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Properties

Label 6-80e3-1.1-c13e3-0-3
Degree $6$
Conductor $512000$
Sign $-1$
Analytic cond. $631291.$
Root an. cond. $9.26200$
Motivic weight $13$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 416·3-s + 4.68e4·5-s − 4.48e5·7-s − 1.66e6·9-s + 6.60e6·11-s − 3.35e7·13-s − 1.95e7·15-s + 8.31e7·17-s − 9.74e7·19-s + 1.86e8·21-s − 3.16e8·23-s + 1.46e9·25-s − 1.23e9·27-s + 2.23e9·29-s − 7.48e9·31-s − 2.74e9·33-s − 2.10e10·35-s + 3.14e10·37-s + 1.39e10·39-s − 1.07e10·41-s − 1.69e10·43-s − 7.79e10·45-s − 3.19e10·47-s − 6.43e10·49-s − 3.45e10·51-s − 2.21e11·53-s + 3.09e11·55-s + ⋯
L(s)  = 1  − 0.329·3-s + 1.34·5-s − 1.44·7-s − 1.04·9-s + 1.12·11-s − 1.92·13-s − 0.442·15-s + 0.835·17-s − 0.475·19-s + 0.474·21-s − 0.445·23-s + 6/5·25-s − 0.614·27-s + 0.698·29-s − 1.51·31-s − 0.370·33-s − 1.93·35-s + 2.01·37-s + 0.634·39-s − 0.353·41-s − 0.408·43-s − 1.39·45-s − 0.432·47-s − 0.663·49-s − 0.275·51-s − 1.37·53-s + 1.50·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512000 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512000 ^{s/2} \, \Gamma_{\C}(s+13/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(512000\)    =    \(2^{12} \cdot 5^{3}\)
Sign: $-1$
Analytic conductor: \(631291.\)
Root analytic conductor: \(9.26200\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 512000,\ (\ :13/2, 13/2, 13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) 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$C_1$ \( ( 1 - p^{6} T )^{3} \)
good3$S_4\times C_2$ \( 1 + 416 T + 611651 p T^{2} + 33225280 p^{4} T^{3} + 611651 p^{14} T^{4} + 416 p^{26} T^{5} + p^{39} T^{6} \)
7$S_4\times C_2$ \( 1 + 448292 T + 265294837557 T^{2} + 9371505504195400 p T^{3} + 265294837557 p^{13} T^{4} + 448292 p^{26} T^{5} + p^{39} T^{6} \)
11$S_4\times C_2$ \( 1 - 600364 p T + 967991042665 p^{2} T^{2} - 349257054550580680 p^{3} T^{3} + 967991042665 p^{15} T^{4} - 600364 p^{27} T^{5} + p^{39} T^{6} \)
13$S_4\times C_2$ \( 1 + 33501974 T + 908837738845763 T^{2} + \)\(18\!\cdots\!40\)\( T^{3} + 908837738845763 p^{13} T^{4} + 33501974 p^{26} T^{5} + p^{39} T^{6} \)
17$S_4\times C_2$ \( 1 - 83129542 T + 15503955856788607 T^{2} - \)\(15\!\cdots\!40\)\( T^{3} + 15503955856788607 p^{13} T^{4} - 83129542 p^{26} T^{5} + p^{39} T^{6} \)
19$S_4\times C_2$ \( 1 + 97491100 T + 62056832238727577 T^{2} + \)\(92\!\cdots\!00\)\( T^{3} + 62056832238727577 p^{13} T^{4} + 97491100 p^{26} T^{5} + p^{39} T^{6} \)
23$S_4\times C_2$ \( 1 + 316255836 T + 30454663727526051 p T^{2} + \)\(45\!\cdots\!40\)\( T^{3} + 30454663727526051 p^{14} T^{4} + 316255836 p^{26} T^{5} + p^{39} T^{6} \)
29$S_4\times C_2$ \( 1 - 2236171850 T + 22868957480011100467 T^{2} - \)\(27\!\cdots\!00\)\( T^{3} + 22868957480011100467 p^{13} T^{4} - 2236171850 p^{26} T^{5} + p^{39} T^{6} \)
31$S_4\times C_2$ \( 1 + 7482994376 T + 71432250099037817565 T^{2} + \)\(35\!\cdots\!20\)\( T^{3} + 71432250099037817565 p^{13} T^{4} + 7482994376 p^{26} T^{5} + p^{39} T^{6} \)
37$S_4\times C_2$ \( 1 - 31447174242 T + \)\(72\!\cdots\!07\)\( T^{2} - \)\(10\!\cdots\!20\)\( T^{3} + \)\(72\!\cdots\!07\)\( p^{13} T^{4} - 31447174242 p^{26} T^{5} + p^{39} T^{6} \)
41$S_4\times C_2$ \( 1 + 262265474 p T + \)\(89\!\cdots\!15\)\( T^{2} + \)\(11\!\cdots\!80\)\( T^{3} + \)\(89\!\cdots\!15\)\( p^{13} T^{4} + 262265474 p^{27} T^{5} + p^{39} T^{6} \)
43$S_4\times C_2$ \( 1 + 16930554856 T + \)\(50\!\cdots\!93\)\( T^{2} + \)\(57\!\cdots\!00\)\( T^{3} + \)\(50\!\cdots\!93\)\( p^{13} T^{4} + 16930554856 p^{26} T^{5} + p^{39} T^{6} \)
47$S_4\times C_2$ \( 1 + 31934201692 T + \)\(98\!\cdots\!57\)\( T^{2} + \)\(15\!\cdots\!60\)\( T^{3} + \)\(98\!\cdots\!57\)\( p^{13} T^{4} + 31934201692 p^{26} T^{5} + p^{39} T^{6} \)
53$S_4\times C_2$ \( 1 + 221149123934 T + \)\(62\!\cdots\!03\)\( T^{2} + \)\(82\!\cdots\!20\)\( T^{3} + \)\(62\!\cdots\!03\)\( p^{13} T^{4} + 221149123934 p^{26} T^{5} + p^{39} T^{6} \)
59$S_4\times C_2$ \( 1 - 55436423900 T + \)\(70\!\cdots\!37\)\( T^{2} - \)\(30\!\cdots\!00\)\( T^{3} + \)\(70\!\cdots\!37\)\( p^{13} T^{4} - 55436423900 p^{26} T^{5} + p^{39} T^{6} \)
61$S_4\times C_2$ \( 1 - 496161392746 T + \)\(31\!\cdots\!15\)\( T^{2} - \)\(76\!\cdots\!20\)\( T^{3} + \)\(31\!\cdots\!15\)\( p^{13} T^{4} - 496161392746 p^{26} T^{5} + p^{39} T^{6} \)
67$S_4\times C_2$ \( 1 + 459297824792 T + \)\(16\!\cdots\!57\)\( T^{2} + \)\(49\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!57\)\( p^{13} T^{4} + 459297824792 p^{26} T^{5} + p^{39} T^{6} \)
71$S_4\times C_2$ \( 1 + 521997878336 T + \)\(21\!\cdots\!65\)\( T^{2} + \)\(16\!\cdots\!20\)\( T^{3} + \)\(21\!\cdots\!65\)\( p^{13} T^{4} + 521997878336 p^{26} T^{5} + p^{39} T^{6} \)
73$S_4\times C_2$ \( 1 - 34315418782 p T + \)\(58\!\cdots\!23\)\( T^{2} - \)\(82\!\cdots\!40\)\( T^{3} + \)\(58\!\cdots\!23\)\( p^{13} T^{4} - 34315418782 p^{27} T^{5} + p^{39} T^{6} \)
79$S_4\times C_2$ \( 1 + 2990636883200 T + \)\(14\!\cdots\!17\)\( T^{2} + \)\(25\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!17\)\( p^{13} T^{4} + 2990636883200 p^{26} T^{5} + p^{39} T^{6} \)
83$S_4\times C_2$ \( 1 + 5137135467696 T + \)\(32\!\cdots\!33\)\( T^{2} + \)\(89\!\cdots\!20\)\( T^{3} + \)\(32\!\cdots\!33\)\( p^{13} T^{4} + 5137135467696 p^{26} T^{5} + p^{39} T^{6} \)
89$S_4\times C_2$ \( 1 + 19423025958450 T + \)\(18\!\cdots\!07\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!07\)\( p^{13} T^{4} + 19423025958450 p^{26} T^{5} + p^{39} T^{6} \)
97$S_4\times C_2$ \( 1 + 11088325396458 T + \)\(15\!\cdots\!07\)\( T^{2} + \)\(92\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!07\)\( p^{13} T^{4} + 11088325396458 p^{26} T^{5} + p^{39} T^{6} \)
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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

