Properties

Label 8-570e4-1.1-c7e4-0-1
Degree $8$
Conductor $105560010000$
Sign $1$
Analytic cond. $1.00521\times 10^{9}$
Root an. cond. $13.3438$
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  + 32·2-s + 108·3-s + 640·4-s − 500·5-s + 3.45e3·6-s − 742·7-s + 1.02e4·8-s + 7.29e3·9-s − 1.60e4·10-s − 354·11-s + 6.91e4·12-s − 6.36e3·13-s − 2.37e4·14-s − 5.40e4·15-s + 1.43e5·16-s − 1.64e4·17-s + 2.33e5·18-s + 2.74e4·19-s − 3.20e5·20-s − 8.01e4·21-s − 1.13e4·22-s − 6.81e4·23-s + 1.10e6·24-s + 1.56e5·25-s − 2.03e5·26-s + 3.93e5·27-s − 4.74e5·28-s + ⋯
L(s)  = 1  + 2.82·2-s + 2.30·3-s + 5·4-s − 1.78·5-s + 6.53·6-s − 0.817·7-s + 7.07·8-s + 10/3·9-s − 5.05·10-s − 0.0801·11-s + 11.5·12-s − 0.803·13-s − 2.31·14-s − 4.13·15-s + 35/4·16-s − 0.810·17-s + 9.42·18-s + 0.917·19-s − 8.94·20-s − 1.88·21-s − 0.226·22-s − 1.16·23-s + 16.3·24-s + 2·25-s − 2.27·26-s + 3.84·27-s − 4.08·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(1.00521\times 10^{9}\)
Root analytic conductor: \(13.3438\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 7/2, 7/2, 7/2, 7/2 ),\ 1 )\)

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)$
bad2$C_1$ \( ( 1 - p^{3} T )^{4} \)
3$C_1$ \( ( 1 - p^{3} T )^{4} \)
5$C_1$ \( ( 1 + p^{3} T )^{4} \)
19$C_1$ \( ( 1 - p^{3} T )^{4} \)
good7$C_2 \wr S_4$ \( 1 + 106 p T + 2338684 T^{2} + 194436962 p T^{3} + 2634613140230 T^{4} + 194436962 p^{8} T^{5} + 2338684 p^{14} T^{6} + 106 p^{22} T^{7} + p^{28} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 354 T + 67045724 T^{2} + 18730143554 T^{3} + 1878732212546166 T^{4} + 18730143554 p^{7} T^{5} + 67045724 p^{14} T^{6} + 354 p^{21} T^{7} + p^{28} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 6366 T + 179272672 T^{2} + 1181916318586 T^{3} + 14624389555824270 T^{4} + 1181916318586 p^{7} T^{5} + 179272672 p^{14} T^{6} + 6366 p^{21} T^{7} + p^{28} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 16412 T + 1005068564 T^{2} + 5014778827044 T^{3} + 411451180067055510 T^{4} + 5014778827044 p^{7} T^{5} + 1005068564 p^{14} T^{6} + 16412 p^{21} T^{7} + p^{28} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 68140 T + 6196949068 T^{2} + 387752338910300 T^{3} + 1568792911108226986 p T^{4} + 387752338910300 p^{7} T^{5} + 6196949068 p^{14} T^{6} + 68140 p^{21} T^{7} + p^{28} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 120486 T + 69839099720 T^{2} + 6036786191409426 T^{3} + \)\(18\!\cdots\!78\)\( T^{4} + 6036786191409426 p^{7} T^{5} + 69839099720 p^{14} T^{6} + 120486 p^{21} T^{7} + p^{28} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 223328 T + 114984675708 T^{2} + 18065451654165856 T^{3} + \)\(48\!\cdots\!54\)\( T^{4} + 18065451654165856 p^{7} T^{5} + 114984675708 p^{14} T^{6} + 223328 p^{21} T^{7} + p^{28} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 409930 T + 7760698120 p T^{2} + 76957070018532014 T^{3} + \)\(36\!\cdots\!42\)\( T^{4} + 76957070018532014 p^{7} T^{5} + 7760698120 p^{15} T^{6} + 409930 p^{21} T^{7} + p^{28} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 5102 p T + 775365730928 T^{2} - 121986939516063234 T^{3} + \)\(22\!\cdots\!74\)\( T^{4} - 121986939516063234 p^{7} T^{5} + 775365730928 p^{14} T^{6} - 5102 p^{22} T^{7} + p^{28} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 983566 T + 1372119527836 T^{2} + 828632687748341070 T^{3} + \)\(59\!\cdots\!62\)\( T^{4} + 828632687748341070 p^{7} T^{5} + 1372119527836 p^{14} T^{6} + 983566 p^{21} T^{7} + p^{28} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 371420 T + 905978481260 T^{2} + 358460562449398764 T^{3} + \)\(73\!\cdots\!02\)\( T^{4} + 358460562449398764 p^{7} T^{5} + 905978481260 p^{14} T^{6} + 371420 p^{21} T^{7} + p^{28} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 1254692 T + 1471984298740 T^{2} + 1850687697017723692 T^{3} + \)\(29\!\cdots\!22\)\( T^{4} + 1850687697017723692 p^{7} T^{5} + 1471984298740 p^{14} T^{6} + 1254692 p^{21} T^{7} + p^{28} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 797084 T + 6614604889772 T^{2} + 7014313634377518012 T^{3} + \)\(20\!\cdots\!54\)\( T^{4} + 7014313634377518012 p^{7} T^{5} + 6614604889772 p^{14} T^{6} + 797084 p^{21} T^{7} + p^{28} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 3424652 T + 10942376307524 T^{2} + 19105496424707249220 T^{3} + \)\(39\!\cdots\!46\)\( T^{4} + 19105496424707249220 p^{7} T^{5} + 10942376307524 p^{14} T^{6} + 3424652 p^{21} T^{7} + p^{28} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 1072972 T + 14294146355068 T^{2} + 29311722415389764748 T^{3} + \)\(98\!\cdots\!70\)\( T^{4} + 29311722415389764748 p^{7} T^{5} + 14294146355068 p^{14} T^{6} + 1072972 p^{21} T^{7} + p^{28} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 2077240 T + 8243098647452 T^{2} + 41779253786200611864 T^{3} + \)\(12\!\cdots\!42\)\( T^{4} + 41779253786200611864 p^{7} T^{5} + 8243098647452 p^{14} T^{6} + 2077240 p^{21} T^{7} + p^{28} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 257780 T + 42038708695732 T^{2} + 7316707107420649132 T^{3} + \)\(68\!\cdots\!02\)\( T^{4} + 7316707107420649132 p^{7} T^{5} + 42038708695732 p^{14} T^{6} + 257780 p^{21} T^{7} + p^{28} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 2112232 T + 53622591469756 T^{2} + 40737889971968523464 T^{3} + \)\(12\!\cdots\!46\)\( T^{4} + 40737889971968523464 p^{7} T^{5} + 53622591469756 p^{14} T^{6} + 2112232 p^{21} T^{7} + p^{28} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 8743304 T + 95765702400556 T^{2} + \)\(56\!\cdots\!12\)\( T^{3} + \)\(35\!\cdots\!50\)\( T^{4} + \)\(56\!\cdots\!12\)\( p^{7} T^{5} + 95765702400556 p^{14} T^{6} + 8743304 p^{21} T^{7} + p^{28} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 18352170 T + 234470138693456 T^{2} + \)\(21\!\cdots\!30\)\( T^{3} + \)\(16\!\cdots\!50\)\( T^{4} + \)\(21\!\cdots\!30\)\( p^{7} T^{5} + 234470138693456 p^{14} T^{6} + 18352170 p^{21} T^{7} + p^{28} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 18150 T + 210806977035416 T^{2} + 67861542544112089590 T^{3} + \)\(23\!\cdots\!78\)\( T^{4} + 67861542544112089590 p^{7} T^{5} + 210806977035416 p^{14} T^{6} - 18150 p^{21} T^{7} + p^{28} T^{8} \)
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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

