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Properties

Label 6-24e3-1.1-c21e3-0-1
Degree $6$
Conductor $13824$
Sign $1$
Analytic cond. $301768.$
Root an. cond. $8.18990$
Motivic weight $21$
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  + 1.77e5·3-s + 5.28e6·5-s + 8.52e8·7-s + 2.09e10·9-s + 6.24e10·11-s + 2.03e11·13-s + 9.35e11·15-s + 6.95e11·17-s + 4.95e12·19-s + 1.51e14·21-s + 1.50e14·23-s − 3.62e14·25-s + 2.05e15·27-s + 3.24e15·29-s + 6.52e14·31-s + 1.10e16·33-s + 4.50e15·35-s − 2.06e16·37-s + 3.60e16·39-s − 4.76e16·41-s − 2.77e17·43-s + 1.10e17·45-s − 5.35e17·47-s + 1.22e17·49-s + 1.23e17·51-s + 2.84e18·53-s + 3.29e17·55-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.241·5-s + 1.14·7-s + 2·9-s + 0.726·11-s + 0.409·13-s + 0.418·15-s + 0.0837·17-s + 0.185·19-s + 1.97·21-s + 0.759·23-s − 0.759·25-s + 1.92·27-s + 1.43·29-s + 0.142·31-s + 1.25·33-s + 0.275·35-s − 0.707·37-s + 0.710·39-s − 0.553·41-s − 1.95·43-s + 0.483·45-s − 1.48·47-s + 0.218·49-s + 0.144·51-s + 2.23·53-s + 0.175·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13824 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13824 ^{s/2} \, \Gamma_{\C}(s+21/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(13824\)    =    \(2^{9} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(301768.\)
Root analytic conductor: \(8.18990\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 13824,\ (\ :21/2, 21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(20.41498707\)
\(L(\frac12)\) \(\approx\) \(20.41498707\)
\(L(\frac{23}{2})\) 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$C_1$ \( ( 1 - p^{10} T )^{3} \)
good5$S_4\times C_2$ \( 1 - 5280498 T + 78020027876751 p T^{2} + 67622353088469656708 p^{3} T^{3} + 78020027876751 p^{22} T^{4} - 5280498 p^{42} T^{5} + p^{63} T^{6} \)
7$S_4\times C_2$ \( 1 - 17398824 p^{2} T + 12343302458050629 p^{2} T^{2} - \)\(19\!\cdots\!48\)\( p^{3} T^{3} + 12343302458050629 p^{23} T^{4} - 17398824 p^{44} T^{5} + p^{63} T^{6} \)
11$S_4\times C_2$ \( 1 - 62490757668 T + \)\(18\!\cdots\!29\)\( T^{2} - \)\(85\!\cdots\!96\)\( p T^{3} + \)\(18\!\cdots\!29\)\( p^{21} T^{4} - 62490757668 p^{42} T^{5} + p^{63} T^{6} \)
13$S_4\times C_2$ \( 1 - 15674246754 p T + \)\(28\!\cdots\!63\)\( p T^{2} - \)\(97\!\cdots\!04\)\( p^{2} T^{3} + \)\(28\!\cdots\!63\)\( p^{22} T^{4} - 15674246754 p^{43} T^{5} + p^{63} T^{6} \)
17$S_4\times C_2$ \( 1 - 695827819926 T + \)\(92\!\cdots\!79\)\( p T^{2} + \)\(15\!\cdots\!52\)\( p^{2} T^{3} + \)\(92\!\cdots\!79\)\( p^{22} T^{4} - 695827819926 p^{42} T^{5} + p^{63} T^{6} \)
19$S_4\times C_2$ \( 1 - 13727158236 p^{2} T + \)\(19\!\cdots\!81\)\( p^{2} T^{2} - \)\(37\!\cdots\!68\)\( p^{3} T^{3} + \)\(19\!\cdots\!81\)\( p^{23} T^{4} - 13727158236 p^{44} T^{5} + p^{63} T^{6} \)
