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Properties

Label 6-200e3-1.1-c5e3-0-1
Degree $6$
Conductor $8000000$
Sign $1$
Analytic cond. $33004.3$
Root an. cond. $5.66363$
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  − 3-s + 70·7-s − 287·9-s − 19·11-s + 196·13-s − 1.22e3·17-s + 1.22e3·19-s − 70·21-s + 2.49e3·23-s − 1.00e3·27-s + 1.19e4·29-s + 7.44e3·31-s + 19·33-s − 1.47e4·37-s − 196·39-s + 3.22e3·41-s + 4.10e4·43-s − 2.91e4·47-s + 1.04e4·49-s + 1.22e3·51-s + 1.28e4·53-s − 1.22e3·57-s + 6.49e4·59-s + 2.24e4·61-s − 2.00e4·63-s − 2.64e4·67-s − 2.49e3·69-s + ⋯
L(s)  = 1  − 0.0641·3-s + 0.539·7-s − 1.18·9-s − 0.0473·11-s + 0.321·13-s − 1.02·17-s + 0.775·19-s − 0.0346·21-s + 0.981·23-s − 0.265·27-s + 2.63·29-s + 1.39·31-s + 0.00303·33-s − 1.77·37-s − 0.0206·39-s + 0.299·41-s + 3.38·43-s − 1.92·47-s + 0.621·49-s + 0.0658·51-s + 0.629·53-s − 0.0497·57-s + 2.42·59-s + 0.773·61-s − 0.637·63-s − 0.721·67-s − 0.0629·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8000000 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8000000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(8000000\)    =    \(2^{9} \cdot 5^{6}\)
Sign: $1$
Analytic conductor: \(33004.3\)
Root analytic conductor: \(5.66363\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 8000000,\ (\ :5/2, 5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(4.878743208\)
\(L(\frac12)\) \(\approx\) \(4.878743208\)
\(L(\frac{7}{2})\) 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 \( 1 \)
good3$S_4\times C_2$ \( 1 + T + 32 p^{2} T^{2} + 527 p T^{3} + 32 p^{7} T^{4} + p^{10} T^{5} + p^{15} T^{6} \)
7$S_4\times C_2$ \( 1 - 10 p T - 793 p T^{2} + 3221636 T^{3} - 793 p^{6} T^{4} - 10 p^{11} T^{5} + p^{15} T^{6} \)
11$S_4\times C_2$ \( 1 + 19 T + 398752 T^{2} + 4577119 T^{3} + 398752 p^{5} T^{4} + 19 p^{10} T^{5} + p^{15} T^{6} \)
13$S_4\times C_2$ \( 1 - 196 T + 642439 T^{2} - 197315624 T^{3} + 642439 p^{5} T^{4} - 196 p^{10} T^{5} + p^{15} T^{6} \)
17$S_4\times C_2$ \( 1 + 1223 T + 2296046 T^{2} + 2818923747 T^{3} + 2296046 p^{5} T^{4} + 1223 p^{10} T^{5} + p^{15} T^{6} \)
19$S_4\times C_2$ \( 1 - 1221 T + 2478936 T^{2} + 986654423 T^{3} + 2478936 p^{5} T^{4} - 1221 p^{10} T^{5} + p^{15} T^{6} \)
23$S_4\times C_2$ \( 1 - 2490 T + 1075617 T^{2} + 18384852604 T^{3} + 1075617 p^{5} T^{4} - 2490 p^{10} T^{5} + p^{15} T^{6} \)
29$S_4\times C_2$ \( 1 - 11912 T + 107337223 T^{2} - 544650638288 T^{3} + 107337223 p^{5} T^{4} - 11912 p^{10} T^{5} + p^{15} T^{6} \)
31$S_4\times C_2$ \( 1 - 7442 T + 82803193 T^{2} - 411681472084 T^{3} + 82803193 p^{5} T^{4} - 7442 p^{10} T^{5} + p^{15} T^{6} \)
37$S_4\times C_2$ \( 1 + 14766 T + 175309995 T^{2} + 1612649497908 T^{3} + 175309995 p^{5} T^{4} + 14766 p^{10} T^{5} + p^{15} T^{6} \)
41$S_4\times C_2$ \( 1 - 3223 T + 204190518 T^{2} - 833781103267 T^{3} + 204190518 p^{5} T^{4} - 3223 p^{10} T^{5} + p^{15} T^{6} \)
43$S_4\times C_2$ \( 1 - 41060 T + 882414881 T^{2} - 12733285389272 T^{3} + 882414881 p^{5} T^{4} - 41060 p^{10} T^{5} + p^{15} T^{6} \)
47$S_4\times C_2$ \( 1 + 29188 T + 677809469 T^{2} + 13063893267768 T^{3} + 677809469 p^{5} T^{4} + 29188 p^{10} T^{5} + p^{15} T^{6} \)
53$S_4\times C_2$ \( 1 - 12878 T + 18166107 T^{2} + 11553992883916 T^{3} + 18166107 p^{5} T^{4} - 12878 p^{10} T^{5} + p^{15} T^{6} \)
59$S_4\times C_2$ \( 1 - 64912 T + 3469849057 T^{2} - 101432669884000 T^{3} + 3469849057 p^{5} T^{4} - 64912 p^{10} T^{5} + p^{15} T^{6} \)
61$S_4\times C_2$ \( 1 - 22478 T + 1331345683 T^{2} - 9784946414356 T^{3} + 1331345683 p^{5} T^{4} - 22478 p^{10} T^{5} + p^{15} T^{6} \)
67$S_4\times C_2$ \( 1 + 26499 T + 3645391656 T^{2} + 72815104048223 T^{3} + 3645391656 p^{5} T^{4} + 26499 p^{10} T^{5} + p^{15} T^{6} \)
71$S_4\times C_2$ \( 1 - 86676 T + 6149511717 T^{2} - 281168787837912 T^{3} + 6149511717 p^{5} T^{4} - 86676 p^{10} T^{5} + p^{15} T^{6} \)
73$S_4\times C_2$ \( 1 + 8305 T - 51125178 T^{2} - 126365418680971 T^{3} - 51125178 p^{5} T^{4} + 8305 p^{10} T^{5} + p^{15} T^{6} \)
79$S_4\times C_2$ \( 1 - 21982 T + 8034849513 T^{2} - 109220050198796 T^{3} + 8034849513 p^{5} T^{4} - 21982 p^{10} T^{5} + p^{15} T^{6} \)
83$S_4\times C_2$ \( 1 - 213353 T + 26132961424 T^{2} - 1969967057862869 T^{3} + 26132961424 p^{5} T^{4} - 213353 p^{10} T^{5} + p^{15} T^{6} \)
89$S_4\times C_2$ \( 1 - 182381 T + 27173913486 T^{2} - 2227663367824697 T^{3} + 27173913486 p^{5} T^{4} - 182381 p^{10} T^{5} + p^{15} T^{6} \)
97$S_4\times C_2$ \( 1 + 76342 T + 16917659759 T^{2} + 1144576356113332 T^{3} + 16917659759 p^{5} T^{4} + 76342 p^{10} T^{5} + p^{15} T^{6} \)
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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

