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Properties

Label 8-280e4-1.1-c5e4-0-0
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  + 5·3-s − 100·5-s − 196·7-s − 190·9-s − 123·11-s + 789·13-s − 500·15-s + 1.05e3·17-s + 958·19-s − 980·21-s − 530·23-s + 6.25e3·25-s − 2.45e3·27-s − 5.54e3·29-s − 1.04e4·31-s − 615·33-s + 1.96e4·35-s − 2.40e4·37-s + 3.94e3·39-s − 3.54e4·41-s + 2.17e3·43-s + 1.90e4·45-s − 2.28e3·47-s + 2.40e4·49-s + 5.25e3·51-s − 2.22e4·53-s + 1.23e4·55-s + ⋯
L(s)  = 1  + 0.320·3-s − 1.78·5-s − 1.51·7-s − 0.781·9-s − 0.306·11-s + 1.29·13-s − 0.573·15-s + 0.882·17-s + 0.608·19-s − 0.484·21-s − 0.208·23-s + 2·25-s − 0.647·27-s − 1.22·29-s − 1.95·31-s − 0.0983·33-s + 2.70·35-s − 2.88·37-s + 0.415·39-s − 3.29·41-s + 0.179·43-s + 1.39·45-s − 0.151·47-s + 10/7·49-s + 0.282·51-s − 1.08·53-s + 0.548·55-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: Trivial
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$C_2 \wr S_4$ \( 1 - 5 T + 215 T^{2} + 428 T^{3} + 29200 p T^{4} + 428 p^{5} T^{5} + 215 p^{10} T^{6} - 5 p^{15} T^{7} + p^{20} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 123 T + 172787 T^{2} + 27386972 T^{3} + 43153011252 T^{4} + 27386972 p^{5} T^{5} + 172787 p^{10} T^{6} + 123 p^{15} T^{7} + p^{20} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 789 T + 485125 T^{2} - 38581642 T^{3} - 54076086342 T^{4} - 38581642 p^{5} T^{5} + 485125 p^{10} T^{6} - 789 p^{15} T^{7} + p^{20} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 1051 T + 4066457 T^{2} - 3404977458 T^{3} + 7501353530190 T^{4} - 3404977458 p^{5} T^{5} + 4066457 p^{10} T^{6} - 1051 p^{15} T^{7} + p^{20} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 958 T + 9387448 T^{2} - 6771508758 T^{3} + 34356193238334 T^{4} - 6771508758 p^{5} T^{5} + 9387448 p^{10} T^{6} - 958 p^{15} T^{7} + p^{20} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 530 T + 20490728 T^{2} + 10459894002 T^{3} + 185608302865806 T^{4} + 10459894002 p^{5} T^{5} + 20490728 p^{10} T^{6} + 530 p^{15} T^{7} + p^{20} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 5541 T + 49341797 T^{2} + 227093137722 T^{3} + 1146321865768026 T^{4} + 227093137722 p^{5} T^{5} + 49341797 p^{10} T^{6} + 5541 p^{15} T^{7} + p^{20} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 10440 T + 94620412 T^{2} + 542919718440 T^{3} + 3224458488249222 T^{4} + 542919718440 p^{5} T^{5} + 94620412 p^{10} T^{6} + 10440 p^{15} T^{7} + p^{20} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 24048 T + 321765148 T^{2} + 2580926443920 T^{3} + 20365849878329718 T^{4} + 2580926443920 p^{5} T^{5} + 321765148 p^{10} T^{6} + 24048 p^{15} T^{7} + p^{20} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 35414 T + 772958996 T^{2} + 11819669454226 T^{3} + 143175729036850886 T^{4} + 11819669454226 p^{5} T^{5} + 772958996 p^{10} T^{6} + 35414 p^{15} T^{7} + p^{20} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 2174 T + 390119152 T^{2} - 1775727181062 T^{3} + 72194412162722958 T^{4} - 1775727181062 p^{5} T^{5} + 390119152 p^{10} T^{6} - 2174 p^{15} T^{7} + p^{20} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 2287 T + 247149407 T^{2} - 513791739024 T^{3} + 14874205588227384 T^{4} - 513791739024 p^{5} T^{5} + 247149407 p^{10} T^{6} + 2287 p^{15} T^{7} + p^{20} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 22238 T + 1495715468 T^{2} + 22190212657122 T^{3} + 880939165865106246 T^{4} + 22190212657122 p^{5} T^{5} + 1495715468 p^{10} T^{6} + 22238 p^{15} T^{7} + p^{20} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 384 p T + 1184280236 T^{2} - 11179995989888 T^{3} + 166820726890790358 T^{4} - 11179995989888 p^{5} T^{5} + 1184280236 p^{10} T^{6} + 384 p^{16} T^{7} + p^{20} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 20250 T + 324507820 T^{2} - 10177013411882 T^{3} - 471272477545255002 T^{4} - 10177013411882 p^{5} T^{5} + 324507820 p^{10} T^{6} + 20250 p^{15} T^{7} + p^{20} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 19456 T + 4909041004 T^{2} + 72846962352640 T^{3} + 9651201241680406966 T^{4} + 72846962352640 p^{5} T^{5} + 4909041004 p^{10} T^{6} + 19456 p^{15} T^{7} + p^{20} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 72288 T + 7606620956 T^{2} + 353978653171168 T^{3} + 20954883099118322598 T^{4} + 353978653171168 p^{5} T^{5} + 7606620956 p^{10} T^{6} + 72288 p^{15} T^{7} + p^{20} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 59464 T + 3234849724 T^{2} - 196016010601912 T^{3} + 7276303444528329766 T^{4} - 196016010601912 p^{5} T^{5} + 3234849724 p^{10} T^{6} - 59464 p^{15} T^{7} + p^{20} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 232001 T + 26640497623 T^{2} + 2104483229338464 T^{3} + \)\(13\!\cdots\!76\)\( T^{4} + 2104483229338464 p^{5} T^{5} + 26640497623 p^{10} T^{6} + 232001 p^{15} T^{7} + p^{20} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 169620 T + 16135930556 T^{2} + 1107681189987060 T^{3} + 71718565102959823446 T^{4} + 1107681189987060 p^{5} T^{5} + 16135930556 p^{10} T^{6} + 169620 p^{15} T^{7} + p^{20} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 141434 T + 18669817124 T^{2} - 1584599232069374 T^{3} + \)\(12\!\cdots\!46\)\( T^{4} - 1584599232069374 p^{5} T^{5} + 18669817124 p^{10} T^{6} - 141434 p^{15} T^{7} + p^{20} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 167159 T + 29462387305 T^{2} - 3299142326815570 T^{3} + \)\(37\!\cdots\!34\)\( T^{4} - 3299142326815570 p^{5} T^{5} + 29462387305 p^{10} T^{6} - 167159 p^{15} T^{7} + p^{20} T^{8} \)
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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

