Properties

Label 12-3e6-1.1-c22e6-0-0
Degree $12$
Conductor $729$
Sign $1$
Analytic cond. $606838.$
Root an. cond. $3.03335$
Motivic weight $22$
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  + 8.66e4·3-s + 6.95e6·4-s − 3.44e9·7-s + 3.24e10·9-s + 6.02e11·12-s + 2.02e12·13-s + 9.94e12·16-s + 1.00e14·19-s − 2.98e14·21-s + 7.85e15·25-s − 9.83e14·27-s − 2.39e16·28-s + 1.14e17·31-s + 2.25e17·36-s + 6.83e17·37-s + 1.75e17·39-s + 2.73e17·43-s + 8.61e17·48-s − 9.93e18·49-s + 1.40e19·52-s + 8.70e18·57-s + 1.21e20·61-s − 1.11e20·63-s − 2.17e18·64-s + 7.57e19·67-s − 5.71e19·73-s + 6.80e20·75-s + ⋯
L(s)  = 1  + 0.489·3-s + 1.65·4-s − 1.74·7-s + 1.03·9-s + 0.811·12-s + 1.12·13-s + 0.565·16-s + 0.862·19-s − 0.852·21-s + 3.29·25-s − 0.176·27-s − 2.89·28-s + 4.49·31-s + 1.71·36-s + 3.84·37-s + 0.552·39-s + 0.294·43-s + 0.276·48-s − 2.54·49-s + 1.87·52-s + 0.422·57-s + 2.79·61-s − 1.80·63-s − 0.0294·64-s + 0.619·67-s − 0.182·73-s + 1.61·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(23-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+11)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $1$
Analytic conductor: \(606838.\)
Root analytic conductor: \(3.03335\)
Motivic weight: \(22\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 729,\ (\ :[11]^{6}),\ 1)\)

