Properties

Label 12-1-1.1-c75e6-0-0
Degree $12$
Conductor $1$
Sign $1$
Analytic cond. $2.04348\times 10^{9}$
Root an. cond. $5.96848$
Motivic weight $75$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.70e10·2-s − 7.85e17·3-s − 2.54e22·4-s − 3.89e25·5-s + 4.48e28·6-s + 1.92e30·7-s + 3.70e33·8-s − 4.56e35·9-s + 2.22e36·10-s − 9.45e38·11-s + 1.99e40·12-s + 5.33e41·13-s − 1.09e41·14-s + 3.06e43·15-s − 7.63e44·16-s + 1.82e46·17-s + 2.60e46·18-s + 1.06e48·19-s + 9.90e47·20-s − 1.51e48·21-s + 5.39e49·22-s + 1.51e51·23-s − 2.91e51·24-s − 6.89e52·25-s − 3.04e52·26-s + 3.21e53·27-s − 4.88e52·28-s + ⋯
L(s)  = 1  − 0.293·2-s − 1.00·3-s − 0.672·4-s − 0.239·5-s + 0.295·6-s + 0.0391·7-s + 0.505·8-s − 0.751·9-s + 0.0703·10-s − 0.838·11-s + 0.676·12-s + 0.899·13-s − 0.0115·14-s + 0.241·15-s − 0.534·16-s + 1.31·17-s + 0.220·18-s + 1.18·19-s + 0.161·20-s − 0.0394·21-s + 0.246·22-s + 1.30·23-s − 0.508·24-s − 2.60·25-s − 0.264·26-s + 0.676·27-s − 0.0263·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(76-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+75/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(2.04348\times 10^{9}\)
Root analytic conductor: \(5.96848\)
Motivic weight: \(75\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 1,\ (\ :[75/2]^{6}),\ 1)\)

Particular Values

\(L(38)\) \(\approx\) \(0.2129286002\)
\(L(\frac12)\) \(\approx\) \(0.2129286002\)
\(L(\frac{77}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 7135102755 p^{3} T + 13996910454673606305 p^{11} T^{2} - \)\(14\!\cdots\!45\)\( p^{22} T^{3} + \)\(45\!\cdots\!13\)\( p^{38} T^{4} - \)\(65\!\cdots\!45\)\( p^{58} T^{5} + \)\(23\!\cdots\!45\)\( p^{80} T^{6} - \)\(65\!\cdots\!45\)\( p^{133} T^{7} + \)\(45\!\cdots\!13\)\( p^{188} T^{8} - \)\(14\!\cdots\!45\)\( p^{247} T^{9} + 13996910454673606305 p^{311} T^{10} + 7135102755 p^{378} T^{11} + p^{450} T^{12} \)
3 \( 1 + 29077494956248520 p^{3} T + \)\(18\!\cdots\!70\)\( p^{10} T^{2} + \)\(25\!\cdots\!80\)\( p^{20} T^{3} + \)\(46\!\cdots\!03\)\( p^{30} T^{4} + \)\(28\!\cdots\!60\)\( p^{43} T^{5} + \)\(53\!\cdots\!80\)\( p^{61} T^{6} + \)\(28\!\cdots\!60\)\( p^{118} T^{7} + \)\(46\!\cdots\!03\)\( p^{180} T^{8} + \)\(25\!\cdots\!80\)\( p^{245} T^{9} + \)\(18\!\cdots\!70\)\( p^{310} T^{10} + 29077494956248520 p^{378} T^{11} + p^{450} T^{12} \)
