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Properties

Label 12-189e6-1.1-c1e6-0-2
Degree $12$
Conductor $4.558\times 10^{13}$
Sign $1$
Analytic cond. $11.8149$
Root an. cond. $1.22848$
Motivic weight $1$
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  + 2·2-s + 3·4-s + 5-s + 2·7-s + 2·10-s + 7·11-s + 4·13-s + 4·14-s − 4·16-s − 5·19-s + 3·20-s + 14·22-s + 6·23-s + 9·25-s + 8·26-s + 6·28-s − 26·29-s + 8·31-s − 7·32-s + 2·35-s + 8·37-s − 10·38-s − 4·41-s − 18·43-s + 21·44-s + 12·46-s + 9·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.447·5-s + 0.755·7-s + 0.632·10-s + 2.11·11-s + 1.10·13-s + 1.06·14-s − 16-s − 1.14·19-s + 0.670·20-s + 2.98·22-s + 1.25·23-s + 9/5·25-s + 1.56·26-s + 1.13·28-s − 4.82·29-s + 1.43·31-s − 1.23·32-s + 0.338·35-s + 1.31·37-s − 1.62·38-s − 0.624·41-s − 2.74·43-s + 3.16·44-s + 1.76·46-s + 1.31·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{18} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(11.8149\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{18} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.586740175\)
\(L(\frac12)\) \(\approx\) \(5.586740175\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - 2 T - 4 T^{2} + 31 T^{3} - 4 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
good2 \( 1 - p T + T^{2} + p^{2} T^{3} - 7 T^{4} + T^{5} + 7 T^{6} + p T^{7} - 7 p^{2} T^{8} + p^{5} T^{9} + p^{4} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 - T - 8 T^{2} + 17 T^{3} + 23 T^{4} - 52 T^{5} - 11 T^{6} - 52 p T^{7} + 23 p^{2} T^{8} + 17 p^{3} T^{9} - 8 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 7 T + 4 T^{2} - T^{3} + 431 T^{4} - 982 T^{5} - 893 T^{6} - 982 p T^{7} + 431 p^{2} T^{8} - p^{3} T^{9} + 4 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 - 2 T + 20 T^{2} - 5 T^{3} + 20 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 - 18 T^{2} + 18 T^{3} + 18 T^{4} - 162 T^{5} + 4399 T^{6} - 162 p T^{7} + 18 p^{2} T^{8} + 18 p^{3} T^{9} - 18 p^{4} T^{10} + p^{6} T^{12} \)
19 \( 1 + 5 T - 28 T^{2} - 3 p T^{3} + 997 T^{4} + 268 T^{5} - 22757 T^{6} + 268 p T^{7} + 997 p^{2} T^{8} - 3 p^{4} T^{9} - 28 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 6 T - 36 T^{2} + 102 T^{3} + 1926 T^{4} - 2526 T^{5} - 42653 T^{6} - 2526 p T^{7} + 1926 p^{2} T^{8} + 102 p^{3} T^{9} - 36 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 + 13 T + 117 T^{2} + 763 T^{3} + 117 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 8 T - 30 T^{2} + 102 T^{3} + 2506 T^{4} - 1202 T^{5} - 93509 T^{6} - 1202 p T^{7} + 2506 p^{2} T^{8} + 102 p^{3} T^{9} - 30 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 8 T - 42 T^{2} + 150 T^{3} + 3322 T^{4} - 1094 T^{5} - 153041 T^{6} - 1094 p T^{7} + 3322 p^{2} T^{8} + 150 p^{3} T^{9} - 42 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 + 2 T + 18 T^{2} - 223 T^{3} + 18 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 + 9 T + 117 T^{2} + 673 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 9 T - 18 T^{2} + 819 T^{3} - 2547 T^{4} - 20772 T^{5} + 299095 T^{6} - 20772 p T^{7} - 2547 p^{2} T^{8} + 819 p^{3} T^{9} - 18 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 24 T + 252 T^{2} - 2202 T^{3} + 20916 T^{4} - 146148 T^{5} + 883411 T^{6} - 146148 p T^{7} + 20916 p^{2} T^{8} - 2202 p^{3} T^{9} + 252 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 15 T - 18 T^{2} - 57 T^{3} + 16947 T^{4} + 71898 T^{5} - 430157 T^{6} + 71898 p T^{7} + 16947 p^{2} T^{8} - 57 p^{3} T^{9} - 18 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - T - 133 T^{2} - 132 T^{3} + 9781 T^{4} + 12493 T^{5} - 645074 T^{6} + 12493 p T^{7} + 9781 p^{2} T^{8} - 132 p^{3} T^{9} - 133 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 14 T - 58 T^{2} - 258 T^{3} + 20800 T^{4} + 70720 T^{5} - 964241 T^{6} + 70720 p T^{7} + 20800 p^{2} T^{8} - 258 p^{3} T^{9} - 58 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 3 T + 105 T^{2} - 669 T^{3} + 105 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 7 T - 36 T^{2} + 513 T^{3} + 733 T^{4} - 46082 T^{5} - 8831 T^{6} - 46082 p T^{7} + 733 p^{2} T^{8} + 513 p^{3} T^{9} - 36 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 6 T - 132 T^{2} - 634 T^{3} + 10026 T^{4} + 16110 T^{5} - 794253 T^{6} + 16110 p T^{7} + 10026 p^{2} T^{8} - 634 p^{3} T^{9} - 132 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 3 T + 69 T^{2} + 1227 T^{3} + 69 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 + 5 T - 149 T^{2} + 68 T^{3} + 12785 T^{4} - 45481 T^{5} - 1321850 T^{6} - 45481 p T^{7} + 12785 p^{2} T^{8} + 68 p^{3} T^{9} - 149 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 + 14 T + 307 T^{2} + 2692 T^{3} + 307 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−7.02424117395271113149223290533, −6.69762645908457850119297206861, −6.57025391529394888713174948974, −6.30855650846792055215385772322, −6.11202225945612341545576712914, −5.79672416169981861450977037760, −5.78888646380277477436126506371, −5.72347437717030477254089544245, −5.47851722756754097897830001174, −5.28103108177296070175151546898, −4.76684561509377538972656150986, −4.66878324491480102281295224713, −4.40244857594651719507545620912, −4.38581712729720874802127500090, −4.04102450783686135612168721122, −3.75142663198476890107936911459, −3.48145875187955992746860293345, −3.36939575975004998159327043845, −3.23232935400843525132708911932, −2.75295905682635696080946726110, −2.40491511976131716383366971973, −2.21574931045802993404890900932, −1.75562215354869559880831019856, −1.44210283564697640358080730706, −1.05382292861248251908236924670, 1.05382292861248251908236924670, 1.44210283564697640358080730706, 1.75562215354869559880831019856, 2.21574931045802993404890900932, 2.40491511976131716383366971973, 2.75295905682635696080946726110, 3.23232935400843525132708911932, 3.36939575975004998159327043845, 3.48145875187955992746860293345, 3.75142663198476890107936911459, 4.04102450783686135612168721122, 4.38581712729720874802127500090, 4.40244857594651719507545620912, 4.66878324491480102281295224713, 4.76684561509377538972656150986, 5.28103108177296070175151546898, 5.47851722756754097897830001174, 5.72347437717030477254089544245, 5.78888646380277477436126506371, 5.79672416169981861450977037760, 6.11202225945612341545576712914, 6.30855650846792055215385772322, 6.57025391529394888713174948974, 6.69762645908457850119297206861, 7.02424117395271113149223290533

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

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