Properties

Label 8-70e4-1.1-c19e4-0-0
Degree $8$
Conductor $24010000$
Sign $1$
Analytic cond. $6.58177\times 10^{8}$
Root an. cond. $12.6558$
Motivic weight $19$
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  + 2.04e3·2-s − 2.20e4·3-s + 2.62e6·4-s − 7.81e6·5-s − 4.52e7·6-s − 1.61e8·7-s + 2.68e9·8-s − 1.46e9·9-s − 1.60e10·10-s + 5.53e9·11-s − 5.78e10·12-s − 1.12e10·13-s − 3.30e11·14-s + 1.72e11·15-s + 2.40e12·16-s + 4.00e11·17-s − 3.00e12·18-s − 2.03e12·19-s − 2.04e13·20-s + 3.56e12·21-s + 1.13e13·22-s + 4.49e12·23-s − 5.92e13·24-s + 3.81e13·25-s − 2.30e13·26-s + 2.62e13·27-s − 4.23e14·28-s + ⋯
L(s)  = 1  + 2.82·2-s − 0.647·3-s + 5·4-s − 1.78·5-s − 1.83·6-s − 1.51·7-s + 7.07·8-s − 1.26·9-s − 5.05·10-s + 0.708·11-s − 3.23·12-s − 0.293·13-s − 4.27·14-s + 1.15·15-s + 35/4·16-s + 0.818·17-s − 3.56·18-s − 1.44·19-s − 8.94·20-s + 0.979·21-s + 2.00·22-s + 0.520·23-s − 4.57·24-s + 2·25-s − 0.831·26-s + 0.661·27-s − 7.55·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+19/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(24010000\)    =    \(2^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(6.58177\times 10^{8}\)
Root analytic conductor: \(12.6558\)
Motivic weight: \(19\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 24010000,\ (\ :19/2, 19/2, 19/2, 19/2),\ 1)\)