−8.417045045335758232368720895402, −8.000039703281484956539634354956, −7.59879261682903142408505148942, −7.50248924216760270492327895053, −7.38103072762469866591470527212, −6.97680628526731628216939458375, −6.76538114322340775120267798631, −6.40519323468951673165037833518, −6.39580200652740478115743020991, −5.67263556922900946812255183540, −5.48977852867457603354689274327, −5.41913611184646114752086281913, −5.33208753085653283680864926882, −4.62799616177016725493611562158, −4.22823051671775034119407665271, −3.96162574036797707854998730945, −3.87380029725334783027252014223, −3.22664176850573131102635705980, −3.15561069952064319220838287622, −3.12291194035451942392429207026, −3.06970809611890698964527742662, −1.90127606825536623180477890571, −1.89986154513509967847327094088, −1.33975937783639410142476151144, −1.09797306873370059569989114053, 0, 0, 0, 0, 1.09797306873370059569989114053, 1.33975937783639410142476151144, 1.89986154513509967847327094088, 1.90127606825536623180477890571, 3.06970809611890698964527742662, 3.12291194035451942392429207026, 3.15561069952064319220838287622, 3.22664176850573131102635705980, 3.87380029725334783027252014223, 3.96162574036797707854998730945, 4.22823051671775034119407665271, 4.62799616177016725493611562158, 5.33208753085653283680864926882, 5.41913611184646114752086281913, 5.48977852867457603354689274327, 5.67263556922900946812255183540, 6.39580200652740478115743020991, 6.40519323468951673165037833518, 6.76538114322340775120267798631, 6.97680628526731628216939458375, 7.38103072762469866591470527212, 7.50248924216760270492327895053, 7.59879261682903142408505148942, 8.000039703281484956539634354956, 8.417045045335758232368720895402

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