−10.43956714602919563245489221873, −9.992791329809895860747027305193, −9.492560724729185885149883270928, −9.252610009439731305291664994944, −8.857465775120736157045749587300, −8.466268485109697450293903820081, −8.392986212885438133793613667387, −8.015774695148119502915425248509, −7.36116287342849505840789081748, −7.34084207754978139546791022978, −6.53146874511071911458249366601, −6.40319313856558786190172319850, −6.24521391524796193672161572479, −5.40872865975549200769875303407, −5.17569205838455585714413381630, −4.97778178012007947005406623840, −4.45843374267866547894081251648, −3.85482310656440687598901617831, −3.56025771413306003020087632332, −2.86707313208384228457885472967, −2.46120088075422355403030224348, −2.28424330054677470567702312894, −1.31059604583427795907521263817, −0.72985944586377455906311787850, −0.60403176976293949201517441481, 0.60403176976293949201517441481, 0.72985944586377455906311787850, 1.31059604583427795907521263817, 2.28424330054677470567702312894, 2.46120088075422355403030224348, 2.86707313208384228457885472967, 3.56025771413306003020087632332, 3.85482310656440687598901617831, 4.45843374267866547894081251648, 4.97778178012007947005406623840, 5.17569205838455585714413381630, 5.40872865975549200769875303407, 6.24521391524796193672161572479, 6.40319313856558786190172319850, 6.53146874511071911458249366601, 7.34084207754978139546791022978, 7.36116287342849505840789081748, 8.015774695148119502915425248509, 8.392986212885438133793613667387, 8.466268485109697450293903820081, 8.857465775120736157045749587300, 9.252610009439731305291664994944, 9.492560724729185885149883270928, 9.992791329809895860747027305193, 10.43956714602919563245489221873

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