23$S_4\times C_2$ \( 1 - 150867407938152 T + \)\(12\!\cdots\!45\)\( T^{2} - \)\(11\!\cdots\!44\)\( T^{3} + \)\(12\!\cdots\!45\)\( p^{21} T^{4} - 150867407938152 p^{42} T^{5} + p^{63} T^{6} \)
29$S_4\times C_2$ \( 1 - 3242224905226746 T + \)\(14\!\cdots\!31\)\( T^{2} - \)\(31\!\cdots\!92\)\( T^{3} + \)\(14\!\cdots\!31\)\( p^{21} T^{4} - 3242224905226746 p^{42} T^{5} + p^{63} T^{6} \)
31$S_4\times C_2$ \( 1 - 652508601550896 T + \)\(14\!\cdots\!37\)\( T^{2} - \)\(11\!\cdots\!72\)\( T^{3} + \)\(14\!\cdots\!37\)\( p^{21} T^{4} - 652508601550896 p^{42} T^{5} + p^{63} T^{6} \)
37$S_4\times C_2$ \( 1 + 20693271836916222 T + \)\(17\!\cdots\!39\)\( T^{2} + \)\(34\!\cdots\!52\)\( T^{3} + \)\(17\!\cdots\!39\)\( p^{21} T^{4} + 20693271836916222 p^{42} T^{5} + p^{63} T^{6} \)
41$S_4\times C_2$ \( 1 + 47608600312552002 T + \)\(75\!\cdots\!39\)\( T^{2} - \)\(21\!\cdots\!96\)\( T^{3} + \)\(75\!\cdots\!39\)\( p^{21} T^{4} + 47608600312552002 p^{42} T^{5} + p^{63} T^{6} \)
43$S_4\times C_2$ \( 1 + 277110211230415548 T + \)\(74\!\cdots\!97\)\( T^{2} + \)\(11\!\cdots\!24\)\( T^{3} + \)\(74\!\cdots\!97\)\( p^{21} T^{4} + 277110211230415548 p^{42} T^{5} + p^{63} T^{6} \)
47$S_4\times C_2$ \( 1 + 535942507326221328 T + \)\(24\!\cdots\!41\)\( T^{2} + \)\(93\!\cdots\!32\)\( T^{3} + \)\(24\!\cdots\!41\)\( p^{21} T^{4} + 535942507326221328 p^{42} T^{5} + p^{63} T^{6} \)
53$S_4\times C_2$ \( 1 - 2849297892348987426 T + \)\(74\!\cdots\!43\)\( T^{2} - \)\(10\!\cdots\!76\)\( T^{3} + \)\(74\!\cdots\!43\)\( p^{21} T^{4} - 2849297892348987426 p^{42} T^{5} + p^{63} T^{6} \)
59$S_4\times C_2$ \( 1 - 881046377434726548 T + \)\(20\!\cdots\!13\)\( T^{2} - \)\(19\!\cdots\!04\)\( T^{3} + \)\(20\!\cdots\!13\)\( p^{21} T^{4} - 881046377434726548 p^{42} T^{5} + p^{63} T^{6} \)
61$S_4\times C_2$ \( 1 - 10620890455615553322 T + \)\(98\!\cdots\!83\)\( T^{2} - \)\(52\!\cdots\!88\)\( T^{3} + \)\(98\!\cdots\!83\)\( p^{21} T^{4} - 10620890455615553322 p^{42} T^{5} + p^{63} T^{6} \)
67$S_4\times C_2$ \( 1 - 14840080728867715116 T + \)\(50\!\cdots\!25\)\( T^{2} - \)\(61\!\cdots\!28\)\( T^{3} + \)\(50\!\cdots\!25\)\( p^{21} T^{4} - 14840080728867715116 p^{42} T^{5} + p^{63} T^{6} \)
71$S_4\times C_2$ \( 1 - 14608944838311941496 T + \)\(57\!\cdots\!57\)\( T^{2} - \)\(26\!\cdots\!52\)\( T^{3} + \)\(57\!\cdots\!57\)\( p^{21} T^{4} - 14608944838311941496 p^{42} T^{5} + p^{63} T^{6} \)
73$S_4\times C_2$ \( 1 - 13575086263949291454 T + \)\(26\!\cdots\!99\)\( T^{2} - \)\(88\!\cdots\!84\)\( T^{3} + \)\(26\!\cdots\!99\)\( p^{21} T^{4} - 13575086263949291454 p^{42} T^{5} + p^{63} T^{6} \)
