Properties

Label 6-1080e3-1.1-c5e3-0-0
Degree $6$
Conductor $1259712000$
Sign $1$
Analytic cond. $5.19700\times 10^{6}$
Root an. cond. $13.1610$
Motivic weight $5$
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  − 75·5-s − 30·7-s + 660·11-s − 1.27e3·13-s − 1.73e3·17-s + 1.01e3·19-s − 2.43e3·23-s + 3.75e3·25-s − 3.78e3·29-s + 4.13e3·31-s + 2.25e3·35-s − 7.12e3·37-s − 1.73e4·41-s + 1.15e4·43-s + 1.95e4·47-s − 2.86e4·49-s + 4.96e3·53-s − 4.95e4·55-s + 3.21e4·59-s − 1.63e4·61-s + 9.58e4·65-s − 5.12e4·67-s + 8.40e4·71-s − 1.61e5·73-s − 1.98e4·77-s + 3.64e4·79-s + 6.45e4·83-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.231·7-s + 1.64·11-s − 2.09·13-s − 1.45·17-s + 0.642·19-s − 0.959·23-s + 6/5·25-s − 0.835·29-s + 0.772·31-s + 0.310·35-s − 0.855·37-s − 1.61·41-s + 0.953·43-s + 1.29·47-s − 1.70·49-s + 0.242·53-s − 2.20·55-s + 1.20·59-s − 0.561·61-s + 2.81·65-s − 1.39·67-s + 1.97·71-s − 3.55·73-s − 0.380·77-s + 0.656·79-s + 1.02·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{9} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(5.19700\times 10^{6}\)
Root analytic conductor: \(13.1610\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 3^{9} \cdot 5^{3} ,\ ( \ : 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(2.873735139\)
\(L(\frac12)\) \(\approx\) \(2.873735139\)
\(L(\frac{7}{2})\) 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 \( 1 \)
3 \( 1 \)
5$C_1$ \( ( 1 + p^{2} T )^{3} \)
good7$S_4\times C_2$ \( 1 + 30 T + 4227 p T^{2} + 1944844 T^{3} + 4227 p^{6} T^{4} + 30 p^{10} T^{5} + p^{15} T^{6} \)
11$S_4\times C_2$ \( 1 - 60 p T + 369069 T^{2} - 195617600 T^{3} + 369069 p^{5} T^{4} - 60 p^{11} T^{5} + p^{15} T^{6} \)
13$S_4\times C_2$ \( 1 + 1278 T + 1399023 T^{2} + 886453940 T^{3} + 1399023 p^{5} T^{4} + 1278 p^{10} T^{5} + p^{15} T^{6} \)
17$S_4\times C_2$ \( 1 + 1731 T + 4195482 T^{2} + 4553493875 T^{3} + 4195482 p^{5} T^{4} + 1731 p^{10} T^{5} + p^{15} T^{6} \)
19$S_4\times C_2$ \( 1 - 1011 T + 4567560 T^{2} - 3940193383 T^{3} + 4567560 p^{5} T^{4} - 1011 p^{10} T^{5} + p^{15} T^{6} \)
23$S_4\times C_2$ \( 1 + 2433 T + 3736224 T^{2} + 15021612913 T^{3} + 3736224 p^{5} T^{4} + 2433 p^{10} T^{5} + p^{15} T^{6} \)
29$S_4\times C_2$ \( 1 + 3786 T + 25703655 T^{2} + 55079231788 T^{3} + 25703655 p^{5} T^{4} + 3786 p^{10} T^{5} + p^{15} T^{6} \)
31$S_4\times C_2$ \( 1 - 4131 T + 76999284 T^{2} - 199758612823 T^{3} + 76999284 p^{5} T^{4} - 4131 p^{10} T^{5} + p^{15} T^{6} \)
37$S_4\times C_2$ \( 1 + 7122 T + 106725675 T^{2} + 976455033036 T^{3} + 106725675 p^{5} T^{4} + 7122 p^{10} T^{5} + p^{15} T^{6} \)
41$S_4\times C_2$ \( 1 + 17352 T + 307979727 T^{2} + 4000851697032 T^{3} + 307979727 p^{5} T^{4} + 17352 p^{10} T^{5} + p^{15} T^{6} \)
43$S_4\times C_2$ \( 1 - 11556 T + 407786805 T^{2} - 3323569921792 T^{3} + 407786805 p^{5} T^{4} - 11556 p^{10} T^{5} + p^{15} T^{6} \)
47$S_4\times C_2$ \( 1 - 19548 T + 757259373 T^{2} - 9014704644872 T^{3} + 757259373 p^{5} T^{4} - 19548 p^{10} T^{5} + p^{15} T^{6} \)
53$S_4\times C_2$ \( 1 - 4965 T + 501644238 T^{2} - 10353376238325 T^{3} + 501644238 p^{5} T^{4} - 4965 p^{10} T^{5} + p^{15} T^{6} \)
