Properties

Degree 12
Conductor $ 2^{6} \cdot 3^{6} $
Sign $1$
Motivic weight 18
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.25e3·3-s − 3.93e5·4-s + 2.82e7·7-s + 3.67e8·9-s + 2.46e9·12-s + 2.95e10·13-s + 1.03e11·16-s − 4.38e11·19-s − 1.76e11·21-s − 1.40e12·25-s + 1.51e12·27-s − 1.11e13·28-s + 2.17e13·31-s − 1.44e14·36-s + 6.38e14·37-s − 1.85e14·39-s − 1.68e15·43-s − 6.45e14·48-s − 4.93e15·49-s − 1.16e16·52-s + 2.74e15·57-s − 1.62e16·61-s + 1.03e16·63-s − 2.25e16·64-s + 1.11e16·67-s − 7.89e16·73-s + 8.78e15·75-s + ⋯
L(s)  = 1  − 0.317·3-s − 3/2·4-s + 0.699·7-s + 0.948·9-s + 0.476·12-s + 2.78·13-s + 3/2·16-s − 1.35·19-s − 0.222·21-s − 0.368·25-s + 0.198·27-s − 1.04·28-s + 0.823·31-s − 1.42·36-s + 4.91·37-s − 0.886·39-s − 3.35·43-s − 0.476·48-s − 3.03·49-s − 4.18·52-s + 0.432·57-s − 1.39·61-s + 0.663·63-s − 5/4·64-s + 0.409·67-s − 1.34·73-s + 0.117·75-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+9)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(46656\)    =    \(2^{6} \cdot 3^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(18\)
character  :  induced by $\chi_{6} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(12,\ 46656,\ (\ :[9]^{6}),\ 1)$
$L(\frac{19}{2})$  $\approx$  $0.000449258$
$L(\frac12)$  $\approx$  $0.000449258$
$L(10)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \(F_p\) is a polynomial of degree 12. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 11.
$p$$F_p$
bad2 \( ( 1 + p^{17} T^{2} )^{3} \)
3 \( 1 + 2086 p T - 1351459 p^{5} T^{2} - 3681356 p^{13} T^{3} - 1351459 p^{23} T^{4} + 2086 p^{37} T^{5} + p^{54} T^{6} \)
good5 \( 1 + 280786512354 p T^{2} + \)\(32\!\cdots\!59\)\( p^{2} T^{4} - \)\(14\!\cdots\!12\)\( p^{8} T^{6} + \)\(32\!\cdots\!59\)\( p^{38} T^{8} + 280786512354 p^{73} T^{10} + p^{108} T^{12} \)
7 \( ( 1 - 14116902 T + 395172261716649 p T^{2} + 10282831385308224244 p^{3} T^{3} + 395172261716649 p^{19} T^{4} - 14116902 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
11 \( 1 - 17996209479583479510 T^{2} + \)\(11\!\cdots\!23\)\( p^{2} T^{4} - \)\(49\!\cdots\!20\)\( p^{4} T^{6} + \)\(11\!\cdots\!23\)\( p^{38} T^{8} - 17996209479583479510 p^{72} T^{10} + p^{108} T^{12} \)
13 \( ( 1 - 14783098110 T + 24957586200982956099 p T^{2} - \)\(16\!\cdots\!20\)\( p^{2} T^{3} + 24957586200982956099 p^{19} T^{4} - 14783098110 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
17 \( 1 + 33818999432808116730 p^{2} T^{2} + \)\(66\!\cdots\!63\)\( p^{4} T^{4} + \)\(14\!\cdots\!60\)\( p^{6} T^{6} + \)\(66\!\cdots\!63\)\( p^{40} T^{8} + 33818999432808116730 p^{74} T^{10} + p^{108} T^{12} \)
19 \( ( 1 + 219407023506 T + \)\(24\!\cdots\!15\)\( T^{2} + \)\(33\!\cdots\!60\)\( T^{3} + \)\(24\!\cdots\!15\)\( p^{18} T^{4} + 219407023506 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
23 \( 1 - \)\(10\!\cdots\!50\)\( T^{2} + \)\(61\!\cdots\!63\)\( T^{4} - \)\(23\!\cdots\!00\)\( T^{6} + \)\(61\!\cdots\!63\)\( p^{36} T^{8} - \)\(10\!\cdots\!50\)\( p^{72} T^{10} + p^{108} T^{12} \)
29 \( 1 - \)\(94\!\cdots\!86\)\( T^{2} + \)\(42\!\cdots\!95\)\( T^{4} - \)\(11\!\cdots\!40\)\( T^{6} + \)\(42\!\cdots\!95\)\( p^{36} T^{8} - \)\(94\!\cdots\!86\)\( p^{72} T^{10} + p^{108} T^{12} \)
31 \( ( 1 - 10887907463574 T + \)\(15\!\cdots\!15\)\( T^{2} - \)\(12\!\cdots\!80\)\( T^{3} + \)\(15\!\cdots\!15\)\( p^{18} T^{4} - 10887907463574 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
37 \( ( 1 - 319223282408718 T + \)\(69\!\cdots\!43\)\( T^{2} - \)\(11\!\cdots\!32\)\( T^{3} + \)\(69\!\cdots\!43\)\( p^{18} T^{4} - 319223282408718 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
41 \( 1 - \)\(24\!\cdots\!70\)\( T^{2} + \)\(50\!\cdots\!43\)\( T^{4} - \)\(59\!\cdots\!40\)\( T^{6} + \)\(50\!\cdots\!43\)\( p^{36} T^{8} - \)\(24\!\cdots\!70\)\( p^{72} T^{10} + p^{108} T^{12} \)
43 \( ( 1 + 844156859441826 T + \)\(94\!\cdots\!71\)\( T^{2} + \)\(43\!\cdots\!12\)\( T^{3} + \)\(94\!\cdots\!71\)\( p^{18} T^{4} + 844156859441826 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