Particular Values

\(L(\frac{23}{2})\) \(\approx\) \(17.65510454\)
\(L(\frac12)\) \(\approx\) \(17.65510454\)
\(L(12)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1070 p^{4} T - 421921 p^{10} T^{2} + 15367100 p^{18} T^{3} - 421921 p^{32} T^{4} - 1070 p^{48} T^{5} + p^{66} T^{6} \)
good2 \( 1 - 217299 p^{5} T^{2} + 18753667383 p^{11} T^{4} - 23334828972403 p^{23} T^{6} + 18753667383 p^{55} T^{8} - 217299 p^{93} T^{10} + p^{132} T^{12} \)
5 \( 1 - 1570168400982654 p T^{2} + \)\(10\!\cdots\!79\)\( p^{5} T^{4} - \)\(99\!\cdots\!32\)\( p^{10} T^{6} + \)\(10\!\cdots\!79\)\( p^{49} T^{8} - 1570168400982654 p^{89} T^{10} + p^{132} T^{12} \)
7 \( ( 1 + 246218790 p T + 27474384614083737 p^{3} T^{2} + \)\(55\!\cdots\!40\)\( p^{4} T^{3} + 27474384614083737 p^{25} T^{4} + 246218790 p^{45} T^{5} + p^{66} T^{6} )^{2} \)
11 \( 1 - \)\(13\!\cdots\!06\)\( p T^{2} + \)\(13\!\cdots\!75\)\( p^{2} T^{4} - \)\(13\!\cdots\!60\)\( p^{3} T^{6} + \)\(13\!\cdots\!75\)\( p^{46} T^{8} - \)\(13\!\cdots\!06\)\( p^{89} T^{10} + p^{132} T^{12} \)
13 \( ( 1 - 1012566248430 T + \)\(44\!\cdots\!47\)\( p T^{2} - \)\(38\!\cdots\!80\)\( p^{2} T^{3} + \)\(44\!\cdots\!47\)\( p^{23} T^{4} - 1012566248430 p^{44} T^{5} + p^{66} T^{6} )^{2} \)
17 \( 1 - \)\(38\!\cdots\!54\)\( p T^{2} + \)\(14\!\cdots\!83\)\( p^{3} T^{4} - \)\(17\!\cdots\!92\)\( p^{5} T^{6} + \)\(14\!\cdots\!83\)\( p^{47} T^{8} - \)\(38\!\cdots\!54\)\( p^{89} T^{10} + p^{132} T^{12} \)
19 \( ( 1 - 2644360228122 p T + \)\(10\!\cdots\!99\)\( p^{2} T^{2} - \)\(18\!\cdots\!56\)\( p^{3} T^{3} + \)\(10\!\cdots\!99\)\( p^{24} T^{4} - 2644360228122 p^{45} T^{5} + p^{66} T^{6} )^{2} \)
23 \( 1 - \)\(91\!\cdots\!18\)\( T^{2} + \)\(15\!\cdots\!59\)\( T^{4} - \)\(71\!\cdots\!24\)\( T^{6} + \)\(15\!\cdots\!59\)\( p^{44} T^{8} - \)\(91\!\cdots\!18\)\( p^{88} T^{10} + p^{132} T^{12} \)
29 \( 1 - \)\(61\!\cdots\!06\)\( T^{2} + \)\(18\!\cdots\!55\)\( T^{4} - \)\(34\!\cdots\!80\)\( T^{6} + \)\(18\!\cdots\!55\)\( p^{44} T^{8} - \)\(61\!\cdots\!06\)\( p^{88} T^{10} + p^{132} T^{12} \)
31 \( ( 1 - 57054831200576646 T + \)\(30\!\cdots\!55\)\( T^{2} - \)\(80\!\cdots\!80\)\( T^{3} + \)\(30\!\cdots\!55\)\( p^{22} T^{4} - 57054831200576646 p^{44} T^{5} + p^{66} T^{6} )^{2} \)
37 \( ( 1 - 341891437040330910 T + \)\(11\!\cdots\!11\)\( T^{2} - \)\(19\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!11\)\( p^{22} T^{4} - 341891437040330910 p^{44} T^{5} + p^{66} T^{6} )^{2} \)
41 \( 1 - \)\(78\!\cdots\!26\)\( T^{2} + \)\(46\!\cdots\!75\)\( T^{4} - \)\(15\!\cdots\!60\)\( T^{6} + \)\(46\!\cdots\!75\)\( p^{44} T^{8} - \)\(78\!\cdots\!26\)\( p^{88} T^{10} + p^{132} T^{12} \)
43 \( ( 1 - 136636665084884670 T + \)\(21\!\cdots\!11\)\( T^{2} - \)\(12\!\cdots\!60\)\( T^{3} + \)\(21\!\cdots\!11\)\( p^{22} T^{4} - 136636665084884670 p^{44} T^{5} + p^{66} T^{6} )^{2} \)
47 \( 1 - \)\(29\!\cdots\!58\)\( T^{2} + \)\(38\!\cdots\!19\)\( T^{4} - \)\(29\!\cdots\!04\)\( T^{6} + \)\(38\!\cdots\!19\)\( p^{44} T^{8} - \)\(29\!\cdots\!58\)\( p^{88} T^{10} + p^{132} T^{12} \)
53 \( 1 - \)\(30\!\cdots\!58\)\( T^{2} + \)\(44\!\cdots\!19\)\( T^{4} - \)\(15\!\cdots\!56\)\( p^{2} T^{6} + \)\(44\!\cdots\!19\)\( p^{44} T^{8} - \)\(30\!\cdots\!58\)\( p^{88} T^{10} + p^{132} T^{12} \)
59 \( 1 - \)\(37\!\cdots\!26\)\( T^{2} + \)\(12\!\cdots\!25\)\( p T^{4} - \)\(80\!\cdots\!60\)\( T^{6} + \)\(12\!\cdots\!25\)\( p^{45} T^{8} - \)\(37\!\cdots\!26\)\( p^{88} T^{10} + p^{132} T^{12} \)
61 \( ( 1 - 60824625194072746446 T + \)\(42\!\cdots\!35\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(42\!\cdots\!35\)\( p^{22} T^{4} - 60824625194072746446 p^{44} T^{5} + p^{66} T^{6} )^{2} \)
67 \( ( 1 - 37850785543087630350 T + \)\(19\!\cdots\!31\)\( T^{2} - \)\(15\!\cdots\!60\)\( T^{3} + \)\(19\!\cdots\!31\)\( p^{22} T^{4} - 37850785543087630350 p^{44} T^{5} + p^{66} T^{6} )^{2} \)
71 \( 1 - \)\(11\!\cdots\!06\)\( T^{2} + \)\(56\!\cdots\!55\)\( T^{4} - \)\(22\!\cdots\!80\)\( T^{6} + \)\(56\!\cdots\!55\)\( p^{44} T^{8} - \)\(11\!\cdots\!06\)\( p^{88} T^{10} + p^{132} T^{12} \)
73 \( ( 1 + 28589240654766993450 T + \)\(16\!\cdots\!31\)\( T^{2} + \)\(13\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!31\)\( p^{22} T^{4} + 28589240654766993450 p^{44} T^{5} + p^{66} T^{6} )^{2} \)
79 \( ( 1 - \)\(19\!\cdots\!98\)\( T + \)\(51\!\cdots\!59\)\( T^{2} - \)\(50\!\cdots\!64\)\( T^{3} + \)\(51\!\cdots\!59\)\( p^{22} T^{4} - \)\(19\!\cdots\!98\)\( p^{44} T^{5} + p^{66} T^{6} )^{2} \)
83 \( 1 - \)\(77\!\cdots\!18\)\( T^{2} + \)\(27\!\cdots\!79\)\( T^{4} - \)\(57\!\cdots\!44\)\( T^{6} + \)\(27\!\cdots\!79\)\( p^{44} T^{8} - \)\(77\!\cdots\!18\)\( p^{88} T^{10} + p^{132} T^{12} \)
89 \( 1 - \)\(40\!\cdots\!66\)\( T^{2} + \)\(70\!\cdots\!75\)\( T^{4} - \)\(70\!\cdots\!60\)\( T^{6} + \)\(70\!\cdots\!75\)\( p^{44} T^{8} - \)\(40\!\cdots\!66\)\( p^{88} T^{10} + p^{132} T^{12} \)
97 \( ( 1 - \)\(78\!\cdots\!70\)\( T + \)\(13\!\cdots\!11\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!11\)\( p^{22} T^{4} - \)\(78\!\cdots\!70\)\( p^{44} T^{5} + p^{66} T^{6} )^{2} \)
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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

