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Properties

Label 12-7600e6-1.1-c1e6-0-7
Degree $12$
Conductor $1.927\times 10^{23}$
Sign $1$
Analytic cond. $4.99510\times 10^{10}$
Root an. cond. $7.79014$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·7-s − 4·9-s − 3·11-s − 13-s − 14·17-s + 6·19-s + 4·21-s − 12·23-s + 10·27-s + 9·29-s − 5·31-s + 6·33-s − 8·37-s + 2·39-s + 3·41-s − 15·43-s − 4·47-s − 8·49-s + 28·51-s + 13·53-s − 12·57-s + 9·61-s + 8·63-s + 3·67-s + 24·69-s − 19·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.755·7-s − 4/3·9-s − 0.904·11-s − 0.277·13-s − 3.39·17-s + 1.37·19-s + 0.872·21-s − 2.50·23-s + 1.92·27-s + 1.67·29-s − 0.898·31-s + 1.04·33-s − 1.31·37-s + 0.320·39-s + 0.468·41-s − 2.28·43-s − 0.583·47-s − 8/7·49-s + 3.92·51-s + 1.78·53-s − 1.58·57-s + 1.15·61-s + 1.00·63-s + 0.366·67-s + 2.88·69-s − 2.25·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 5^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(4.99510\times 10^{10}\)
Root analytic conductor: \(7.79014\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{24} \cdot 5^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( ( 1 - T )^{6} \)
good3 \( 1 + 2 T + 8 T^{2} + 14 T^{3} + 10 p T^{4} + 50 T^{5} + 29 p T^{6} + 50 p T^{7} + 10 p^{3} T^{8} + 14 p^{3} T^{9} + 8 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 2 T + 12 T^{2} + 22 T^{3} + 118 T^{4} + 18 p T^{5} + 779 T^{6} + 18 p^{2} T^{7} + 118 p^{2} T^{8} + 22 p^{3} T^{9} + 12 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 3 T + 26 T^{2} + 26 T^{3} + 65 T^{4} - 749 T^{5} - 208 p T^{6} - 749 p T^{7} + 65 p^{2} T^{8} + 26 p^{3} T^{9} + 26 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + T + p T^{2} + 50 T^{3} + 38 T^{4} - 516 T^{5} + 1813 T^{6} - 516 p T^{7} + 38 p^{2} T^{8} + 50 p^{3} T^{9} + p^{5} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 14 T + 124 T^{2} + 848 T^{3} + 4842 T^{4} + 24018 T^{5} + 105929 T^{6} + 24018 p T^{7} + 4842 p^{2} T^{8} + 848 p^{3} T^{9} + 124 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 12 T + 112 T^{2} + 778 T^{3} + 4884 T^{4} + 25856 T^{5} + 128015 T^{6} + 25856 p T^{7} + 4884 p^{2} T^{8} + 778 p^{3} T^{9} + 112 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 9 T + 107 T^{2} - 600 T^{3} + 5158 T^{4} - 26640 T^{5} + 189919 T^{6} - 26640 p T^{7} + 5158 p^{2} T^{8} - 600 p^{3} T^{9} + 107 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 5 T + 50 T^{2} + 176 T^{3} + 771 T^{4} + 5843 T^{5} + 22284 T^{6} + 5843 p T^{7} + 771 p^{2} T^{8} + 176 p^{3} T^{9} + 50 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 8 T + 132 T^{2} + 743 T^{3} + 7797 T^{4} + 33802 T^{5} + 309503 T^{6} + 33802 p T^{7} + 7797 p^{2} T^{8} + 743 p^{3} T^{9} + 132 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 3 T + 66 T^{2} - 146 T^{3} + 4087 T^{4} - 5943 T^{5} + 137612 T^{6} - 5943 p T^{7} + 4087 p^{2} T^{8} - 146 p^{3} T^{9} + 66 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 15 T + 271 T^{2} + 2530 T^{3} + 26829 T^{4} + 185919 T^{5} + 1470270 T^{6} + 185919 p T^{7} + 26829 p^{2} T^{8} + 2530 p^{3} T^{9} + 271 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 4 T + 230 T^{2} + 749 T^{3} + 23811 T^{4} + 1352 p T^{5} + 1431531 T^{6} + 1352 p^{2} T^{7} + 23811 p^{2} T^{8} + 749 p^{3} T^{9} + 230 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 13 T + 126 T^{2} - 1554 T^{3} + 12921 T^{4} - 108347 T^{5} + 968797 T^{6} - 108347 p T^{7} + 12921 p^{2} T^{8} - 1554 p^{3} T^{9} + 126 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 161 T^{2} - 255 T^{3} + 12605 T^{4} - 44407 T^{5} + 744938 T^{6} - 44407 p T^{7} + 12605 p^{2} T^{8} - 255 p^{3} T^{9} + 161 p^{4} T^{10} + p^{6} T^{12} \)