Particular Values

\(L(10)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{21}{2})\) 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$C_1$ \( ( 1 - p^{9} T )^{4} \)
5$C_1$ \( ( 1 + p^{9} T )^{4} \)
7$C_1$ \( ( 1 + p^{9} T )^{4} \)
good3$C_2 \wr S_4$ \( 1 + 2453 p^{2} T + 72328973 p^{3} T^{2} + 7508420876 p^{8} T^{3} + 133265385401912 p^{9} T^{4} + 7508420876 p^{27} T^{5} + 72328973 p^{41} T^{6} + 2453 p^{59} T^{7} + p^{76} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 5538515879 T + \)\(14\!\cdots\!59\)\( T^{2} - \)\(21\!\cdots\!72\)\( p T^{3} + \)\(71\!\cdots\!56\)\( p^{2} T^{4} - \)\(21\!\cdots\!72\)\( p^{20} T^{5} + \)\(14\!\cdots\!59\)\( p^{38} T^{6} - 5538515879 p^{57} T^{7} + p^{76} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 11235701187 T + \)\(57\!\cdots\!37\)\( T^{2} + \)\(37\!\cdots\!10\)\( p T^{3} + \)\(74\!\cdots\!34\)\( p^{2} T^{4} + \)\(37\!\cdots\!10\)\( p^{20} T^{5} + \)\(57\!\cdots\!37\)\( p^{38} T^{6} + 11235701187 p^{57} T^{7} + p^{76} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 23530589879 p T + \)\(21\!\cdots\!45\)\( p^{2} T^{2} - \)\(18\!\cdots\!34\)\( p^{3} T^{3} + \)\(18\!\cdots\!70\)\( p^{4} T^{4} - \)\(18\!\cdots\!34\)\( p^{22} T^{5} + \)\(21\!\cdots\!45\)\( p^{40} T^{6} - 23530589879 p^{58} T^{7} + p^{76} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 107275744734 p T + \)\(55\!\cdots\!68\)\( T^{2} + \)\(49\!\cdots\!10\)\( p T^{3} + \)\(13\!\cdots\!82\)\( T^{4} + \)\(49\!\cdots\!10\)\( p^{20} T^{5} + \)\(55\!\cdots\!68\)\( p^{38} T^{6} + 107275744734 p^{58} T^{7} + p^{76} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 4494017033550 T + \)\(11\!\cdots\!48\)\( T^{2} - \)\(16\!\cdots\!50\)\( T^{3} + \)\(65\!\cdots\!14\)\( T^{4} - \)\(16\!\cdots\!50\)\( p^{19} T^{5} + \)\(11\!\cdots\!48\)\( p^{38} T^{6} - 4494017033550 p^{57} T^{7} + p^{76} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 31103529955593 T + \)\(16\!\cdots\!81\)\( T^{2} + \)\(52\!\cdots\!58\)\( T^{3} + \)\(13\!\cdots\!50\)\( T^{4} + \)\(52\!\cdots\!58\)\( p^{19} T^{5} + \)\(16\!\cdots\!81\)\( p^{38} T^{6} + 31103529955593 p^{57} T^{7} + p^{76} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 134946719293796 T + \)\(74\!\cdots\!56\)\( T^{2} - \)\(77\!\cdots\!08\)\( T^{3} + \)\(23\!\cdots\!78\)\( T^{4} - \)\(77\!\cdots\!08\)\( p^{19} T^{5} + \)\(74\!\cdots\!56\)\( p^{38} T^{6} - 134946719293796 p^{57} T^{7} + p^{76} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 22089779837716 p T + \)\(21\!\cdots\!20\)\( T^{2} + \)\(35\!\cdots\!76\)\( p T^{3} + \)\(19\!\cdots\!02\)\( T^{4} + \)\(35\!\cdots\!76\)\( p^{20} T^{5} + \)\(21\!\cdots\!20\)\( p^{38} T^{6} + 22089779837716 p^{58} T^{7} + p^{76} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 1878775500738310 T + \)\(92\!\cdots\!08\)\( T^{2} - \)\(15\!\cdots\!82\)\( T^{3} + \)\(62\!\cdots\!06\)\( T^{4} - \)\(15\!\cdots\!82\)\( p^{19} T^{5} + \)\(92\!\cdots\!08\)\( p^{38} T^{6} - 1878775500738310 p^{57} T^{7} + p^{76} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 11088400756930 p T + \)\(26\!\cdots\!32\)\( T^{2} - \)\(85\!\cdots\!82\)\( T^{3} + \)\(39\!\cdots\!02\)\( T^{4} - \)\(85\!\cdots\!82\)\( p^{19} T^{5} + \)\(26\!\cdots\!32\)\( p^{38} T^{6} - 11088400756930 p^{58} T^{7} + p^{76} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 10362757757417717 T + \)\(13\!\cdots\!47\)\( T^{2} + \)\(11\!\cdots\!16\)\( T^{3} + \)\(12\!\cdots\!08\)\( T^{4} + \)\(11\!\cdots\!16\)\( p^{19} T^{5} + \)\(13\!\cdots\!47\)\( p^{38} T^{6} + 10362757757417717 p^{57} T^{7} + p^{76} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 1842309486673150 T + \)\(15\!\cdots\!28\)\( p T^{2} + \)\(18\!\cdots\!54\)\( T^{3} + \)\(38\!\cdots\!90\)\( T^{4} + \)\(18\!\cdots\!54\)\( p^{19} T^{5} + \)\(15\!\cdots\!28\)\( p^{39} T^{6} + 1842309486673150 p^{57} T^{7} + p^{76} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 30461811110595296 T + \)\(49\!\cdots\!48\)\( T^{2} - \)\(18\!\cdots\!32\)\( T^{3} + \)\(22\!\cdots\!30\)\( T^{4} - \)\(18\!\cdots\!32\)\( p^{19} T^{5} + \)\(49\!\cdots\!48\)\( p^{38} T^{6} - 30461811110595296 p^{57} T^{7} + p^{76} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 3107276310076190 T + \)\(66\!\cdots\!80\)\( T^{2} + \)\(92\!\cdots\!86\)\( T^{3} - \)\(43\!\cdots\!30\)\( T^{4} + \)\(92\!\cdots\!86\)\( p^{19} T^{5} + \)\(66\!\cdots\!80\)\( p^{38} T^{6} + 3107276310076190 p^{57} T^{7} + p^{76} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 413771765114474040 T + \)\(14\!\cdots\!08\)\( T^{2} + \)\(53\!\cdots\!88\)\( p T^{3} + \)\(90\!\cdots\!22\)\( T^{4} + \)\(53\!\cdots\!88\)\( p^{20} T^{5} + \)\(14\!\cdots\!08\)\( p^{38} T^{6} + 413771765114474040 p^{57} T^{7} + p^{76} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 886111785494070776 T + \)\(79\!\cdots\!64\)\( T^{2} + \)\(39\!\cdots\!68\)\( T^{3} + \)\(19\!\cdots\!46\)\( T^{4} + \)\(39\!\cdots\!68\)\( p^{19} T^{5} + \)\(79\!\cdots\!64\)\( p^{38} T^{6} + 886111785494070776 p^{57} T^{7} + p^{76} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 476373403365997448 T + \)\(10\!\cdots\!28\)\( T^{2} - \)\(36\!\cdots\!20\)\( T^{3} + \)\(41\!\cdots\!38\)\( T^{4} - \)\(36\!\cdots\!20\)\( p^{19} T^{5} + \)\(10\!\cdots\!28\)\( p^{38} T^{6} - 476373403365997448 p^{57} T^{7} + p^{76} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 784782689990867831 T + \)\(37\!\cdots\!31\)\( T^{2} + \)\(25\!\cdots\!68\)\( T^{3} + \)\(59\!\cdots\!20\)\( T^{4} + \)\(25\!\cdots\!68\)\( p^{19} T^{5} + \)\(37\!\cdots\!31\)\( p^{38} T^{6} + 784782689990867831 p^{57} T^{7} + p^{76} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 46750680434822548 T + \)\(10\!\cdots\!96\)\( T^{2} - \)\(82\!\cdots\!28\)\( T^{3} + \)\(45\!\cdots\!54\)\( T^{4} - \)\(82\!\cdots\!28\)\( p^{19} T^{5} + \)\(10\!\cdots\!96\)\( p^{38} T^{6} - 46750680434822548 p^{57} T^{7} + p^{76} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 3709797894788888842 T + \)\(40\!\cdots\!24\)\( T^{2} + \)\(96\!\cdots\!78\)\( T^{3} + \)\(62\!\cdots\!70\)\( T^{4} + \)\(96\!\cdots\!78\)\( p^{19} T^{5} + \)\(40\!\cdots\!24\)\( p^{38} T^{6} + 3709797894788888842 p^{57} T^{7} + p^{76} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 3031357972138151581 T + \)\(92\!\cdots\!57\)\( T^{2} + \)\(77\!\cdots\!94\)\( T^{3} + \)\(52\!\cdots\!34\)\( T^{4} + \)\(77\!\cdots\!94\)\( p^{19} T^{5} + \)\(92\!\cdots\!57\)\( p^{38} T^{6} + 3031357972138151581 p^{57} T^{7} + p^{76} T^{8} \)
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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