5 \( 1 + \)\(77\!\cdots\!88\)\( p T + \)\(22\!\cdots\!94\)\( p^{5} T^{2} + \)\(11\!\cdots\!64\)\( p^{10} T^{3} + \)\(37\!\cdots\!47\)\( p^{17} T^{4} + \)\(41\!\cdots\!48\)\( p^{26} T^{5} + \)\(61\!\cdots\!44\)\( p^{36} T^{6} + \)\(41\!\cdots\!48\)\( p^{101} T^{7} + \)\(37\!\cdots\!47\)\( p^{167} T^{8} + \)\(11\!\cdots\!64\)\( p^{235} T^{9} + \)\(22\!\cdots\!94\)\( p^{305} T^{10} + \)\(77\!\cdots\!88\)\( p^{376} T^{11} + p^{450} T^{12} \)
7 \( 1 - \)\(27\!\cdots\!00\)\( p T + \)\(42\!\cdots\!50\)\( p^{4} T^{2} + \)\(16\!\cdots\!00\)\( p^{9} T^{3} + \)\(10\!\cdots\!29\)\( p^{15} T^{4} + \)\(11\!\cdots\!00\)\( p^{22} T^{5} + \)\(89\!\cdots\!00\)\( p^{31} T^{6} + \)\(11\!\cdots\!00\)\( p^{97} T^{7} + \)\(10\!\cdots\!29\)\( p^{165} T^{8} + \)\(16\!\cdots\!00\)\( p^{234} T^{9} + \)\(42\!\cdots\!50\)\( p^{304} T^{10} - \)\(27\!\cdots\!00\)\( p^{376} T^{11} + p^{450} T^{12} \)
11 \( 1 + \)\(85\!\cdots\!08\)\( p T + \)\(44\!\cdots\!86\)\( p^{3} T^{2} + \)\(29\!\cdots\!80\)\( p^{5} T^{3} + \)\(75\!\cdots\!95\)\( p^{8} T^{4} + \)\(31\!\cdots\!68\)\( p^{13} T^{5} + \)\(42\!\cdots\!64\)\( p^{19} T^{6} + \)\(31\!\cdots\!68\)\( p^{88} T^{7} + \)\(75\!\cdots\!95\)\( p^{158} T^{8} + \)\(29\!\cdots\!80\)\( p^{230} T^{9} + \)\(44\!\cdots\!86\)\( p^{303} T^{10} + \)\(85\!\cdots\!08\)\( p^{376} T^{11} + p^{450} T^{12} \)
13 \( 1 - \)\(41\!\cdots\!40\)\( p T + \)\(26\!\cdots\!70\)\( p^{5} T^{2} - \)\(19\!\cdots\!20\)\( p^{3} T^{3} + \)\(56\!\cdots\!83\)\( p^{6} T^{4} - \)\(10\!\cdots\!40\)\( p^{10} T^{5} + \)\(74\!\cdots\!40\)\( p^{15} T^{6} - \)\(10\!\cdots\!40\)\( p^{85} T^{7} + \)\(56\!\cdots\!83\)\( p^{156} T^{8} - \)\(19\!\cdots\!20\)\( p^{228} T^{9} + \)\(26\!\cdots\!70\)\( p^{305} T^{10} - \)\(41\!\cdots\!40\)\( p^{376} T^{11} + p^{450} T^{12} \)
17 \( 1 - \)\(18\!\cdots\!80\)\( T + \)\(59\!\cdots\!10\)\( p T^{2} - \)\(28\!\cdots\!80\)\( p^{3} T^{3} + \)\(17\!\cdots\!63\)\( p^{6} T^{4} - \)\(40\!\cdots\!20\)\( p^{9} T^{5} + \)\(10\!\cdots\!80\)\( p^{13} T^{6} - \)\(40\!\cdots\!20\)\( p^{84} T^{7} + \)\(17\!\cdots\!63\)\( p^{156} T^{8} - \)\(28\!\cdots\!80\)\( p^{228} T^{9} + \)\(59\!\cdots\!10\)\( p^{301} T^{10} - \)\(18\!\cdots\!80\)\( p^{375} T^{11} + p^{450} T^{12} \)
