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Properties

Label 12-304e6-1.1-c1e6-0-3
Degree $12$
Conductor $7.893\times 10^{14}$
Sign $1$
Analytic cond. $204.599$
Root an. cond. $1.55802$
Motivic weight $1$
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  + 3·3-s − 6·5-s + 6·9-s − 3·13-s − 18·15-s + 3·17-s + 12·19-s − 6·23-s + 18·25-s + 10·27-s − 3·29-s − 9·31-s − 9·39-s + 21·41-s + 3·43-s − 36·45-s + 3·47-s + 18·49-s + 9·51-s − 3·53-s + 36·57-s − 12·59-s − 12·61-s + 18·65-s + 30·67-s − 18·69-s + 6·71-s + ⋯
L(s)  = 1  + 1.73·3-s − 2.68·5-s + 2·9-s − 0.832·13-s − 4.64·15-s + 0.727·17-s + 2.75·19-s − 1.25·23-s + 18/5·25-s + 1.92·27-s − 0.557·29-s − 1.61·31-s − 1.44·39-s + 3.27·41-s + 0.457·43-s − 5.36·45-s + 0.437·47-s + 18/7·49-s + 1.26·51-s − 0.412·53-s + 4.76·57-s − 1.56·59-s − 1.53·61-s + 2.23·65-s + 3.66·67-s − 2.16·69-s + 0.712·71-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(204.599\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.017079476\)
\(L(\frac12)\) \(\approx\) \(3.017079476\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 - 12 T + 78 T^{2} - 385 T^{3} + 78 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
good3 \( 1 - p T + p T^{2} - T^{3} - 4 p T^{4} + 10 p T^{5} - 35 T^{6} + 10 p^{2} T^{7} - 4 p^{3} T^{8} - p^{3} T^{9} + p^{5} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 + 6 T + 18 T^{2} + 9 p T^{3} + 81 T^{4} + 87 T^{5} + 109 T^{6} + 87 p T^{7} + 81 p^{2} T^{8} + 9 p^{4} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 18 T^{2} - 2 T^{3} + 198 T^{4} + 18 T^{5} - 1581 T^{6} + 18 p T^{7} + 198 p^{2} T^{8} - 2 p^{3} T^{9} - 18 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 24 T^{2} + 18 T^{3} + 312 T^{4} - 216 T^{5} - 3593 T^{6} - 216 p T^{7} + 312 p^{2} T^{8} + 18 p^{3} T^{9} - 24 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 + 3 T + 24 T^{2} + 2 p T^{3} + 315 T^{4} + 261 T^{5} + 4905 T^{6} + 261 p T^{7} + 315 p^{2} T^{8} + 2 p^{4} T^{9} + 24 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 3 T - 72 T^{3} - 117 T^{4} + 75 p T^{5} + 1369 T^{6} + 75 p^{2} T^{7} - 117 p^{2} T^{8} - 72 p^{3} T^{9} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 6 T + 36 T^{2} + 54 T^{3} + 576 T^{4} - 516 T^{5} + 4969 T^{6} - 516 p T^{7} + 576 p^{2} T^{8} + 54 p^{3} T^{9} + 36 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 3 T + 36 T^{2} + 378 T^{3} + 1872 T^{4} + 9921 T^{5} + 94159 T^{6} + 9921 p T^{7} + 1872 p^{2} T^{8} + 378 p^{3} T^{9} + 36 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 9 T - 18 T^{2} - 119 T^{3} + 2187 T^{4} + 3402 T^{5} - 67065 T^{6} + 3402 p T^{7} + 2187 p^{2} T^{8} - 119 p^{3} T^{9} - 18 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 90 T^{2} - 17 T^{3} + 90 p T^{4} + p^{3} T^{6} )^{2} \)
41 \( 1 - 21 T + 162 T^{2} - 180 T^{3} - 4707 T^{4} + 28401 T^{5} - 103463 T^{6} + 28401 p T^{7} - 4707 p^{2} T^{8} - 180 p^{3} T^{9} + 162 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 3 T + 60 T^{2} - 8 T^{3} + 2691 T^{4} - 2799 T^{5} + 147141 T^{6} - 2799 p T^{7} + 2691 p^{2} T^{8} - 8 p^{3} T^{9} + 60 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 3 T + 54 T^{2} - 306 T^{3} + 4635 T^{4} - 537 p T^{5} + 190171 T^{6} - 537 p^{2} T^{7} + 4635 p^{2} T^{8} - 306 p^{3} T^{9} + 54 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 3 T + 234 T^{3} - 1521 T^{4} - 27771 T^{5} - 29411 T^{6} - 27771 p T^{7} - 1521 p^{2} T^{8} + 234 p^{3} T^{9} + 3 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 12 T + 18 T^{2} - 1080 T^{3} - 7614 T^{4} + 29982 T^{5} + 754273 T^{6} + 29982 p T^{7} - 7614 p^{2} T^{8} - 1080 p^{3} T^{9} + 18 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 12 T + 24 T^{2} - 586 T^{3} - 4140 T^{4} + 44676 T^{5} + 736335 T^{6} + 44676 p T^{7} - 4140 p^{2} T^{8} - 586 p^{3} T^{9} + 24 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 30 T + 348 T^{2} - 1322 T^{3} - 6408 T^{4} + 55008 T^{5} - 101691 T^{6} + 55008 p T^{7} - 6408 p^{2} T^{8} - 1322 p^{3} T^{9} + 348 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 6 T - 36 T^{2} - 594 T^{3} + 3240 T^{4} - 14892 T^{5} + 665785 T^{6} - 14892 p T^{7} + 3240 p^{2} T^{8} - 594 p^{3} T^{9} - 36 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 12 T + 96 T^{2} + 512 T^{3} + 7776 T^{4} + 102924 T^{5} + 981639 T^{6} + 102924 p T^{7} + 7776 p^{2} T^{8} + 512 p^{3} T^{9} + 96 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 39 T + 708 T^{2} - 8198 T^{3} + 67068 T^{4} - 403623 T^{5} + 2596617 T^{6} - 403623 p T^{7} + 67068 p^{2} T^{8} - 8198 p^{3} T^{9} + 708 p^{4} T^{10} - 39 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 60 T^{2} - 918 T^{3} - 1380 T^{4} + 27540 T^{5} + 1055455 T^{6} + 27540 p T^{7} - 1380 p^{2} T^{8} - 918 p^{3} T^{9} - 60 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 + 12 T + 54 T^{2} + 1035 T^{3} - 279 T^{4} - 69891 T^{5} + 3961 T^{6} - 69891 p T^{7} - 279 p^{2} T^{8} + 1035 p^{3} T^{9} + 54 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 18 T + 234 T^{2} - 3310 T^{3} + 39204 T^{4} - 371520 T^{5} + 3748107 T^{6} - 371520 p T^{7} + 39204 p^{2} T^{8} - 3310 p^{3} T^{9} + 234 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
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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