−10.19298631718085347215285989361, −9.976103913997127153057445156112, −9.821209243303868121360767550388, −9.539657149998931929046855752156, −8.869254268699354946280466420355, −8.475722430578308538404930475249, −8.168967554166307530684594486665, −7.50144732091555667112179073425, −7.43624434275993137945998008811, −6.54451035472687341667245464386, −6.42387384495468101028146302980, −6.41172389356607934760918331858, −6.31652993605159054143179566966, −5.05559911470025601919041955807, −4.92285971224108343039870424030, −4.26724804491130504538189631432, −3.82232981166134874814831847924, −3.24423073600163050121881869767, −2.82755296976513199969678474019, −2.68081540151200740157366115719, −2.46204756692539834131464786000, −1.60383384740802629757000207669, −1.11857598096344048042108222658, −0.74115108378253705518423164001, −0.69335001857621320114681693846, 0.69335001857621320114681693846, 0.74115108378253705518423164001, 1.11857598096344048042108222658, 1.60383384740802629757000207669, 2.46204756692539834131464786000, 2.68081540151200740157366115719, 2.82755296976513199969678474019, 3.24423073600163050121881869767, 3.82232981166134874814831847924, 4.26724804491130504538189631432, 4.92285971224108343039870424030, 5.05559911470025601919041955807, 6.31652993605159054143179566966, 6.41172389356607934760918331858, 6.42387384495468101028146302980, 6.54451035472687341667245464386, 7.43624434275993137945998008811, 7.50144732091555667112179073425, 8.168967554166307530684594486665, 8.475722430578308538404930475249, 8.869254268699354946280466420355, 9.539657149998931929046855752156, 9.821209243303868121360767550388, 9.976103913997127153057445156112, 10.19298631718085347215285989361

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

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