19 \( 1 - \)\(29\!\cdots\!80\)\( p^{2} T + \)\(67\!\cdots\!54\)\( p^{2} T^{2} - \)\(33\!\cdots\!00\)\( p^{3} T^{3} + \)\(27\!\cdots\!15\)\( p^{4} T^{4} - \)\(31\!\cdots\!00\)\( p^{7} T^{5} + \)\(56\!\cdots\!80\)\( p^{10} T^{6} - \)\(31\!\cdots\!00\)\( p^{82} T^{7} + \)\(27\!\cdots\!15\)\( p^{154} T^{8} - \)\(33\!\cdots\!00\)\( p^{228} T^{9} + \)\(67\!\cdots\!54\)\( p^{302} T^{10} - \)\(29\!\cdots\!80\)\( p^{377} T^{11} + p^{450} T^{12} \)
23 \( 1 - \)\(65\!\cdots\!60\)\( p T + \)\(11\!\cdots\!10\)\( p^{2} T^{2} - \)\(22\!\cdots\!60\)\( p^{4} T^{3} + \)\(11\!\cdots\!23\)\( p^{6} T^{4} - \)\(76\!\cdots\!80\)\( p^{9} T^{5} + \)\(12\!\cdots\!20\)\( p^{12} T^{6} - \)\(76\!\cdots\!80\)\( p^{84} T^{7} + \)\(11\!\cdots\!23\)\( p^{156} T^{8} - \)\(22\!\cdots\!60\)\( p^{229} T^{9} + \)\(11\!\cdots\!10\)\( p^{302} T^{10} - \)\(65\!\cdots\!60\)\( p^{376} T^{11} + p^{450} T^{12} \)
29 \( 1 - \)\(14\!\cdots\!20\)\( T + \)\(10\!\cdots\!86\)\( p T^{2} - \)\(35\!\cdots\!00\)\( p^{2} T^{3} + \)\(14\!\cdots\!35\)\( p^{3} T^{4} - \)\(12\!\cdots\!00\)\( p^{5} T^{5} + \)\(12\!\cdots\!20\)\( p^{7} T^{6} - \)\(12\!\cdots\!00\)\( p^{80} T^{7} + \)\(14\!\cdots\!35\)\( p^{153} T^{8} - \)\(35\!\cdots\!00\)\( p^{227} T^{9} + \)\(10\!\cdots\!86\)\( p^{301} T^{10} - \)\(14\!\cdots\!20\)\( p^{375} T^{11} + p^{450} T^{12} \)
31 \( 1 + \)\(41\!\cdots\!88\)\( T + \)\(51\!\cdots\!86\)\( p T^{2} + \)\(77\!\cdots\!80\)\( p^{2} T^{3} + \)\(17\!\cdots\!95\)\( p^{4} T^{4} + \)\(11\!\cdots\!68\)\( p^{6} T^{5} + \)\(15\!\cdots\!64\)\( p^{8} T^{6} + \)\(11\!\cdots\!68\)\( p^{81} T^{7} + \)\(17\!\cdots\!95\)\( p^{154} T^{8} + \)\(77\!\cdots\!80\)\( p^{227} T^{9} + \)\(51\!\cdots\!86\)\( p^{301} T^{10} + \)\(41\!\cdots\!88\)\( p^{375} T^{11} + p^{450} T^{12} \)
37 \( 1 - \)\(26\!\cdots\!20\)\( p T + \)\(11\!\cdots\!90\)\( p^{2} T^{2} - \)\(28\!\cdots\!40\)\( p^{3} T^{3} + \)\(74\!\cdots\!27\)\( p^{4} T^{4} - \)\(13\!\cdots\!60\)\( p^{5} T^{5} + \)\(29\!\cdots\!20\)\( p^{6} T^{6} - \)\(13\!\cdots\!60\)\( p^{80} T^{7} + \)\(74\!\cdots\!27\)\( p^{154} T^{8} - \)\(28\!\cdots\!40\)\( p^{228} T^{9} + \)\(11\!\cdots\!90\)\( p^{302} T^{10} - \)\(26\!\cdots\!20\)\( p^{376} T^{11} + p^{450} T^{12} \)
41 \( 1 - \)\(12\!\cdots\!32\)\( p T + \)\(29\!\cdots\!66\)\( T^{2} - \)\(66\!\cdots\!20\)\( T^{3} + \)\(59\!\cdots\!95\)\( p T^{4} - \)\(11\!\cdots\!32\)\( p^{2} T^{5} + \)\(18\!\cdots\!44\)\( p^{3} T^{6} - \)\(11\!\cdots\!32\)\( p^{77} T^{7} + \)\(59\!\cdots\!95\)\( p^{151} T^{8} - \)\(66\!\cdots\!20\)\( p^{225} T^{9} + \)\(29\!\cdots\!66\)\( p^{300} T^{10} - \)\(12\!\cdots\!32\)\( p^{376} T^{11} + p^{450} T^{12} \)