−7.87424267477376079884208535569, −7.11237297784708256408130922412, −7.09587913114360333529702956878, −6.86492961847730246636442140471, −6.86328480828758954109883295026, −6.18685941282790878012824518185, −5.94669340551890035047683451752, −5.94535074388117138781387476202, −5.74358472091592984880262774387, −5.12866996312660082286026823209, −4.94104840871878832469663798766, −4.81966159251562733300012527759, −4.33917068879663746773252130257, −4.00994489973912039563765536055, −3.78930675359119594086121530208, −3.76710024273659699632983972056, −3.47350048013393975364564233800, −2.90664898709280145472351067876, −2.74149359637372235781584790434, −2.62454303877689518497284199165, −2.60927779997378779234325414749, −1.75076922395410607488613610314, −1.41545626141758930470728396806, −1.17153598710796079759490536504, −0.955476608757601494985329849892, 0, 0, 0, 0, 0.955476608757601494985329849892, 1.17153598710796079759490536504, 1.41545626141758930470728396806, 1.75076922395410607488613610314, 2.60927779997378779234325414749, 2.62454303877689518497284199165, 2.74149359637372235781584790434, 2.90664898709280145472351067876, 3.47350048013393975364564233800, 3.76710024273659699632983972056, 3.78930675359119594086121530208, 4.00994489973912039563765536055, 4.33917068879663746773252130257, 4.81966159251562733300012527759, 4.94104840871878832469663798766, 5.12866996312660082286026823209, 5.74358472091592984880262774387, 5.94535074388117138781387476202, 5.94669340551890035047683451752, 6.18685941282790878012824518185, 6.86328480828758954109883295026, 6.86492961847730246636442140471, 7.09587913114360333529702956878, 7.11237297784708256408130922412, 7.87424267477376079884208535569

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