Properties

Label 8-280e4-1.1-c5e4-0-1
Degree $8$
Conductor $6146560000$
Sign $1$
Analytic cond. $4.06700\times 10^{6}$
Root an. cond. $6.70130$
Motivic weight $5$
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  − 13·3-s + 100·5-s + 196·7-s − 498·9-s − 595·11-s − 969·13-s − 1.30e3·15-s − 1.31e3·17-s − 1.09e3·19-s − 2.54e3·21-s − 1.53e3·23-s + 6.25e3·25-s + 8.31e3·27-s + 4.09e3·29-s − 4.82e3·31-s + 7.73e3·33-s + 1.96e4·35-s + 7.69e3·37-s + 1.25e4·39-s − 9.72e3·41-s − 2.06e4·43-s − 4.98e4·45-s − 1.66e3·47-s + 2.40e4·49-s + 1.70e4·51-s − 2.88e4·53-s − 5.95e4·55-s + ⋯
L(s)  = 1  − 0.833·3-s + 1.78·5-s + 1.51·7-s − 2.04·9-s − 1.48·11-s − 1.59·13-s − 1.49·15-s − 1.10·17-s − 0.692·19-s − 1.26·21-s − 0.604·23-s + 2·25-s + 2.19·27-s + 0.905·29-s − 0.900·31-s + 1.23·33-s + 2.70·35-s + 0.923·37-s + 1.32·39-s − 0.903·41-s − 1.69·43-s − 3.66·45-s − 0.109·47-s + 10/7·49-s + 0.920·51-s − 1.41·53-s − 2.65·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(4.06700\times 10^{6}\)
Root analytic conductor: \(6.70130\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{280} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{12} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - p^{2} T )^{4} \)
7$C_1$ \( ( 1 - p^{2} T )^{4} \)
good3$C_2 \wr S_4$ \( 1 + 13 T + 667 T^{2} + 2276 p T^{3} + 76388 p T^{4} + 2276 p^{6} T^{5} + 667 p^{10} T^{6} + 13 p^{15} T^{7} + p^{20} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 595 T + 479323 T^{2} + 184901500 T^{3} + 103801011804 T^{4} + 184901500 p^{5} T^{5} + 479323 p^{10} T^{6} + 595 p^{15} T^{7} + p^{20} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 969 T + 854765 T^{2} + 536517786 T^{3} + 371765074362 T^{4} + 536517786 p^{5} T^{5} + 854765 p^{10} T^{6} + 969 p^{15} T^{7} + p^{20} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 1315 T + 1974309 T^{2} + 143930922 p T^{3} + 5076229612050 T^{4} + 143930922 p^{6} T^{5} + 1974309 p^{10} T^{6} + 1315 p^{15} T^{7} + p^{20} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 1090 T + 2213952 T^{2} + 5017886506 T^{3} + 8543938734926 T^{4} + 5017886506 p^{5} T^{5} + 2213952 p^{10} T^{6} + 1090 p^{15} T^{7} + p^{20} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 1534 T - 7960136 T^{2} + 1798331998 T^{3} + 101800084820526 T^{4} + 1798331998 p^{5} T^{5} - 7960136 p^{10} T^{6} + 1534 p^{15} T^{7} + p^{20} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 4099 T + 29793309 T^{2} - 136314828534 T^{3} + 1164813566189586 T^{4} - 136314828534 p^{5} T^{5} + 29793309 p^{10} T^{6} - 4099 p^{15} T^{7} + p^{20} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 4820 T + 31753708 T^{2} + 19759859012 T^{3} - 226092297250394 T^{4} + 19759859012 p^{5} T^{5} + 31753708 p^{10} T^{6} + 4820 p^{15} T^{7} + p^{20} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 7692 T + 175500260 T^{2} - 1180476718404 T^{3} + 17588568445541798 T^{4} - 1180476718404 p^{5} T^{5} + 175500260 p^{10} T^{6} - 7692 p^{15} T^{7} + p^{20} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 9722 T + 203055180 T^{2} - 518969844162 T^{3} + 7131234578147638 T^{4} - 518969844162 p^{5} T^{5} + 203055180 p^{10} T^{6} + 9722 p^{15} T^{7} + p^{20} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 20610 T + 527846400 T^{2} + 6849954949306 T^{3} + 104599319533166382 T^{4} + 6849954949306 p^{5} T^{5} + 527846400 p^{10} T^{6} + 20610 p^{15} T^{7} + p^{20} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 1661 T + 199788015 T^{2} - 4693752118344 T^{3} - 18939889526085840 T^{4} - 4693752118344 p^{5} T^{5} + 199788015 p^{10} T^{6} + 1661 p^{15} T^{7} + p^{20} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 28898 T + 1173378868 T^{2} + 24323876824062 T^{3} + 702577854187252310 T^{4} + 24323876824062 p^{5} T^{5} + 1173378868 p^{10} T^{6} + 28898 p^{15} T^{7} + p^{20} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 101872 T + 6076981004 T^{2} + 251992326420208 T^{3} + 7775015642428985110 T^{4} + 251992326420208 p^{5} T^{5} + 6076981004 p^{10} T^{6} + 101872 p^{15} T^{7} + p^{20} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 24742 T + 2928124044 T^{2} + 59013636688234 T^{3} + 3563650890261191270 T^{4} + 59013636688234 p^{5} T^{5} + 2928124044 p^{10} T^{6} + 24742 p^{15} T^{7} + p^{20} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 82060 T + 5506352876 T^{2} + 257005495952556 T^{3} + 10264691538936675766 T^{4} + 257005495952556 p^{5} T^{5} + 5506352876 p^{10} T^{6} + 82060 p^{15} T^{7} + p^{20} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 102784 T + 9937198492 T^{2} + 559933878886272 T^{3} + 28970927280329238182 T^{4} + 559933878886272 p^{5} T^{5} + 9937198492 p^{10} T^{6} + 102784 p^{15} T^{7} + p^{20} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 80652 T + 6440600308 T^{2} + 247522632961428 T^{3} + 13529295119417271990 T^{4} + 247522632961428 p^{5} T^{5} + 6440600308 p^{10} T^{6} + 80652 p^{15} T^{7} + p^{20} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 117801 T + 11615719199 T^{2} + 636554664220896 T^{3} + 40673802658268600936 T^{4} + 636554664220896 p^{5} T^{5} + 11615719199 p^{10} T^{6} + 117801 p^{15} T^{7} + p^{20} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 155440 T + 13584353516 T^{2} + 1214675548111792 T^{3} + 92232922420922867830 T^{4} + 1214675548111792 p^{5} T^{5} + 13584353516 p^{10} T^{6} + 155440 p^{15} T^{7} + p^{20} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 634 p T + 13747608804 T^{2} - 774146578408782 T^{3} + 93055736557698405670 T^{4} - 774146578408782 p^{5} T^{5} + 13747608804 p^{10} T^{6} - 634 p^{16} T^{7} + p^{20} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 261031 T + 57221980661 T^{2} + 7496978473830090 T^{3} + \)\(84\!\cdots\!62\)\( T^{4} + 7496978473830090 p^{5} T^{5} + 57221980661 p^{10} T^{6} + 261031 p^{15} T^{7} + p^{20} T^{8} \)
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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

