Properties

Label 12-10e6-1.1-c14e6-0-0
Degree $12$
Conductor $1000000$
Sign $1$
Analytic cond. $3.69346\times 10^{6}$
Root an. cond. $3.52603$
Motivic weight $14$
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  − 384·2-s + 2.91e3·3-s + 7.37e4·4-s + 8.25e4·5-s − 1.11e6·6-s + 9.43e5·7-s − 8.38e6·8-s + 4.23e6·9-s − 3.16e7·10-s − 4.55e7·11-s + 2.14e8·12-s − 5.21e7·13-s − 3.62e8·14-s + 2.40e8·15-s + 4.02e8·16-s − 2.94e8·17-s − 1.62e9·18-s + 6.08e9·20-s + 2.74e9·21-s + 1.74e10·22-s − 9.43e9·23-s − 2.44e10·24-s + 5.72e9·25-s + 2.00e10·26-s + 1.29e10·27-s + 6.95e10·28-s − 9.22e10·30-s + ⋯
L(s)  = 1  − 3·2-s + 1.33·3-s + 9/2·4-s + 1.05·5-s − 3.99·6-s + 1.14·7-s − 4·8-s + 0.886·9-s − 3.16·10-s − 2.33·11-s + 5.99·12-s − 0.831·13-s − 3.43·14-s + 1.40·15-s + 3/2·16-s − 0.717·17-s − 2.65·18-s + 4.75·20-s + 1.52·21-s + 7.01·22-s − 2.76·23-s − 5.32·24-s + 0.938·25-s + 2.49·26-s + 1.23·27-s + 5.15·28-s − 4.21·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000000 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000000 ^{s/2} \, \Gamma_{\C}(s+7)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(1000000\)    =    \(2^{6} \cdot 5^{6}\)
Sign: $1$
Analytic conductor: \(3.69346\times 10^{6}\)
Root analytic conductor: \(3.52603\)
Motivic weight: \(14\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 1000000,\ (\ :[7]^{6}),\ 1)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.003046654530\)
\(L(\frac12)\) \(\approx\) \(0.003046654530\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p^{7} T + p^{13} T^{2} )^{3} \)
5 \( 1 - 132 p^{4} T + 69159 p^{6} T^{2} + 10256088 p^{11} T^{3} + 69159 p^{20} T^{4} - 132 p^{32} T^{5} + p^{42} T^{6} \)
good3 \( 1 - 2912 T + 4239872 T^{2} - 480395992 p^{3} T^{3} - 17182907273 p^{6} T^{4} + 43637510352856 p^{7} T^{5} - 21441534272480992 p^{8} T^{6} + 43637510352856 p^{21} T^{7} - 17182907273 p^{34} T^{8} - 480395992 p^{45} T^{9} + 4239872 p^{56} T^{10} - 2912 p^{70} T^{11} + p^{84} T^{12} \)
7 \( 1 - 943128 T + 444745212192 T^{2} + 20096526034742432 p T^{3} - \)\(16\!\cdots\!93\)\( p^{2} T^{4} - \)\(10\!\cdots\!68\)\( p^{2} T^{5} + \)\(10\!\cdots\!72\)\( p^{2} T^{6} - \)\(10\!\cdots\!68\)\( p^{16} T^{7} - \)\(16\!\cdots\!93\)\( p^{30} T^{8} + 20096526034742432 p^{43} T^{9} + 444745212192 p^{56} T^{10} - 943128 p^{70} T^{11} + p^{84} T^{12} \)
11 \( ( 1 + 22783284 T + 116422088665725 p T^{2} + \)\(14\!\cdots\!40\)\( p^{2} T^{3} + 116422088665725 p^{15} T^{4} + 22783284 p^{28} T^{5} + p^{42} T^{6} )^{2} \)