−7.23292748746037705930195957775, −6.66236403730246694662269627491, −6.49242486515091343838297449064, −6.45391145876748504887885651466, −6.37849252173983539700933162344, −5.57698868077872580091961151767, −5.43890003251423860442586389228, −5.24761184101300703248539972530, −5.17872034138073281394735607653, −4.52199045694804609671540363494, −4.47192048773607028348689079261, −4.32055868228538483939500938402, −4.17953198432872310478317503355, −3.57532871182441955531562138569, −3.51766847087062678459268242867, −3.48802883227351020198115352877, −3.44127182674164568704932858520, −2.73953990665106362507874738814, −2.72435818668792170036762758670, −2.55911140465551547707084928388, −2.43999183218334796916715669065, −1.58686644838972866615834414066, −1.51510950888036526639336495180, −1.42015912507782759707320877582, −1.39614863251284107240475838564, 0, 0, 0, 0, 1.39614863251284107240475838564, 1.42015912507782759707320877582, 1.51510950888036526639336495180, 1.58686644838972866615834414066, 2.43999183218334796916715669065, 2.55911140465551547707084928388, 2.72435818668792170036762758670, 2.73953990665106362507874738814, 3.44127182674164568704932858520, 3.48802883227351020198115352877, 3.51766847087062678459268242867, 3.57532871182441955531562138569, 4.17953198432872310478317503355, 4.32055868228538483939500938402, 4.47192048773607028348689079261, 4.52199045694804609671540363494, 5.17872034138073281394735607653, 5.24761184101300703248539972530, 5.43890003251423860442586389228, 5.57698868077872580091961151767, 6.37849252173983539700933162344, 6.45391145876748504887885651466, 6.49242486515091343838297449064, 6.66236403730246694662269627491, 7.23292748746037705930195957775

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