47 \( 1 - \)\(39\!\cdots\!22\)\( p T^{2} + \)\(24\!\cdots\!15\)\( T^{4} - \)\(37\!\cdots\!80\)\( T^{6} + \)\(24\!\cdots\!15\)\( p^{36} T^{8} - \)\(39\!\cdots\!22\)\( p^{73} T^{10} + p^{108} T^{12} \)
53 \( 1 - \)\(38\!\cdots\!30\)\( T^{2} + \)\(67\!\cdots\!43\)\( T^{4} - \)\(81\!\cdots\!60\)\( T^{6} + \)\(67\!\cdots\!43\)\( p^{36} T^{8} - \)\(38\!\cdots\!30\)\( p^{72} T^{10} + p^{108} T^{12} \)
59 \( 1 - \)\(40\!\cdots\!70\)\( T^{2} + \)\(71\!\cdots\!03\)\( T^{4} - \)\(69\!\cdots\!40\)\( T^{6} + \)\(71\!\cdots\!03\)\( p^{36} T^{8} - \)\(40\!\cdots\!70\)\( p^{72} T^{10} + p^{108} T^{12} \)
61 \( ( 1 + 8139798638850018 T + \)\(35\!\cdots\!71\)\( T^{2} + \)\(22\!\cdots\!52\)\( T^{3} + \)\(35\!\cdots\!71\)\( p^{18} T^{4} + 8139798638850018 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
67 \( ( 1 - 5576862157306638 T + \)\(27\!\cdots\!23\)\( T^{2} + \)\(20\!\cdots\!48\)\( T^{3} + \)\(27\!\cdots\!23\)\( p^{18} T^{4} - 5576862157306638 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
71 \( 1 - \)\(34\!\cdots\!86\)\( T^{2} + \)\(14\!\cdots\!95\)\( T^{4} - \)\(30\!\cdots\!40\)\( T^{6} + \)\(14\!\cdots\!95\)\( p^{36} T^{8} - \)\(34\!\cdots\!86\)\( p^{72} T^{10} + p^{108} T^{12} \)
73 \( ( 1 + 39455121673890570 T + \)\(49\!\cdots\!27\)\( T^{2} + \)\(22\!\cdots\!60\)\( T^{3} + \)\(49\!\cdots\!27\)\( p^{18} T^{4} + 39455121673890570 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
79 \( ( 1 + 238259488214463114 T + \)\(52\!\cdots\!95\)\( T^{2} + \)\(66\!\cdots\!20\)\( T^{3} + \)\(52\!\cdots\!95\)\( p^{18} T^{4} + 238259488214463114 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
83 \( 1 - \)\(14\!\cdots\!58\)\( T^{2} + \)\(10\!\cdots\!51\)\( T^{4} - \)\(43\!\cdots\!72\)\( T^{6} + \)\(10\!\cdots\!51\)\( p^{36} T^{8} - \)\(14\!\cdots\!58\)\( p^{72} T^{10} + p^{108} T^{12} \)
89 \( 1 - \)\(36\!\cdots\!50\)\( T^{2} + \)\(54\!\cdots\!63\)\( T^{4} - \)\(59\!\cdots\!00\)\( T^{6} + \)\(54\!\cdots\!63\)\( p^{36} T^{8} - \)\(36\!\cdots\!50\)\( p^{72} T^{10} + p^{108} T^{12} \)
97 \( ( 1 - 417347708621655174 T + \)\(13\!\cdots\!91\)\( T^{2} - \)\(53\!\cdots\!88\)\( T^{3} + \)\(13\!\cdots\!91\)\( p^{18} T^{4} - 417347708621655174 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.355958781691056624680816120701, −8.808047610760058026605869492421, −8.308936127550290206990373150178, −8.256047643991086991922305836148, −8.098142462528363529437747176132, −7.929462688212163478526248611833, −6.95601463872881762348848010645, −6.92455988168256863984314958708, −6.21812247830934507349338685393, −6.04050552282655351767962832158, −6.00557962546592460698305099099, −5.35665333584062995755861993809, −4.78617867487877891476828947179, −4.44192665180808016114122343240, −4.33019921875572607085754437859, −4.24984511827496006736782550584, −3.51252611718057511956965275021, −3.17749461110040617873476189407, −2.91397138753411590374262218766, −2.06752858939011897345490473136, −1.60603000496686569773600133234, −1.36447798711501239219946032905, −1.06296280784780493999662929962, −0.73820954359297864085451684902, −0.00306737089505830738251974423, 0.00306737089505830738251974423, 0.73820954359297864085451684902, 1.06296280784780493999662929962, 1.36447798711501239219946032905, 1.60603000496686569773600133234, 2.06752858939011897345490473136, 2.91397138753411590374262218766, 3.17749461110040617873476189407, 3.51252611718057511956965275021, 4.24984511827496006736782550584, 4.33019921875572607085754437859, 4.44192665180808016114122343240, 4.78617867487877891476828947179, 5.35665333584062995755861993809, 6.00557962546592460698305099099, 6.04050552282655351767962832158, 6.21812247830934507349338685393, 6.92455988168256863984314958708, 6.95601463872881762348848010645, 7.929462688212163478526248611833, 8.098142462528363529437747176132, 8.256047643991086991922305836148, 8.308936127550290206990373150178, 8.808047610760058026605869492421, 9.355958781691056624680816120701

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

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