−8.476693466952630636160414095496, −7.83464341061390383836449758844, −7.66667684949598559133056679475, −7.64679699090966369404837863051, −7.40812181585284365082860264846, −6.63400957874845551085782197836, −6.60115478532026277543450807366, −6.43155283216311769250923835560, −6.12077550500831032985390606591, −5.69601646029471376814478397937, −5.43008287654672575419986607457, −5.40567821544252933332267483474, −5.37817334688003719224045933239, −4.74625704202239531172674058320, −4.58687640325731253714074586486, −4.32986042385450295655262629403, −4.24315584067897383890586769303, −2.96644043825605861330697200898, −2.94438265540481431091055211129, −2.84381915186886150968011683042, −2.74245815212572737521802825782, −2.00852005888322678381302962654, −1.81229171657368070915484761308, −1.39474790836835800227809513082, −1.37505942126249257328326042987, 0, 0, 0, 0, 1.37505942126249257328326042987, 1.39474790836835800227809513082, 1.81229171657368070915484761308, 2.00852005888322678381302962654, 2.74245815212572737521802825782, 2.84381915186886150968011683042, 2.94438265540481431091055211129, 2.96644043825605861330697200898, 4.24315584067897383890586769303, 4.32986042385450295655262629403, 4.58687640325731253714074586486, 4.74625704202239531172674058320, 5.37817334688003719224045933239, 5.40567821544252933332267483474, 5.43008287654672575419986607457, 5.69601646029471376814478397937, 6.12077550500831032985390606591, 6.43155283216311769250923835560, 6.60115478532026277543450807366, 6.63400957874845551085782197836, 7.40812181585284365082860264846, 7.64679699090966369404837863051, 7.66667684949598559133056679475, 7.83464341061390383836449758844, 8.476693466952630636160414095496

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