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Properties

Label 12-12e6-1.1-c18e6-0-0
Degree $12$
Conductor $2985984$
Sign $1$
Analytic cond. $2.24137\times 10^{8}$
Root an. cond. $4.96450$
Motivic weight $18$
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  + 2.39e4·3-s + 1.10e7·7-s + 2.69e8·9-s + 1.77e9·13-s − 2.02e11·19-s + 2.63e11·21-s + 8.46e12·25-s − 8.45e11·27-s − 5.60e13·31-s − 1.67e14·37-s + 4.24e13·39-s + 4.48e14·43-s − 2.91e15·49-s − 4.83e15·57-s + 1.33e16·61-s + 2.96e15·63-s + 3.75e16·67-s + 1.49e17·73-s + 2.02e17·75-s + 6.46e17·79-s − 2.20e17·81-s + 1.95e16·91-s − 1.34e18·93-s + 2.98e18·97-s + 5.93e18·103-s + 4.19e18·109-s − 4.01e18·111-s + ⋯
L(s)  = 1  + 1.21·3-s + 0.273·7-s + 0.695·9-s + 0.167·13-s − 0.626·19-s + 0.332·21-s + 2.21·25-s − 0.110·27-s − 2.12·31-s − 1.28·37-s + 0.203·39-s + 0.892·43-s − 1.79·49-s − 0.761·57-s + 1.14·61-s + 0.189·63-s + 1.38·67-s + 2.53·73-s + 2.69·75-s + 5.39·79-s − 1.46·81-s + 0.0457·91-s − 2.57·93-s + 3.92·97-s + 4.55·103-s + 1.93·109-s − 1.56·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+9)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2985984\)    =    \(2^{12} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(2.24137\times 10^{8}\)
Root analytic conductor: \(4.96450\)
Motivic weight: \(18\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2985984,\ (\ :[9]^{6}),\ 1)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(6.925427210\)
\(L(\frac12)\) \(\approx\) \(6.925427210\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 7978 p T + 1248797 p^{5} T^{2} + 18644 p^{13} T^{3} + 1248797 p^{23} T^{4} - 7978 p^{37} T^{5} + p^{54} T^{6} \)
good5 \( 1 - 1693015849182 p T^{2} + \)\(40\!\cdots\!39\)\( p^{3} T^{4} - \)\(61\!\cdots\!92\)\( p^{8} T^{6} + \)\(40\!\cdots\!39\)\( p^{39} T^{8} - 1693015849182 p^{73} T^{10} + p^{108} T^{12} \)
7 \( ( 1 - 5512182 T + 214854874596969 p T^{2} - \)\(12\!\cdots\!16\)\( p^{3} T^{3} + 214854874596969 p^{19} T^{4} - 5512182 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
11 \( 1 - 1642954602874985346 p T^{2} + \)\(17\!\cdots\!45\)\( p T^{4} - \)\(77\!\cdots\!40\)\( p^{5} T^{6} + \)\(17\!\cdots\!45\)\( p^{37} T^{8} - 1642954602874985346 p^{73} T^{10} + p^{108} T^{12} \)
13 \( ( 1 - 887796798 T + 13505973677807746371 p T^{2} - \)\(23\!\cdots\!48\)\( p^{2} T^{3} + 13505973677807746371 p^{19} T^{4} - 887796798 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
17 \( 1 - \)\(19\!\cdots\!22\)\( p T^{2} + \)\(17\!\cdots\!95\)\( p^{3} T^{4} - \)\(94\!\cdots\!00\)\( p^{5} T^{6} + \)\(17\!\cdots\!95\)\( p^{39} T^{8} - \)\(19\!\cdots\!22\)\( p^{73} T^{10} + p^{108} T^{12} \)
19 \( ( 1 + 101017266402 T + \)\(27\!\cdots\!91\)\( T^{2} + \)\(22\!\cdots\!68\)\( T^{3} + \)\(27\!\cdots\!91\)\( p^{18} T^{4} + 101017266402 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
23 \( 1 - \)\(58\!\cdots\!94\)\( T^{2} + \)\(12\!\cdots\!95\)\( T^{4} - \)\(26\!\cdots\!60\)\( T^{6} + \)\(12\!\cdots\!95\)\( p^{36} T^{8} - \)\(58\!\cdots\!94\)\( p^{72} T^{10} + p^{108} T^{12} \)
29 \( 1 + \)\(25\!\cdots\!54\)\( T^{2} + \)\(60\!\cdots\!35\)\( T^{4} + \)\(66\!\cdots\!00\)\( T^{6} + \)\(60\!\cdots\!35\)\( p^{36} T^{8} + \)\(25\!\cdots\!54\)\( p^{72} T^{10} + p^{108} T^{12} \)
31 \( ( 1 + 28029735971418 T + \)\(64\!\cdots\!31\)\( T^{2} + \)\(16\!\cdots\!92\)\( T^{3} + \)\(64\!\cdots\!31\)\( p^{18} T^{4} + 28029735971418 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
37 \( ( 1 + 83798052757554 T + \)\(37\!\cdots\!91\)\( T^{2} + \)\(18\!\cdots\!48\)\( T^{3} + \)\(37\!\cdots\!91\)\( p^{18} T^{4} + 83798052757554 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
41 \( 1 - \)\(30\!\cdots\!06\)\( T^{2} + \)\(54\!\cdots\!35\)\( T^{4} - \)\(70\!\cdots\!00\)\( T^{6} + \)\(54\!\cdots\!35\)\( p^{36} T^{8} - \)\(30\!\cdots\!06\)\( p^{72} T^{10} + p^{108} T^{12} \)
43 \( ( 1 - 224295963387918 T + \)\(73\!\cdots\!03\)\( T^{2} - \)\(10\!\cdots\!72\)\( T^{3} + \)\(73\!\cdots\!03\)\( p^{18} T^{4} - 224295963387918 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
47 \( 1 - \)\(43\!\cdots\!14\)\( T^{2} + \)\(83\!\cdots\!95\)\( T^{4} - \)\(11\!\cdots\!60\)\( T^{6} + \)\(83\!\cdots\!95\)\( p^{36} T^{8} - \)\(43\!\cdots\!14\)\( p^{72} T^{10} + p^{108} T^{12} \)
53 \( 1 - \)\(27\!\cdots\!14\)\( T^{2} + \)\(36\!\cdots\!95\)\( T^{4} - \)\(40\!\cdots\!60\)\( T^{6} + \)\(36\!\cdots\!95\)\( p^{36} T^{8} - \)\(27\!\cdots\!14\)\( p^{72} T^{10} + p^{108} T^{12} \)
59 \( 1 - \)\(30\!\cdots\!46\)\( T^{2} + \)\(47\!\cdots\!95\)\( T^{4} - \)\(44\!\cdots\!40\)\( T^{6} + \)\(47\!\cdots\!95\)\( p^{36} T^{8} - \)\(30\!\cdots\!46\)\( p^{72} T^{10} + p^{108} T^{12} \)
61 \( ( 1 - 6675236098341342 T + \)\(30\!\cdots\!31\)\( T^{2} - \)\(10\!\cdots\!48\)\( T^{3} + \)\(30\!\cdots\!31\)\( p^{18} T^{4} - 6675236098341342 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
67 \( ( 1 - 18780225153818622 T + \)\(16\!\cdots\!63\)\( T^{2} - \)\(25\!\cdots\!88\)\( T^{3} + \)\(16\!\cdots\!63\)\( p^{18} T^{4} - 18780225153818622 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
71 \( 1 - \)\(10\!\cdots\!86\)\( T^{2} + \)\(48\!\cdots\!95\)\( T^{4} - \)\(12\!\cdots\!40\)\( T^{6} + \)\(48\!\cdots\!95\)\( p^{36} T^{8} - \)\(10\!\cdots\!86\)\( p^{72} T^{10} + p^{108} T^{12} \)
73 \( ( 1 - 74709087087837174 T + \)\(98\!\cdots\!71\)\( T^{2} - \)\(46\!\cdots\!48\)\( T^{3} + \)\(98\!\cdots\!71\)\( p^{18} T^{4} - 74709087087837174 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
79 \( ( 1 - 323261066809369734 T + \)\(71\!\cdots\!35\)\( T^{2} - \)\(97\!\cdots\!00\)\( T^{3} + \)\(71\!\cdots\!35\)\( p^{18} T^{4} - 323261066809369734 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
83 \( 1 - \)\(58\!\cdots\!74\)\( T^{2} + \)\(39\!\cdots\!35\)\( T^{4} - \)\(13\!\cdots\!00\)\( T^{6} + \)\(39\!\cdots\!35\)\( p^{36} T^{8} - \)\(58\!\cdots\!74\)\( p^{72} T^{10} + p^{108} T^{12} \)
89 \( 1 - \)\(64\!\cdots\!66\)\( T^{2} + \)\(18\!\cdots\!35\)\( T^{4} - \)\(28\!\cdots\!00\)\( T^{6} + \)\(18\!\cdots\!35\)\( p^{36} T^{8} - \)\(64\!\cdots\!66\)\( p^{72} T^{10} + p^{108} T^{12} \)
97 \( ( 1 - 1492299152191129542 T + \)\(65\!\cdots\!03\)\( T^{2} - \)\(11\!\cdots\!88\)\( T^{3} + \)\(65\!\cdots\!03\)\( p^{18} T^{4} - 1492299152191129542 p^{36} T^{5} + p^{54} T^{6} )^{2} \)
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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

