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Properties

Label 12-90e6-1.1-c11e6-0-0
Degree $12$
Conductor $531441000000$
Sign $1$
Analytic cond. $1.09341\times 10^{11}$
Root an. cond. $8.31570$
Motivic weight $11$
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  − 3.07e3·4-s − 530·5-s + 6.42e5·11-s + 6.29e6·16-s − 2.41e7·19-s + 1.62e6·20-s − 9.07e7·25-s + 2.56e8·29-s + 4.58e8·31-s + 1.64e8·41-s − 1.97e9·44-s + 5.59e9·49-s − 3.40e8·55-s − 1.76e10·59-s − 5.02e9·61-s − 1.07e10·64-s − 5.67e10·71-s + 7.40e10·76-s + 2.60e9·79-s − 3.33e9·80-s − 2.49e11·89-s + 1.27e10·95-s + 2.78e11·100-s + 4.10e11·101-s + 7.02e11·109-s − 7.87e11·116-s − 5.92e11·121-s + ⋯
L(s)  = 1  − 3/2·4-s − 0.0758·5-s + 1.20·11-s + 3/2·16-s − 2.23·19-s + 0.113·20-s − 1.85·25-s + 2.32·29-s + 2.87·31-s + 0.222·41-s − 1.80·44-s + 2.82·49-s − 0.0912·55-s − 3.21·59-s − 0.761·61-s − 5/4·64-s − 3.73·71-s + 3.35·76-s + 0.0951·79-s − 0.113·80-s − 4.73·89-s + 0.169·95-s + 2.78·100-s + 3.88·101-s + 4.37·109-s − 3.48·116-s − 2.07·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{12} \cdot 5^{6}\)
Sign: $1$
Analytic conductor: \(1.09341\times 10^{11}\)
Root analytic conductor: \(8.31570\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{12} \cdot 5^{6} ,\ ( \ : [11/2]^{6} ),\ 1 )\)

