014028 T + 8333080346880 T^{2} - 5185804612165979436 T^{3} + \)\(28\!\cdots\!18\)\( T^{4} - 5185804612165979436 p^{7} T^{5} + 8333080346880 p^{14} T^{6} - 1014028 p^{21} T^{7} + p^{28} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 3634268 T + 14896116837455 T^{2} - 33671427358198254864 T^{3} + \)\(73\!\cdots\!32\)\( T^{4} - 33671427358198254864 p^{7} T^{5} + 14896116837455 p^{14} T^{6} - 3634268 p^{21} T^{7} + p^{28} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 163644 T + 7781719154972 T^{2} - 9599754812523518148 T^{3} + \)\(55\!\cdots\!98\)\( T^{4} - 9599754812523518148 p^{7} T^{5} + 7781719154972 p^{14} T^{6} + 163644 p^{21} T^{7} + p^{28} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 5547896 T + 19647900551875 T^{2} + 1204016691069584200 p T^{3} + \)\(33\!\cdots\!76\)\( T^{4} + 1204016691069584200 p^{8} T^{5} + 19647900551875 p^{14} T^{6} + 5547896 p^{21} T^{7} + p^{28} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 3614404 T + 32828483432096 T^{2} + 1561303419533186076 p T^{3} + \)\(49\!\cdots\!78\)\( T^{4} + 1561303419533186076 p^{8} T^{5} + 32828483432096 p^{14} T^{6} + 3614404 p^{21} T^{7} + p^{28} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 8995368 T + 81205401792872 T^{2} + \)\(44\!\cdots\!56\)\( T^{3} + \)\(24\!\cdots\!34\)\( T^{4} + \)\(44\!\cdots\!56\)\( p^{7} T^{5} + 81205401792872 p^{14} T^{6} + 8995368 p^{21} T^{7} + p^{28} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 5010356 T + 768816539556 p T^{2} + \)\(35\!\cdots\!72\)\( T^{3} + \)\(19\!\cdots\!38\)\( T^{4} + \)\(35\!\cdots\!72\)\( p^{7} T^{5} + 768816539556 p^{15} T^{6} + 5010356 p^{21} T^{7} + p^{28} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 8759946 T + 171260395444777 T^{2} + \)\(96\!\cdots\!82\)\( T^{3} + \)\(10\!\cdots\!84\)\( T^{4} + \)\(96\!\cdots\!82\)\( p^{7} T^{5} + 171260395444777 p^{14} T^{6} + 8759946 p^{21} T^{7} + p^{28} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 11731552 T + 177363252636584 T^{2} - \)\(11\!\cdots\!96\)\( T^{3} + \)\(15\!\cdots\!58\)\( T^{4} - \)\(11\!\cdots\!96\)\( p^{7} T^{5} + 177363252636584 p^{14} T^{6} - 11731552 p^{21} T^{7} + p^{28} T^{8} \)
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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

−10.93107790097366066583790987799, −9.852375774559868359097429286819, −9.589300936743841186974592179350, −9.367586104600243029857456350582, −9.174634965481528778346575354363, −8.711887087780111170132146835467, −8.440340678361060538994253485271, −8.421933122790500817160406794113, −8.380147307194267517356830418742, −7.62081425662995163234546922761, −7.58133252814792888819632372954, −7.19573581111375426245578585622, −6.70230913011172615084266729764, −6.22555788665065379564655841993, −5.64766006376535784070430918219, −5.40092384939221085607508251981, −5.17604890007400262539903399600, −4.29609574720223273717097343511, −4.15713284189185302046280482925, −3.87260741306743468814306974645, −3.36777246446695416267514667524, −2.82554681063799410103717552042, −2.04135842966798014217551516817, −1.62807277431565529556780033854, −1.22806364425746137842961393928, 0, 0, 0, 0, 1.22806364425746137842961393928, 1.62807277431565529556780033854, 2.04135842966798014217551516817, 2.82554681063799410103717552042, 3.36777246446695416267514667524, 3.87260741306743468814306974645, 4.15713284189185302046280482925, 4.29609574720223273717097343511, 5.17604890007400262539903399600, 5.40092384939221085607508251981, 5.64766006376535784070430918219, 6.22555788665065379564655841993, 6.70230913011172615084266729764, 7.19573581111375426245578585622, 7.58133252814792888819632372954, 7.62081425662995163234546922761, 8.380147307194267517356830418742, 8.421933122790500817160406794113, 8.440340678361060538994253485271, 8.711887087780111170132146835467, 9.174634965481528778346575354363, 9.367586104600243029857456350582, 9.589300936743841186974592179350, 9.852375774559868359097429286819, 10.93107790097366066583790987799

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