−6.56283176082568778730590435577, −6.39445832425647321245574011081, −5.82106117924257123185685988122, −5.75882909421681805225593036056, −5.73639091453123901622686686063, −5.34890992854053397536350559769, −5.29352857702139037246157294023, −5.19299902389590848596557018842, −4.60090867201522536964174611802, −4.44707007143136812035931794302, −4.43402312012843698946608803679, −4.29948201197179074005250819340, −3.94641791643319593983671862706, −3.62819893555800189020907483548, −3.58340430887794600867611002422, −3.57496162976818016502204048641, −3.28094742860074663976547872762, −2.89830795757615260192454112119, −2.84084348296607215372602000614, −2.41367861495872853687075273129, −2.19256869397601100123996133621, −1.98674300160102995630767287382, −1.42098548456252627329797119248, −0.805948549935591561084802088504, −0.71901188043701531708829781776, 0.71901188043701531708829781776, 0.805948549935591561084802088504, 1.42098548456252627329797119248, 1.98674300160102995630767287382, 2.19256869397601100123996133621, 2.41367861495872853687075273129, 2.84084348296607215372602000614, 2.89830795757615260192454112119, 3.28094742860074663976547872762, 3.57496162976818016502204048641, 3.58340430887794600867611002422, 3.62819893555800189020907483548, 3.94641791643319593983671862706, 4.29948201197179074005250819340, 4.43402312012843698946608803679, 4.44707007143136812035931794302, 4.60090867201522536964174611802, 5.19299902389590848596557018842, 5.29352857702139037246157294023, 5.34890992854053397536350559769, 5.73639091453123901622686686063, 5.75882909421681805225593036056, 5.82106117924257123185685988122, 6.39445832425647321245574011081, 6.56283176082568778730590435577

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

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