59$S_4\times C_2$ \( 1 - 32106 T + 1846016361 T^{2} - 36041350434772 T^{3} + 1846016361 p^{5} T^{4} - 32106 p^{10} T^{5} + p^{15} T^{6} \)
61$S_4\times C_2$ \( 1 + 16317 T + 1644398874 T^{2} + 11869127424641 T^{3} + 1644398874 p^{5} T^{4} + 16317 p^{10} T^{5} + p^{15} T^{6} \)
67$S_4\times C_2$ \( 1 + 51258 T + 3521678901 T^{2} + 118259129900212 T^{3} + 3521678901 p^{5} T^{4} + 51258 p^{10} T^{5} + p^{15} T^{6} \)
71$S_4\times C_2$ \( 1 - 84072 T + 5874688977 T^{2} - 256382468458392 T^{3} + 5874688977 p^{5} T^{4} - 84072 p^{10} T^{5} + p^{15} T^{6} \)
73$S_4\times C_2$ \( 1 + 161892 T + 14207002167 T^{2} + 791040688088256 T^{3} + 14207002167 p^{5} T^{4} + 161892 p^{10} T^{5} + p^{15} T^{6} \)
79$S_4\times C_2$ \( 1 - 36441 T + 1760859156 T^{2} + 82354833751563 T^{3} + 1760859156 p^{5} T^{4} - 36441 p^{10} T^{5} + p^{15} T^{6} \)
83$S_4\times C_2$ \( 1 - 64581 T + 2437979532 T^{2} + 114994527029099 T^{3} + 2437979532 p^{5} T^{4} - 64581 p^{10} T^{5} + p^{15} T^{6} \)
89$S_4\times C_2$ \( 1 - 61584 T + 9105757071 T^{2} - 534875973508248 T^{3} + 9105757071 p^{5} T^{4} - 61584 p^{10} T^{5} + p^{15} T^{6} \)
97$S_4\times C_2$ \( 1 - 14760 T + 23538340899 T^{2} - 216826936384336 T^{3} + 23538340899 p^{5} T^{4} - 14760 p^{10} T^{5} + p^{15} T^{6} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.149839599418389511038569327749, −7.66535795887336683356050990195, −7.38907837657033388298980340375, −7.30902997778231139322859755664, −6.95500535530519427996176824227, −6.71525733074551654870431487595, −6.48291828083945979288874565780, −6.17422166084883586164732994932, −5.70942281304417575380547189069, −5.46600333759543717600437427884, −4.99787449787747221705708825558, −4.69785180096006452000372907832, −4.59774921686873085830096382274, −4.09742521654689651168480830288, −3.85404621426850174119680186013, −3.83347715860599641662108652779, −3.08082478695045753368361038256, −2.93810192034218144635380288659, −2.70514692304616837714073436936, −1.94149230544641305632542682824, −1.71456388000264169419200823035, −1.70185568096874842747160965096, −0.65150680805430469930778660361, −0.49822402033047344274155518459, −0.40236643327620184646604400456, 0.40236643327620184646604400456, 0.49822402033047344274155518459, 0.65150680805430469930778660361, 1.70185568096874842747160965096, 1.71456388000264169419200823035, 1.94149230544641305632542682824, 2.70514692304616837714073436936, 2.93810192034218144635380288659, 3.08082478695045753368361038256, 3.83347715860599641662108652779, 3.85404621426850174119680186013, 4.09742521654689651168480830288, 4.59774921686873085830096382274, 4.69785180096006452000372907832, 4.99787449787747221705708825558, 5.46600333759543717600437427884, 5.70942281304417575380547189069, 6.17422166084883586164732994932, 6.48291828083945979288874565780, 6.71525733074551654870431487595, 6.95500535530519427996176824227, 7.30902997778231139322859755664, 7.38907837657033388298980340375, 7.66535795887336683356050990195, 8.149839599418389511038569327749

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