13 \( 1 + 4011486 p T + 8046009964098 p^{2} T^{2} + \)\(24\!\cdots\!78\)\( p^{3} T^{3} + \)\(12\!\cdots\!23\)\( p^{4} T^{4} - \)\(13\!\cdots\!76\)\( p^{5} T^{5} + \)\(24\!\cdots\!52\)\( p^{6} T^{6} - \)\(13\!\cdots\!76\)\( p^{19} T^{7} + \)\(12\!\cdots\!23\)\( p^{32} T^{8} + \)\(24\!\cdots\!78\)\( p^{45} T^{9} + 8046009964098 p^{58} T^{10} + 4011486 p^{71} T^{11} + p^{84} T^{12} \)
17 \( 1 + 294348942 T + 43320649828259682 T^{2} - \)\(29\!\cdots\!06\)\( T^{3} - \)\(28\!\cdots\!37\)\( T^{4} - \)\(19\!\cdots\!32\)\( T^{5} + \)\(10\!\cdots\!08\)\( T^{6} - \)\(19\!\cdots\!32\)\( p^{14} T^{7} - \)\(28\!\cdots\!37\)\( p^{28} T^{8} - \)\(29\!\cdots\!06\)\( p^{42} T^{9} + 43320649828259682 p^{56} T^{10} + 294348942 p^{70} T^{11} + p^{84} T^{12} \)
19 \( 1 - 2616540268763896326 T^{2} + \)\(35\!\cdots\!15\)\( T^{4} - \)\(33\!\cdots\!20\)\( T^{6} + \)\(35\!\cdots\!15\)\( p^{28} T^{8} - 2616540268763896326 p^{56} T^{10} + p^{84} T^{12} \)
23 \( 1 + 9431163408 T + 44473421614199087232 T^{2} + \)\(17\!\cdots\!96\)\( T^{3} + \)\(45\!\cdots\!63\)\( T^{4} + \)\(71\!\cdots\!52\)\( T^{5} + \)\(15\!\cdots\!08\)\( T^{6} + \)\(71\!\cdots\!52\)\( p^{14} T^{7} + \)\(45\!\cdots\!63\)\( p^{28} T^{8} + \)\(17\!\cdots\!96\)\( p^{42} T^{9} + 44473421614199087232 p^{56} T^{10} + 9431163408 p^{70} T^{11} + p^{84} T^{12} \)
29 \( 1 - \)\(10\!\cdots\!86\)\( T^{2} + \)\(56\!\cdots\!15\)\( T^{4} - \)\(19\!\cdots\!20\)\( T^{6} + \)\(56\!\cdots\!15\)\( p^{28} T^{8} - \)\(10\!\cdots\!86\)\( p^{56} T^{10} + p^{84} T^{12} \)
31 \( ( 1 - 1860702696 T + \)\(77\!\cdots\!35\)\( T^{2} - \)\(66\!\cdots\!00\)\( T^{3} + \)\(77\!\cdots\!35\)\( p^{14} T^{4} - 1860702696 p^{28} T^{5} + p^{42} T^{6} )^{2} \)
37 \( 1 + 429898030002 T + \)\(92\!\cdots\!02\)\( T^{2} + \)\(14\!\cdots\!94\)\( T^{3} + \)\(18\!\cdots\!23\)\( T^{4} + \)\(22\!\cdots\!88\)\( T^{5} + \)\(23\!\cdots\!48\)\( T^{6} + \)\(22\!\cdots\!88\)\( p^{14} T^{7} + \)\(18\!\cdots\!23\)\( p^{28} T^{8} + \)\(14\!\cdots\!94\)\( p^{42} T^{9} + \)\(92\!\cdots\!02\)\( p^{56} T^{10} + 429898030002 p^{70} T^{11} + p^{84} T^{12} \)
41 \( ( 1 - 22840528956 T + \)\(76\!\cdots\!95\)\( T^{2} - \)\(72\!\cdots\!40\)\( T^{3} + \)\(76\!\cdots\!95\)\( p^{14} T^{4} - 22840528956 p^{28} T^{5} + p^{42} T^{6} )^{2} \)
43 \( 1 + 935465548368 T + \)\(43\!\cdots\!12\)\( T^{2} + \)\(13\!\cdots\!96\)\( T^{3} + \)\(24\!\cdots\!03\)\( T^{4} + \)\(83\!\cdots\!72\)\( T^{5} - \)\(43\!\cdots\!32\)\( T^{6} + \)\(83\!\cdots\!72\)\( p^{14} T^{7} + \)\(24\!\cdots\!03\)\( p^{28} T^{8} + \)\(13\!\cdots\!96\)\( p^{42} T^{9} + \)\(43\!\cdots\!12\)\( p^{56} T^{10} + 935465548368 p^{70} T^{11} + p^{84} T^{12} \)