Particular Values

\(L(6)\) \(\approx\) \(0.002234218937\)
\(L(\frac12)\) \(\approx\) \(0.002234218937\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p^{10} T^{2} )^{3} \)
3 \( 1 \)
5 \( 1 + 106 p T + 727999 p^{3} T^{2} - 1256292 p^{7} T^{3} + 727999 p^{14} T^{4} + 106 p^{23} T^{5} + p^{33} T^{6} \)
good7 \( 1 - 5594223150 T^{2} + 282174762670457103 p^{2} T^{4} - \)\(10\!\cdots\!00\)\( p^{4} T^{6} + 282174762670457103 p^{24} T^{8} - 5594223150 p^{44} T^{10} + p^{66} T^{12} \)
11 \( ( 1 - 321364 T + 451069377065 T^{2} - 68139027321485880 T^{3} + 451069377065 p^{11} T^{4} - 321364 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
13 \( 1 - 577710785670 p T^{2} + \)\(27\!\cdots\!07\)\( T^{4} - \)\(60\!\cdots\!80\)\( T^{6} + \)\(27\!\cdots\!07\)\( p^{22} T^{8} - 577710785670 p^{45} T^{10} + p^{66} T^{12} \)
17 \( 1 - 132432185976870 T^{2} + \)\(83\!\cdots\!67\)\( T^{4} - \)\(33\!\cdots\!60\)\( T^{6} + \)\(83\!\cdots\!67\)\( p^{22} T^{8} - 132432185976870 p^{44} T^{10} + p^{66} T^{12} \)
19 \( ( 1 + 12054540 T + 30008106286257 T^{2} - \)\(13\!\cdots\!80\)\( T^{3} + 30008106286257 p^{11} T^{4} + 12054540 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
23 \( 1 - 4235045141210190 T^{2} + \)\(36\!\cdots\!69\)\( p T^{4} - \)\(10\!\cdots\!20\)\( T^{6} + \)\(36\!\cdots\!69\)\( p^{23} T^{8} - 4235045141210190 p^{44} T^{10} + p^{66} T^{12} \)
29 \( ( 1 - 128204910 T + 22822131113381787 T^{2} - \)\(33\!\cdots\!80\)\( T^{3} + 22822131113381787 p^{11} T^{4} - 128204910 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
31 \( ( 1 - 229240896 T + 64386954291598365 T^{2} - \)\(90\!\cdots\!20\)\( T^{3} + 64386954291598365 p^{11} T^{4} - 229240896 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
37 \( 1 - 345211813058554110 T^{2} + \)\(91\!\cdots\!07\)\( T^{4} - \)\(18\!\cdots\!80\)\( T^{6} + \)\(91\!\cdots\!07\)\( p^{22} T^{8} - 345211813058554110 p^{44} T^{10} + p^{66} T^{12} \)
41 \( ( 1 - 82384474 T + 1346507617380213815 T^{2} - \)\(14\!\cdots\!80\)\( T^{3} + 1346507617380213815 p^{11} T^{4} - 82384474 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
43 \( 1 - 2824235125166049750 T^{2} + \)\(42\!\cdots\!47\)\( T^{4} - \)\(44\!\cdots\!00\)\( T^{6} + \)\(42\!\cdots\!47\)\( p^{22} T^{8} - 2824235125166049750 p^{44} T^{10} + p^{66} T^{12} \)
47 \( 1 - 4774521242059704030 T^{2} + \)\(20\!\cdots\!27\)\( T^{4} - \)\(47\!\cdots\!40\)\( T^{6} + \)\(20\!\cdots\!27\)\( p^{22} T^{8} - 4774521242059704030 p^{44} T^{10} + p^{66} T^{12} \)
53 \( 1 - 32371227025452806430 T^{2} + \)\(55\!\cdots\!27\)\( T^{4} - \)\(63\!\cdots\!40\)\( T^{6} + \)\(55\!\cdots\!27\)\( p^{22} T^{8} - 32371227025452806430 p^{44} T^{10} + p^{66} T^{12} \)
59 \( ( 1 + 8831981180 T + 96619062073692665177 T^{2} + \)\(50\!\cdots\!40\)\( T^{3} + 96619062073692665177 p^{11} T^{4} + 8831981180 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
61 \( ( 1 + 2510396214 T + \)\(12\!\cdots\!15\)\( T^{2} + \)\(21\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!15\)\( p^{11} T^{4} + 2510396214 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
67 \( 1 - \)\(21\!\cdots\!70\)\( T^{2} + \)\(31\!\cdots\!67\)\( T^{4} - \)\(49\!\cdots\!60\)\( T^{6} + \)\(31\!\cdots\!67\)\( p^{22} T^{8} - \)\(21\!\cdots\!70\)\( p^{44} T^{10} + p^{66} T^{12} \)
71 \( ( 1 + 28394209416 T + \)\(79\!\cdots\!65\)\( T^{2} + \)\(13\!\cdots\!20\)\( T^{3} + \)\(79\!\cdots\!65\)\( p^{11} T^{4} + 28394209416 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
73 \( 1 - \)\(14\!\cdots\!90\)\( T^{2} + \)\(10\!\cdots\!87\)\( T^{4} - \)\(40\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!87\)\( p^{22} T^{8} - \)\(14\!\cdots\!90\)\( p^{44} T^{10} + p^{66} T^{12} \)
79 \( ( 1 - 1301275440 T - \)\(13\!\cdots\!63\)\( T^{2} + \)\(24\!\cdots\!80\)\( T^{3} - \)\(13\!\cdots\!63\)\( p^{11} T^{4} - 1301275440 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
83 \( 1 - \)\(32\!\cdots\!70\)\( T^{2} + \)\(56\!\cdots\!67\)\( T^{4} - \)\(71\!\cdots\!60\)\( T^{6} + \)\(56\!\cdots\!67\)\( p^{22} T^{8} - \)\(32\!\cdots\!70\)\( p^{44} T^{10} + p^{66} T^{12} \)
89 \( ( 1 + 124724206270 T + \)\(86\!\cdots\!67\)\( T^{2} + \)\(49\!\cdots\!60\)\( T^{3} + \)\(86\!\cdots\!67\)\( p^{11} T^{4} + 124724206270 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
97 \( 1 - \)\(18\!\cdots\!30\)\( T^{2} + \)\(17\!\cdots\!27\)\( T^{4} - \)\(12\!\cdots\!40\)\( T^{6} + \)\(17\!\cdots\!27\)\( p^{22} T^{8} - \)\(18\!\cdots\!30\)\( p^{44} T^{10} + p^{66} T^{12} \)
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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

−5.87556113447697916601821059412, −5.35743771022946872012425560597, −5.22937840549314275375068484350, −5.03122504460321889433494359031, −4.57740009242435411928897645595, −4.44781325814573020366841795928, −4.26079090623659775648624184034, −4.24430626936263991067189235473, −4.13266275639568582749202377862, −4.12004518561818957646215864295, −3.29376622841631521943042706125, −3.26900967114692319043395509563, −3.06255668592363662493299044802, −3.01722989907651524256087263202, −2.38327810420513601856358871597, −2.36514707007973775794879101515, −2.22012095383061859640036446526, −1.68470335602232376504839993326, −1.44512073409427520975060799277, −1.32842556457810966832493401699, −1.22201978876642939518905347834, −0.61506349032387871736958840334, −0.59582743890034101072970743803, −0.51405972189965656966632650225, −0.00435174253972824685457881126, 0.00435174253972824685457881126, 0.51405972189965656966632650225, 0.59582743890034101072970743803, 0.61506349032387871736958840334, 1.22201978876642939518905347834, 1.32842556457810966832493401699, 1.44512073409427520975060799277, 1.68470335602232376504839993326, 2.22012095383061859640036446526, 2.36514707007973775794879101515, 2.38327810420513601856358871597, 3.01722989907651524256087263202, 3.06255668592363662493299044802, 3.26900967114692319043395509563, 3.29376622841631521943042706125, 4.12004518561818957646215864295, 4.13266275639568582749202377862, 4.24430626936263991067189235473, 4.26079090623659775648624184034, 4.44781325814573020366841795928, 4.57740009242435411928897645595, 5.03122504460321889433494359031, 5.22937840549314275375068484350, 5.35743771022946872012425560597, 5.87556113447697916601821059412

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

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