43 \( 1 - \)\(27\!\cdots\!00\)\( T + \)\(10\!\cdots\!50\)\( T^{2} - \)\(40\!\cdots\!00\)\( p T^{3} + \)\(28\!\cdots\!03\)\( p^{2} T^{4} - \)\(10\!\cdots\!00\)\( p^{3} T^{5} + \)\(60\!\cdots\!00\)\( p^{4} T^{6} - \)\(10\!\cdots\!00\)\( p^{78} T^{7} + \)\(28\!\cdots\!03\)\( p^{152} T^{8} - \)\(40\!\cdots\!00\)\( p^{226} T^{9} + \)\(10\!\cdots\!50\)\( p^{300} T^{10} - \)\(27\!\cdots\!00\)\( p^{375} T^{11} + p^{450} T^{12} \)
47 \( 1 + \)\(13\!\cdots\!80\)\( T + \)\(16\!\cdots\!30\)\( T^{2} + \)\(30\!\cdots\!20\)\( p T^{3} + \)\(47\!\cdots\!83\)\( p^{2} T^{4} + \)\(62\!\cdots\!80\)\( p^{3} T^{5} + \)\(72\!\cdots\!40\)\( p^{4} T^{6} + \)\(62\!\cdots\!80\)\( p^{78} T^{7} + \)\(47\!\cdots\!83\)\( p^{152} T^{8} + \)\(30\!\cdots\!20\)\( p^{226} T^{9} + \)\(16\!\cdots\!30\)\( p^{300} T^{10} + \)\(13\!\cdots\!80\)\( p^{375} T^{11} + p^{450} T^{12} \)
53 \( 1 - \)\(64\!\cdots\!60\)\( T + \)\(76\!\cdots\!30\)\( T^{2} - \)\(54\!\cdots\!40\)\( p T^{3} + \)\(64\!\cdots\!83\)\( p^{2} T^{4} - \)\(25\!\cdots\!40\)\( p^{3} T^{5} + \)\(34\!\cdots\!40\)\( p^{4} T^{6} - \)\(25\!\cdots\!40\)\( p^{78} T^{7} + \)\(64\!\cdots\!83\)\( p^{152} T^{8} - \)\(54\!\cdots\!40\)\( p^{226} T^{9} + \)\(76\!\cdots\!30\)\( p^{300} T^{10} - \)\(64\!\cdots\!60\)\( p^{375} T^{11} + p^{450} T^{12} \)
59 \( 1 + \)\(24\!\cdots\!60\)\( T + \)\(35\!\cdots\!66\)\( p T^{2} + \)\(17\!\cdots\!00\)\( p^{2} T^{3} + \)\(12\!\cdots\!85\)\( p^{3} T^{4} + \)\(49\!\cdots\!00\)\( p^{4} T^{5} + \)\(51\!\cdots\!80\)\( p^{6} T^{6} + \)\(49\!\cdots\!00\)\( p^{79} T^{7} + \)\(12\!\cdots\!85\)\( p^{153} T^{8} + \)\(17\!\cdots\!00\)\( p^{227} T^{9} + \)\(35\!\cdots\!66\)\( p^{301} T^{10} + \)\(24\!\cdots\!60\)\( p^{375} T^{11} + p^{450} T^{12} \)
61 \( 1 + \)\(25\!\cdots\!88\)\( T + \)\(90\!\cdots\!06\)\( p T^{2} + \)\(22\!\cdots\!80\)\( p^{2} T^{3} + \)\(50\!\cdots\!95\)\( p^{3} T^{4} + \)\(90\!\cdots\!88\)\( p^{4} T^{5} + \)\(14\!\cdots\!24\)\( p^{5} T^{6} + \)\(90\!\cdots\!88\)\( p^{79} T^{7} + \)\(50\!\cdots\!95\)\( p^{153} T^{8} + \)\(22\!\cdots\!80\)\( p^{227} T^{9} + \)\(90\!\cdots\!06\)\( p^{301} T^{10} + \)\(25\!\cdots\!88\)\( p^{375} T^{11} + p^{450} T^{12} \)