47 \( 1 + 966227586192 T + \)\(46\!\cdots\!32\)\( T^{2} + \)\(92\!\cdots\!24\)\( T^{3} - \)\(65\!\cdots\!37\)\( T^{4} - \)\(82\!\cdots\!92\)\( T^{5} - \)\(45\!\cdots\!92\)\( T^{6} - \)\(82\!\cdots\!92\)\( p^{14} T^{7} - \)\(65\!\cdots\!37\)\( p^{28} T^{8} + \)\(92\!\cdots\!24\)\( p^{42} T^{9} + \)\(46\!\cdots\!32\)\( p^{56} T^{10} + 966227586192 p^{70} T^{11} + p^{84} T^{12} \)
53 \( 1 + 665070162 p^{2} T + 221159160191353122 p^{4} T^{2} + \)\(23\!\cdots\!26\)\( T^{3} - \)\(15\!\cdots\!37\)\( T^{4} - \)\(63\!\cdots\!08\)\( T^{5} - \)\(62\!\cdots\!92\)\( T^{6} - \)\(63\!\cdots\!08\)\( p^{14} T^{7} - \)\(15\!\cdots\!37\)\( p^{28} T^{8} + \)\(23\!\cdots\!26\)\( p^{42} T^{9} + 221159160191353122 p^{60} T^{10} + 665070162 p^{72} T^{11} + p^{84} T^{12} \)
59 \( 1 - \)\(25\!\cdots\!66\)\( T^{2} + \)\(32\!\cdots\!15\)\( T^{4} - \)\(25\!\cdots\!20\)\( T^{6} + \)\(32\!\cdots\!15\)\( p^{28} T^{8} - \)\(25\!\cdots\!66\)\( p^{56} T^{10} + p^{84} T^{12} \)
61 \( ( 1 - 1055549965236 T + \)\(27\!\cdots\!55\)\( T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(27\!\cdots\!55\)\( p^{14} T^{4} - 1055549965236 p^{28} T^{5} + p^{42} T^{6} )^{2} \)
67 \( 1 + 8480735447712 T + \)\(35\!\cdots\!72\)\( T^{2} + \)\(14\!\cdots\!04\)\( T^{3} + \)\(10\!\cdots\!83\)\( T^{4} + \)\(11\!\cdots\!88\)\( T^{5} + \)\(71\!\cdots\!88\)\( T^{6} + \)\(11\!\cdots\!88\)\( p^{14} T^{7} + \)\(10\!\cdots\!83\)\( p^{28} T^{8} + \)\(14\!\cdots\!04\)\( p^{42} T^{9} + \)\(35\!\cdots\!72\)\( p^{56} T^{10} + 8480735447712 p^{70} T^{11} + p^{84} T^{12} \)
71 \( ( 1 - 11166824728056 T + \)\(28\!\cdots\!55\)\( T^{2} - \)\(18\!\cdots\!80\)\( T^{3} + \)\(28\!\cdots\!55\)\( p^{14} T^{4} - 11166824728056 p^{28} T^{5} + p^{42} T^{6} )^{2} \)
73 \( 1 + 6994307700378 T + \)\(24\!\cdots\!42\)\( T^{2} - \)\(15\!\cdots\!94\)\( T^{3} - \)\(20\!\cdots\!33\)\( p^{2} T^{4} + \)\(53\!\cdots\!64\)\( p T^{5} + \)\(30\!\cdots\!28\)\( T^{6} + \)\(53\!\cdots\!64\)\( p^{15} T^{7} - \)\(20\!\cdots\!33\)\( p^{30} T^{8} - \)\(15\!\cdots\!94\)\( p^{42} T^{9} + \)\(24\!\cdots\!42\)\( p^{56} T^{10} + 6994307700378 p^{70} T^{11} + p^{84} T^{12} \)
79 \( 1 - \)\(21\!\cdots\!86\)\( T^{2} + \)\(19\!\cdots\!15\)\( T^{4} - \)\(94\!\cdots\!20\)\( T^{6} + \)\(19\!\cdots\!15\)\( p^{28} T^{8} - \)\(21\!\cdots\!86\)\( p^{56} T^{10} + p^{84} T^{12} \)
83 \( 1 + 60521791593048 T + \)\(18\!\cdots\!52\)\( T^{2} + \)\(40\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!23\)\( T^{4} + \)\(43\!\cdots\!72\)\( T^{5} + \)\(13\!\cdots\!48\)\( T^{6} + \)\(43\!\cdots\!72\)\( p^{14} T^{7} + \)\(11\!\cdots\!23\)\( p^{28} T^{8} + \)\(40\!\cdots\!76\)\( p^{42} T^{9} + \)\(18\!\cdots\!52\)\( p^{56} T^{10} + 60521791593048 p^{70} T^{11} + p^{84} T^{12} \)
