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Properties

Label 12-931e6-1.1-c1e6-0-11
Degree $12$
Conductor $6.512\times 10^{17}$
Sign $1$
Analytic cond. $168795.$
Root an. cond. $2.72654$
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  + 3·2-s + 6·3-s + 3·5-s + 18·6-s − 9·8-s + 24·9-s + 9·10-s − 3·13-s + 18·15-s − 9·16-s − 6·17-s + 72·18-s + 6·19-s − 12·23-s − 54·24-s − 9·26-s + 71·27-s − 3·29-s + 54·30-s − 18·31-s + 3·32-s − 18·34-s + 18·38-s − 18·39-s − 27·40-s + 21·41-s − 3·43-s + ⋯
L(s)  = 1  + 2.12·2-s + 3.46·3-s + 1.34·5-s + 7.34·6-s − 3.18·8-s + 8·9-s + 2.84·10-s − 0.832·13-s + 4.64·15-s − 9/4·16-s − 1.45·17-s + 16.9·18-s + 1.37·19-s − 2.50·23-s − 11.0·24-s − 1.76·26-s + 13.6·27-s − 0.557·29-s + 9.85·30-s − 3.23·31-s + 0.530·32-s − 3.08·34-s + 2.91·38-s − 2.88·39-s − 4.26·40-s + 3.27·41-s − 0.457·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(168795.\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 7^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(29.17140293\)
\(L(\frac12)\) \(\approx\) \(29.17140293\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 - 6 T - 12 T^{2} + 169 T^{3} - 12 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
good2 \( 1 - 3 T + 9 T^{2} - 9 p T^{3} + 9 p^{2} T^{4} - 57 T^{5} + 91 T^{6} - 57 p T^{7} + 9 p^{4} T^{8} - 9 p^{4} T^{9} + 9 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
3 \( 1 - 2 p T + 4 p T^{2} + T^{3} - 13 p T^{4} + 17 p T^{5} - 35 T^{6} + 17 p^{2} T^{7} - 13 p^{3} T^{8} + p^{3} T^{9} + 4 p^{5} T^{10} - 2 p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 - 3 T + 9 T^{2} - 9 T^{3} + 36 T^{4} - 12 T^{5} + 109 T^{6} - 12 p T^{7} + 36 p^{2} T^{8} - 9 p^{3} T^{9} + 9 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 + 6 T + 9 T^{2} - 99 T^{3} - 423 T^{4} - 435 T^{5} + 2746 T^{6} - 435 p T^{7} - 423 p^{2} T^{8} - 99 p^{3} T^{9} + 9 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 12 T + 72 T^{2} + 378 T^{3} + 1404 T^{4} + 2658 T^{5} + 4969 T^{6} + 2658 p T^{7} + 1404 p^{2} T^{8} + 378 p^{3} T^{9} + 72 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
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 \( ( 1 + 9 T + 99 T^{2} + 505 T^{3} + 99 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 90 T^{2} - 34 T^{3} + 4770 T^{4} + 1530 T^{5} - 198105 T^{6} + 1530 p T^{7} + 4770 p^{2} T^{8} - 34 p^{3} T^{9} - 90 p^{4} T^{10} + p^{6} T^{12} \)
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 \( 1 + 12 T + 63 T^{2} + 333 T^{3} - 1233 T^{4} - 36915 T^{5} - 283166 T^{6} - 36915 p T^{7} - 1233 p^{2} T^{8} + 333 p^{3} T^{9} + 63 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 12 T + 99 T^{2} + 855 T^{3} + 7965 T^{4} + 56001 T^{5} + 369838 T^{6} + 56001 p T^{7} + 7965 p^{2} T^{8} + 855 p^{3} T^{9} + 99 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 6 T + 108 T^{2} + 1296 T^{3} + 12006 T^{4} + 92490 T^{5} + 996409 T^{6} + 92490 p T^{7} + 12006 p^{2} T^{8} + 1296 p^{3} T^{9} + 108 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 12 T + 132 T^{2} + 710 T^{3} + 6768 T^{4} + 15840 T^{5} + 182943 T^{6} + 15840 p T^{7} + 6768 p^{2} T^{8} + 710 p^{3} T^{9} + 132 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 6 T - 48 T^{2} + 242 T^{3} - 1980 T^{4} + 11358 T^{5} + 151569 T^{6} + 11358 p T^{7} - 1980 p^{2} T^{8} + 242 p^{3} T^{9} - 48 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 + 12 T + 96 T^{2} + 512 T^{3} - 432 T^{4} - 79704 T^{5} - 815913 T^{6} - 79704 p T^{7} - 432 p^{2} T^{8} + 512 p^{3} T^{9} + 96 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 24 T + 366 T^{2} - 3817 T^{3} + 31347 T^{4} - 203283 T^{5} + 1436265 T^{6} - 203283 p T^{7} + 31347 p^{2} T^{8} - 3817 p^{3} T^{9} + 366 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 - 15 T + 36 T^{2} + 1548 T^{3} - 15786 T^{4} - 7827 T^{5} + 1123759 T^{6} - 7827 p T^{7} - 15786 p^{2} T^{8} + 1548 p^{3} T^{9} + 36 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
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

−5.41239999136231072578483362425, −4.77724014221889734149023414270, −4.76224342765053481316921125936, −4.75903368363101350916921299218, −4.69330153683977633428451442454, −4.53161314307450842654593699136, −4.25763890619475400463400735457, −4.08288235682164715252209366502, −4.02528548521930849192568521689, −3.91930155285866054091947484615, −3.58958616896188215915044747859, −3.55105801616852685997559128149, −3.39461281099982133181775041390, −3.29379977364942042148502686329, −2.93145904575010962292582139878, −2.67259185438021378642036306117, −2.64388898703161239866980257293, −2.43633505136514477321800896601, −1.93742995674283923658214964682, −1.85097575677684125571095232983, −1.77324919451383477838692930606, −1.71616315466757542283765515157, −1.63462231734185639520291041377, −0.72819173804449992488007459748, −0.37838179817233165937486789548, 0.37838179817233165937486789548, 0.72819173804449992488007459748, 1.63462231734185639520291041377, 1.71616315466757542283765515157, 1.77324919451383477838692930606, 1.85097575677684125571095232983, 1.93742995674283923658214964682, 2.43633505136514477321800896601, 2.64388898703161239866980257293, 2.67259185438021378642036306117, 2.93145904575010962292582139878, 3.29379977364942042148502686329, 3.39461281099982133181775041390, 3.55105801616852685997559128149, 3.58958616896188215915044747859, 3.91930155285866054091947484615, 4.02528548521930849192568521689, 4.08288235682164715252209366502, 4.25763890619475400463400735457, 4.53161314307450842654593699136, 4.69330153683977633428451442454, 4.75903368363101350916921299218, 4.76224342765053481316921125936, 4.77724014221889734149023414270, 5.41239999136231072578483362425

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

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