67 \( 1 - \)\(14\!\cdots\!40\)\( p T + \)\(16\!\cdots\!30\)\( p^{2} T^{2} - \)\(12\!\cdots\!80\)\( p^{3} T^{3} + \)\(83\!\cdots\!07\)\( p^{4} T^{4} - \)\(44\!\cdots\!20\)\( p^{5} T^{5} + \)\(22\!\cdots\!40\)\( p^{6} T^{6} - \)\(44\!\cdots\!20\)\( p^{80} T^{7} + \)\(83\!\cdots\!07\)\( p^{154} T^{8} - \)\(12\!\cdots\!80\)\( p^{228} T^{9} + \)\(16\!\cdots\!30\)\( p^{302} T^{10} - \)\(14\!\cdots\!40\)\( p^{376} T^{11} + p^{450} T^{12} \)
71 \( 1 + \)\(36\!\cdots\!28\)\( p T + \)\(47\!\cdots\!26\)\( p^{2} T^{2} + \)\(44\!\cdots\!80\)\( p^{3} T^{3} + \)\(12\!\cdots\!95\)\( p^{4} T^{4} + \)\(10\!\cdots\!08\)\( p^{5} T^{5} + \)\(21\!\cdots\!44\)\( p^{6} T^{6} + \)\(10\!\cdots\!08\)\( p^{80} T^{7} + \)\(12\!\cdots\!95\)\( p^{154} T^{8} + \)\(44\!\cdots\!80\)\( p^{228} T^{9} + \)\(47\!\cdots\!26\)\( p^{302} T^{10} + \)\(36\!\cdots\!28\)\( p^{376} T^{11} + p^{450} T^{12} \)
73 \( 1 + \)\(41\!\cdots\!40\)\( p T + \)\(11\!\cdots\!10\)\( p^{2} T^{2} + \)\(20\!\cdots\!20\)\( p^{3} T^{3} + \)\(31\!\cdots\!67\)\( p^{4} T^{4} + \)\(40\!\cdots\!20\)\( p^{5} T^{5} + \)\(43\!\cdots\!80\)\( p^{6} T^{6} + \)\(40\!\cdots\!20\)\( p^{80} T^{7} + \)\(31\!\cdots\!67\)\( p^{154} T^{8} + \)\(20\!\cdots\!20\)\( p^{228} T^{9} + \)\(11\!\cdots\!10\)\( p^{302} T^{10} + \)\(41\!\cdots\!40\)\( p^{376} T^{11} + p^{450} T^{12} \)
79 \( 1 - \)\(11\!\cdots\!20\)\( T + \)\(92\!\cdots\!94\)\( T^{2} - \)\(90\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!15\)\( T^{4} - \)\(33\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!80\)\( T^{6} - \)\(33\!\cdots\!00\)\( p^{75} T^{7} + \)\(40\!\cdots\!15\)\( p^{150} T^{8} - \)\(90\!\cdots\!00\)\( p^{225} T^{9} + \)\(92\!\cdots\!94\)\( p^{300} T^{10} - \)\(11\!\cdots\!20\)\( p^{375} T^{11} + p^{450} T^{12} \)
83 \( 1 + \)\(79\!\cdots\!60\)\( T + \)\(37\!\cdots\!70\)\( T^{2} + \)\(25\!\cdots\!20\)\( T^{3} + \)\(64\!\cdots\!47\)\( T^{4} + \)\(36\!\cdots\!80\)\( T^{5} + \)\(67\!\cdots\!60\)\( T^{6} + \)\(36\!\cdots\!80\)\( p^{75} T^{7} + \)\(64\!\cdots\!47\)\( p^{150} T^{8} + \)\(25\!\cdots\!20\)\( p^{225} T^{9} + \)\(37\!\cdots\!70\)\( p^{300} T^{10} + \)\(79\!\cdots\!60\)\( p^{375} T^{11} + p^{450} T^{12} \)
89 \( 1 - \)\(53\!\cdots\!60\)\( T + \)\(17\!\cdots\!94\)\( T^{2} - \)\(40\!\cdots\!00\)\( T^{3} + \)\(76\!\cdots\!15\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!80\)\( T^{6} - \)\(11\!\cdots\!00\)\( p^{75} T^{7} + \)\(76\!\cdots\!15\)\( p^{150} T^{8} - \)\(40\!\cdots\!00\)\( p^{225} T^{9} + \)\(17\!\cdots\!94\)\( p^{300} T^{10} - \)\(53\!\cdots\!60\)\( p^{375} T^{11} + p^{450} T^{12} \)
