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Properties

Label 8-14e8-1.1-c9e4-0-3
Degree $8$
Conductor $1475789056$
Sign $1$
Analytic cond. $1.03842\times 10^{8}$
Root an. cond. $10.0472$
Motivic weight $9$
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  + 70·3-s − 1.55e3·5-s + 2.93e4·9-s − 6.23e4·11-s + 2.45e5·13-s − 1.08e5·15-s − 7.35e4·17-s − 1.17e6·19-s − 2.26e6·23-s + 4.63e5·25-s + 3.07e6·27-s − 3.84e6·29-s − 2.97e6·31-s − 4.36e6·33-s + 1.34e7·37-s + 1.71e7·39-s − 7.27e7·41-s − 4.39e7·43-s − 4.56e7·45-s + 1.36e6·47-s − 5.15e6·51-s + 1.78e7·53-s + 9.69e7·55-s − 8.19e7·57-s − 2.24e8·59-s + 8.58e7·61-s − 3.81e8·65-s + ⋯
L(s)  = 1  + 0.498·3-s − 1.11·5-s + 1.49·9-s − 1.28·11-s + 2.38·13-s − 0.554·15-s − 0.213·17-s − 2.06·19-s − 1.68·23-s + 0.237·25-s + 1.11·27-s − 1.00·29-s − 0.579·31-s − 0.641·33-s + 1.17·37-s + 1.18·39-s − 4.01·41-s − 1.95·43-s − 1.65·45-s + 0.0407·47-s − 0.106·51-s + 0.311·53-s + 1.42·55-s − 1.02·57-s − 2.41·59-s + 0.793·61-s − 2.65·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.03842\times 10^{8}\)
Root analytic conductor: \(10.0472\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 7^{8} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\)

Particular Values

\(L(5)\) \(\approx\) \(2.211439081\)
\(L(\frac12)\) \(\approx\) \(2.211439081\)
\(L(\frac{11}{2})\) 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 \( 1 \)
7 \( 1 \)
good3$D_4\times C_2$ \( 1 - 70 T - 24482 T^{2} + 232960 p T^{3} + 42159415 p^{2} T^{4} + 232960 p^{10} T^{5} - 24482 p^{18} T^{6} - 70 p^{27} T^{7} + p^{36} T^{8} \)
5$D_4\times C_2$ \( 1 + 1554 T + 1951386 T^{2} - 1069997376 p T^{3} - 332658800689 p^{2} T^{4} - 1069997376 p^{10} T^{5} + 1951386 p^{18} T^{6} + 1554 p^{27} T^{7} + p^{36} T^{8} \)
11$D_4\times C_2$ \( 1 + 62388 T - 1425411558 T^{2} + 37543770783360 T^{3} + 13539989063198689019 T^{4} + 37543770783360 p^{9} T^{5} - 1425411558 p^{18} T^{6} + 62388 p^{27} T^{7} + p^{36} T^{8} \)
13$D_{4}$ \( ( 1 - 122766 T + 1914771578 p T^{2} - 122766 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 73584 T - 133173158238 T^{2} - 7254498634084800 T^{3} + \)\(45\!\cdots\!95\)\( T^{4} - 7254498634084800 p^{9} T^{5} - 133173158238 p^{18} T^{6} + 73584 p^{27} T^{7} + p^{36} T^{8} \)
19$D_4\times C_2$ \( 1 + 61642 p T + 429095101294 T^{2} + 18322114153333984 p T^{3} + \)\(34\!\cdots\!43\)\( T^{4} + 18322114153333984 p^{10} T^{5} + 429095101294 p^{18} T^{6} + 61642 p^{28} T^{7} + p^{36} T^{8} \)
23$D_4\times C_2$ \( 1 + 2262384 T + 858274238802 T^{2} + 64704359626113024 p T^{3} + \)\(10\!\cdots\!11\)\( p^{2} T^{4} + 64704359626113024 p^{10} T^{5} + 858274238802 p^{18} T^{6} + 2262384 p^{27} T^{7} + p^{36} T^{8} \)
29$D_{4}$ \( ( 1 + 1923360 T + 23001703439382 T^{2} + 1923360 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 2977884 T - 718062640754 p T^{2} - 64773471640961236608 T^{3} + \)\(35\!\cdots\!83\)\( T^{4} - 64773471640961236608 p^{9} T^{5} - 718062640754 p^{19} T^{6} + 2977884 p^{27} T^{7} + p^{36} T^{8} \)
37$D_4\times C_2$ \( 1 - 13418528 T + 82812980965690 T^{2} + \)\(21\!\cdots\!80\)\( T^{3} - \)\(30\!\cdots\!61\)\( T^{4} + \)\(21\!\cdots\!80\)\( p^{9} T^{5} + 82812980965690 p^{18} T^{6} - 13418528 p^{27} T^{7} + p^{36} T^{8} \)
41$D_{4}$ \( ( 1 + 36367800 T + 982922223862366 T^{2} + 36367800 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 510812 p T + 1094756328321366 T^{2} + 510812 p^{10} T^{3} + p^{18} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 1362732 T - 1920844220694766 T^{2} + \)\(43\!\cdots\!08\)\( T^{3} + \)\(24\!\cdots\!03\)\( T^{4} + \)\(43\!\cdots\!08\)\( p^{9} T^{5} - 1920844220694766 p^{18} T^{6} - 1362732 p^{27} T^{7} + p^{36} T^{8} \)
53$D_4\times C_2$ \( 1 - 17898612 T - 831639283680958 T^{2} + \)\(97\!\cdots\!68\)\( T^{3} - \)\(10\!\cdots\!37\)\( T^{4} + \)\(97\!\cdots\!68\)\( p^{9} T^{5} - 831639283680958 p^{18} T^{6} - 17898612 p^{27} T^{7} + p^{36} T^{8} \)
59$D_4\times C_2$ \( 1 + 224710542 T + 21638406365289486 T^{2} + \)\(25\!\cdots\!00\)\( T^{3} + \)\(32\!\cdots\!67\)\( T^{4} + \)\(25\!\cdots\!00\)\( p^{9} T^{5} + 21638406365289486 p^{18} T^{6} + 224710542 p^{27} T^{7} + p^{36} T^{8} \)
61$D_4\times C_2$ \( 1 - 85847118 T + 1519911503438362 T^{2} + \)\(15\!\cdots\!60\)\( T^{3} - \)\(18\!\cdots\!21\)\( T^{4} + \)\(15\!\cdots\!60\)\( p^{9} T^{5} + 1519911503438362 p^{18} T^{6} - 85847118 p^{27} T^{7} + p^{36} T^{8} \)
67$D_4\times C_2$ \( 1 + 179568872 T - 5051839357092950 T^{2} - \)\(30\!\cdots\!20\)\( T^{3} - \)\(22\!\cdots\!01\)\( T^{4} - \)\(30\!\cdots\!20\)\( p^{9} T^{5} - 5051839357092950 p^{18} T^{6} + 179568872 p^{27} T^{7} + p^{36} T^{8} \)
71$D_{4}$ \( ( 1 - 231378168 T + 105070505575629582 T^{2} - 231378168 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 88098332 T - 111919578863792822 T^{2} + \)\(17\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!35\)\( T^{4} + \)\(17\!\cdots\!40\)\( p^{9} T^{5} - 111919578863792822 p^{18} T^{6} + 88098332 p^{27} T^{7} + p^{36} T^{8} \)
79$D_4\times C_2$ \( 1 - 184274184 T - 123209524679777246 T^{2} + \)\(15\!\cdots\!24\)\( T^{3} + \)\(62\!\cdots\!79\)\( T^{4} + \)\(15\!\cdots\!24\)\( p^{9} T^{5} - 123209524679777246 p^{18} T^{6} - 184274184 p^{27} T^{7} + p^{36} T^{8} \)
83$D_{4}$ \( ( 1 - 624641094 T + 301231392696541014 T^{2} - 624641094 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 1574777148 T + 1208344199101627034 T^{2} - \)\(89\!\cdots\!96\)\( T^{3} + \)\(62\!\cdots\!83\)\( T^{4} - \)\(89\!\cdots\!96\)\( p^{9} T^{5} + 1208344199101627034 p^{18} T^{6} - 1574777148 p^{27} T^{7} + p^{36} T^{8} \)
97$D_{4}$ \( ( 1 - 213665984 T + 1447970079084154398 T^{2} - 213665984 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
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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