79$S_4\times C_2$ \( 1 - \)\(27\!\cdots\!60\)\( T + \)\(42\!\cdots\!85\)\( T^{2} - \)\(42\!\cdots\!44\)\( T^{3} + \)\(42\!\cdots\!85\)\( p^{21} T^{4} - \)\(27\!\cdots\!60\)\( p^{42} T^{5} + p^{63} T^{6} \)
83$S_4\times C_2$ \( 1 - \)\(58\!\cdots\!60\)\( T + \)\(16\!\cdots\!21\)\( T^{2} - \)\(28\!\cdots\!48\)\( T^{3} + \)\(16\!\cdots\!21\)\( p^{21} T^{4} - \)\(58\!\cdots\!60\)\( p^{42} T^{5} + p^{63} T^{6} \)
89$S_4\times C_2$ \( 1 - \)\(47\!\cdots\!30\)\( T + \)\(23\!\cdots\!99\)\( T^{2} - \)\(68\!\cdots\!52\)\( T^{3} + \)\(23\!\cdots\!99\)\( p^{21} T^{4} - \)\(47\!\cdots\!30\)\( p^{42} T^{5} + p^{63} T^{6} \)
97$S_4\times C_2$ \( 1 - \)\(72\!\cdots\!02\)\( T + \)\(14\!\cdots\!87\)\( T^{2} - \)\(63\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!87\)\( p^{21} T^{4} - \)\(72\!\cdots\!02\)\( p^{42} T^{5} + p^{63} T^{6} \)
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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

−11.66954180465573952377203086287, −10.69613369679589813169423232568, −10.53616176521790789069003177479, −10.02068701766496791631639238167, −9.552997313669869539024598391341, −9.011128758526150905752310866511, −8.919380742769244515081597249875, −8.202847281696970620950001677149, −7.952461587429409395727375844113, −7.906190898629386068097696633591, −6.92402663277837857128896666014, −6.56709444262571735979837336660, −6.48090680874521262859276456712, −5.20991534354749667434179922198, −5.12253762421639653757111922168, −4.73914838199475970225262588449, −3.77067262604501734440173626088, −3.72094058930182633044603103482, −3.40550180442753395537849298506, −2.43850099094929870349059792233, −2.40436483292797185410503477636, −1.73379655959949628485345976515, −1.50277133174584964055773702453, −0.844104728546681090579063769177, −0.62248433116497439205336338380, 0.62248433116497439205336338380, 0.844104728546681090579063769177, 1.50277133174584964055773702453, 1.73379655959949628485345976515, 2.40436483292797185410503477636, 2.43850099094929870349059792233, 3.40550180442753395537849298506, 3.72094058930182633044603103482, 3.77067262604501734440173626088, 4.73914838199475970225262588449, 5.12253762421639653757111922168, 5.20991534354749667434179922198, 6.48090680874521262859276456712, 6.56709444262571735979837336660, 6.92402663277837857128896666014, 7.906190898629386068097696633591, 7.952461587429409395727375844113, 8.202847281696970620950001677149, 8.919380742769244515081597249875, 9.011128758526150905752310866511, 9.552997313669869539024598391341, 10.02068701766496791631639238167, 10.53616176521790789069003177479, 10.69613369679589813169423232568, 11.66954180465573952377203086287

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