89 \( 1 - \)\(10\!\cdots\!46\)\( T^{2} + \)\(45\!\cdots\!15\)\( T^{4} - \)\(11\!\cdots\!20\)\( T^{6} + \)\(45\!\cdots\!15\)\( p^{28} T^{8} - \)\(10\!\cdots\!46\)\( p^{56} T^{10} + p^{84} T^{12} \)
97 \( 1 + 307307370113562 T + \)\(47\!\cdots\!22\)\( T^{2} + \)\(46\!\cdots\!34\)\( T^{3} + \)\(29\!\cdots\!83\)\( T^{4} + \)\(11\!\cdots\!28\)\( T^{5} + \)\(44\!\cdots\!88\)\( T^{6} + \)\(11\!\cdots\!28\)\( p^{14} T^{7} + \)\(29\!\cdots\!83\)\( p^{28} T^{8} + \)\(46\!\cdots\!34\)\( p^{42} T^{9} + \)\(47\!\cdots\!22\)\( p^{56} T^{10} + 307307370113562 p^{70} T^{11} + p^{84} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.940683592608203308953280654916, −8.394487767579140120516758800473, −8.215798201808670735207438097380, −8.101424552332672172604745377593, −8.046203289802761964478675780374, −7.69271094052507720444879161973, −7.18134936099970527170259826125, −6.85402815199494331227929449361, −6.59106126734779167303545087104, −6.39519610993865067928942176765, −5.39597994778234086553115408257, −5.29408584535899818843583703261, −5.04457446735568551162274536855, −4.77120784739518636035285519998, −4.06651761486263025638619788656, −3.57568543225484773816208414635, −3.09095388557392059260902943383, −2.49849083420840402233099454941, −2.49432145868979562775025812050, −2.03163485314213588340663230836, −1.57356108671588668059683306035, −1.56967147870654896428590433510, −1.31002183821587873281820054010, −0.087014868131661656380547411213, −0.07033703289042689368084490191, 0.07033703289042689368084490191, 0.087014868131661656380547411213, 1.31002183821587873281820054010, 1.56967147870654896428590433510, 1.57356108671588668059683306035, 2.03163485314213588340663230836, 2.49432145868979562775025812050, 2.49849083420840402233099454941, 3.09095388557392059260902943383, 3.57568543225484773816208414635, 4.06651761486263025638619788656, 4.77120784739518636035285519998, 5.04457446735568551162274536855, 5.29408584535899818843583703261, 5.39597994778234086553115408257, 6.39519610993865067928942176765, 6.59106126734779167303545087104, 6.85402815199494331227929449361, 7.18134936099970527170259826125, 7.69271094052507720444879161973, 8.046203289802761964478675780374, 8.101424552332672172604745377593, 8.215798201808670735207438097380, 8.394487767579140120516758800473, 8.940683592608203308953280654916

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

Plot not available for L-functions of degree greater than 10.