−11.09766326486535470085612114012, −10.05312748906773532211064158190, −10.01583367910452723193646490594, −9.820858258352169516417944735430, −9.552708630756337514955293780030, −9.089045204781870310743987576336, −8.892809187366903845310907755216, −8.044189848955267732782549356862, −7.996473788924319002631776168333, −7.35667514880267290453008278239, −6.63051268245743834167687415439, −6.62059806041643392298559170557, −6.46026467001506161678346468759, −5.55290673607762340189600260628, −5.52291484014544458153576966012, −5.40809449023314205055472603292, −4.52755022151551450314237299079, −4.09849449831518453723984818829, −3.72469432397198523262958644742, −2.91898109674791511098151481432, −2.72701537880672340240181243951, −2.63518323146673209415708262657, −1.63944693555714987593749515431, −1.61753497332193712697640038787, −0.994885375333150637637594269300, 0, 0, 0, 0.994885375333150637637594269300, 1.61753497332193712697640038787, 1.63944693555714987593749515431, 2.63518323146673209415708262657, 2.72701537880672340240181243951, 2.91898109674791511098151481432, 3.72469432397198523262958644742, 4.09849449831518453723984818829, 4.52755022151551450314237299079, 5.40809449023314205055472603292, 5.52291484014544458153576966012, 5.55290673607762340189600260628, 6.46026467001506161678346468759, 6.62059806041643392298559170557, 6.63051268245743834167687415439, 7.35667514880267290453008278239, 7.996473788924319002631776168333, 8.044189848955267732782549356862, 8.892809187366903845310907755216, 9.089045204781870310743987576336, 9.552708630756337514955293780030, 9.820858258352169516417944735430, 10.01583367910452723193646490594, 10.05312748906773532211064158190, 11.09766326486535470085612114012

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