−7.52392889060128304579637412261, −7.32416454799237665569887838524, −7.30250505996233745821602521984, −6.44470376444093978883204694362, −6.43173920076587753336401914773, −6.36245932732027335253397327024, −5.96478996294218134630711538366, −5.14898263973615923900285929294, −5.10961040342557388780990812895, −5.03494589168739918179038446549, −4.79393443911367223292649875916, −4.20523876182244769939483520346, −3.77634515330044774595719005034, −3.53681259507437808425794133072, −3.41459318496762096937772569327, −3.33319274781805692480228860971, −2.60848452003615410805467559518, −2.43938439935853465915442663729, −1.96914281548185552579192369406, −1.90792694921063900882412163401, −1.82753913265100572938166269103, −0.915192874751371619902568423125, −0.907543810677703936488021819019, −0.70097503350911959696629463528, −0.20443181432220343312469528473, 0.20443181432220343312469528473, 0.70097503350911959696629463528, 0.907543810677703936488021819019, 0.915192874751371619902568423125, 1.82753913265100572938166269103, 1.90792694921063900882412163401, 1.96914281548185552579192369406, 2.43938439935853465915442663729, 2.60848452003615410805467559518, 3.33319274781805692480228860971, 3.41459318496762096937772569327, 3.53681259507437808425794133072, 3.77634515330044774595719005034, 4.20523876182244769939483520346, 4.79393443911367223292649875916, 5.03494589168739918179038446549, 5.10961040342557388780990812895, 5.14898263973615923900285929294, 5.96478996294218134630711538366, 6.36245932732027335253397327024, 6.43173920076587753336401914773, 6.44470376444093978883204694362, 7.30250505996233745821602521984, 7.32416454799237665569887838524, 7.52392889060128304579637412261

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

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