61 \( 1 - 9 T + 234 T^{2} - 2302 T^{3} + 30205 T^{4} - 3881 p T^{5} + 2409968 T^{6} - 3881 p^{2} T^{7} + 30205 p^{2} T^{8} - 2302 p^{3} T^{9} + 234 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 3 T + 203 T^{2} - 820 T^{3} + 23862 T^{4} - 100928 T^{5} + 1840467 T^{6} - 100928 p T^{7} + 23862 p^{2} T^{8} - 820 p^{3} T^{9} + 203 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 19 T + 351 T^{2} + 3254 T^{3} + 31709 T^{4} + 169223 T^{5} + 1652142 T^{6} + 169223 p T^{7} + 31709 p^{2} T^{8} + 3254 p^{3} T^{9} + 351 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 3 T + 133 T^{2} - 274 T^{3} + 18744 T^{4} - 31092 T^{5} + 1356111 T^{6} - 31092 p T^{7} + 18744 p^{2} T^{8} - 274 p^{3} T^{9} + 133 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 16 T + 189 T^{2} - 1815 T^{3} + 19499 T^{4} - 139691 T^{5} + 1154286 T^{6} - 139691 p T^{7} + 19499 p^{2} T^{8} - 1815 p^{3} T^{9} + 189 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 31 T + 824 T^{2} + 13802 T^{3} + 205891 T^{4} + 2327315 T^{5} + 23897248 T^{6} + 2327315 p T^{7} + 205891 p^{2} T^{8} + 13802 p^{3} T^{9} + 824 p^{4} T^{10} + 31 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 14 T + 467 T^{2} - 5503 T^{3} + 93933 T^{4} - 916667 T^{5} + 10735038 T^{6} - 916667 p T^{7} + 93933 p^{2} T^{8} - 5503 p^{3} T^{9} + 467 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 11 T + 374 T^{2} + 3862 T^{3} + 74175 T^{4} + 650639 T^{5} + 8888572 T^{6} + 650639 p T^{7} + 74175 p^{2} T^{8} + 3862 p^{3} T^{9} + 374 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.36248026131235495478560677018, −4.22119685744969334121500885388, −3.96180573318158933165713647039, −3.96023836069807826518508487374, −3.86882505245860189016958756184, −3.85596890245412095532682002163, −3.83628037580636372920698586051, −3.40762663573411455041373874920, −3.13295763112600551633933995688, −3.09474303984619965388238504576, −3.05883331486150828370753259194, −3.04749440188583651494434610165, −2.92454298028065355397106180435, −2.46207099836660768785245629665, −2.42232386141092767121161833592, −2.23200998671857837340803908899, −2.19803017826799435681990141822, −2.18747725391138967077014601063, −2.04039875632863891560039312445, −1.82529571928063356520382359159, −1.42678379183848712607059082937, −1.29737931626942746944142033686, −1.06230887718891052529893363417, −1.04701189594523705657282763502, −0.862405898825182011215777121633, 0, 0, 0, 0, 0, 0, 0.862405898825182011215777121633, 1.04701189594523705657282763502, 1.06230887718891052529893363417, 1.29737931626942746944142033686, 1.42678379183848712607059082937, 1.82529571928063356520382359159, 2.04039875632863891560039312445, 2.18747725391138967077014601063, 2.19803017826799435681990141822, 2.23200998671857837340803908899, 2.42232386141092767121161833592, 2.46207099836660768785245629665, 2.92454298028065355397106180435, 3.04749440188583651494434610165, 3.05883331486150828370753259194, 3.09474303984619965388238504576, 3.13295763112600551633933995688, 3.40762663573411455041373874920, 3.83628037580636372920698586051, 3.85596890245412095532682002163, 3.86882505245860189016958756184, 3.96023836069807826518508487374, 3.96180573318158933165713647039, 4.22119685744969334121500885388, 4.36248026131235495478560677018

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

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