97 \( 1 + \)\(74\!\cdots\!80\)\( T + \)\(63\!\cdots\!30\)\( T^{2} + \)\(32\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!47\)\( T^{4} + \)\(61\!\cdots\!40\)\( T^{5} + \)\(22\!\cdots\!40\)\( T^{6} + \)\(61\!\cdots\!40\)\( p^{75} T^{7} + \)\(16\!\cdots\!47\)\( p^{150} T^{8} + \)\(32\!\cdots\!40\)\( p^{225} T^{9} + \)\(63\!\cdots\!30\)\( p^{300} T^{10} + \)\(74\!\cdots\!80\)\( p^{375} T^{11} + p^{450} T^{12} \)
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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

−7.18004539728659984538861799288, −6.74678126540167852088474308539, −6.51922180212410194763033250794, −6.16233581584241431351625569540, −5.83359990030070811993798679782, −5.75025107802660670760758764342, −5.58577562851286952894089240118, −5.03437170072694803087955500738, −4.85290185031793539369583572704, −4.57885962660034381012798481313, −4.48881368106090201687271821890, −3.91373573657624779746893925842, −3.72886977494453316318830446873, −3.26020102931758188694564734759, −3.03941673987890677993991111104, −2.91172135971785883778611554615, −2.55408352758006985195857895610, −2.27943924184356855781281266347, −1.53357852812329393726544591288, −1.45137740352345554509119398267, −1.44169842225286580724395584792, −0.874762885394372842178720411755, −0.58393523381531499715284794994, −0.46629182134089608875322244810, −0.07425016818954904161437391674, 0.07425016818954904161437391674, 0.46629182134089608875322244810, 0.58393523381531499715284794994, 0.874762885394372842178720411755, 1.44169842225286580724395584792, 1.45137740352345554509119398267, 1.53357852812329393726544591288, 2.27943924184356855781281266347, 2.55408352758006985195857895610, 2.91172135971785883778611554615, 3.03941673987890677993991111104, 3.26020102931758188694564734759, 3.72886977494453316318830446873, 3.91373573657624779746893925842, 4.48881368106090201687271821890, 4.57885962660034381012798481313, 4.85290185031793539369583572704, 5.03437170072694803087955500738, 5.58577562851286952894089240118, 5.75025107802660670760758764342, 5.83359990030070811993798679782, 6.16233581584241431351625569540, 6.51922180212410194763033250794, 6.74678126540167852088474308539, 7.18004539728659984538861799288

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

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