−7.59194617543975541571343367198, −7.21630958020324484066865258597, −6.83399470437684797554881655036, −6.55101893499701331974146735428, −6.41342769870334318757379013329, −6.30714048224781784030648273482, −5.87989905733383526252258343058, −5.40753286844716506623351654904, −5.25449327463585577834210600385, −4.71229251533415153628856500930, −4.66447690176451136578936002376, −4.25950778702289320124249076475, −4.02649522237042228609553207860, −3.73812347967985982936480129391, −3.39891570977218936422147971611, −3.24347076403563182609761348236, −3.19236850225186319297868120169, −2.20499753403342280407792948461, −2.10423154754266116592315789337, −1.81115844424215960839656241568, −1.77631637808232919401573560849, −1.27268762104314135867632715757, −0.70913419069952470909877937684, −0.47099619852822137998370259350, −0.20367731758026579618982498544, 0.20367731758026579618982498544, 0.47099619852822137998370259350, 0.70913419069952470909877937684, 1.27268762104314135867632715757, 1.77631637808232919401573560849, 1.81115844424215960839656241568, 2.10423154754266116592315789337, 2.20499753403342280407792948461, 3.19236850225186319297868120169, 3.24347076403563182609761348236, 3.39891570977218936422147971611, 3.73812347967985982936480129391, 4.02649522237042228609553207860, 4.25950778702289320124249076475, 4.66447690176451136578936002376, 4.71229251533415153628856500930, 5.25449327463585577834210600385, 5.40753286844716506623351654904, 5.87989905733383526252258343058, 6.30714048224781784030648273482, 6.41342769870334318757379013329, 6.55101893499701331974146735428, 6.83399470437684797554881655036, 7.21630958020324484066865258597, 7.59194617543975541571343367198

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