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Properties

Label 12-5415e6-1.1-c1e6-0-4
Degree $12$
Conductor $2.521\times 10^{22}$
Sign $1$
Analytic cond. $6.53511\times 10^{9}$
Root an. cond. $6.57563$
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  + 3·2-s − 6·3-s + 6·5-s − 18·6-s − 6·7-s − 9·8-s + 21·9-s + 18·10-s − 12·11-s − 3·13-s − 18·14-s − 36·15-s − 9·16-s − 18·17-s + 63·18-s + 36·21-s − 36·22-s + 9·23-s + 54·24-s + 21·25-s − 9·26-s − 56·27-s + 9·29-s − 108·30-s + 3·32-s + 72·33-s − 54·34-s + ⋯
L(s)  = 1  + 2.12·2-s − 3.46·3-s + 2.68·5-s − 7.34·6-s − 2.26·7-s − 3.18·8-s + 7·9-s + 5.69·10-s − 3.61·11-s − 0.832·13-s − 4.81·14-s − 9.29·15-s − 9/4·16-s − 4.36·17-s + 14.8·18-s + 7.85·21-s − 7.67·22-s + 1.87·23-s + 11.0·24-s + 21/5·25-s − 1.76·26-s − 10.7·27-s + 1.67·29-s − 19.7·30-s + 0.530·32-s + 12.5·33-s − 9.26·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{6} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(6.53511\times 10^{9}\)
Root analytic conductor: \(6.57563\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 3^{6} \cdot 5^{6} \cdot 19^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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)$
bad3 \( ( 1 + T )^{6} \)
5 \( ( 1 - T )^{6} \)
19 \( 1 \)
good2 \( 1 - 3 T + 9 T^{2} - 9 p T^{3} + 9 p^{2} T^{4} - 57 T^{5} + 91 T^{6} - 57 p T^{7} + 9 p^{4} T^{8} - 9 p^{4} T^{9} + 9 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 6 T + 45 T^{2} + 191 T^{3} + 807 T^{4} + 2544 T^{5} + 7575 T^{6} + 2544 p T^{7} + 807 p^{2} T^{8} + 191 p^{3} T^{9} + 45 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 12 T + 117 T^{2} + 747 T^{3} + 4095 T^{4} + 17364 T^{5} + 64441 T^{6} + 17364 p T^{7} + 4095 p^{2} T^{8} + 747 p^{3} T^{9} + 117 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 3 T + 63 T^{2} + 151 T^{3} + 1752 T^{4} + 3360 T^{5} + 28653 T^{6} + 3360 p T^{7} + 1752 p^{2} T^{8} + 151 p^{3} T^{9} + 63 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 + 9 T + 69 T^{2} + 315 T^{3} + 69 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 9 T + 90 T^{2} - 495 T^{3} + 2853 T^{4} - 11970 T^{5} + 60085 T^{6} - 11970 p T^{7} + 2853 p^{2} T^{8} - 495 p^{3} T^{9} + 90 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 9 T + 165 T^{2} - 1071 T^{3} + 11112 T^{4} - 55656 T^{5} + 416743 T^{6} - 55656 p T^{7} + 11112 p^{2} T^{8} - 1071 p^{3} T^{9} + 165 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 132 T^{2} - 44 T^{3} + 8223 T^{4} - 4083 T^{5} + 315021 T^{6} - 4083 p T^{7} + 8223 p^{2} T^{8} - 44 p^{3} T^{9} + 132 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 - 6 T + 75 T^{2} - 482 T^{3} + 3546 T^{4} - 25911 T^{5} + 164193 T^{6} - 25911 p T^{7} + 3546 p^{2} T^{8} - 482 p^{3} T^{9} + 75 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 6 T + 144 T^{2} + 702 T^{3} + 261 p T^{4} + 47211 T^{5} + 535843 T^{6} + 47211 p T^{7} + 261 p^{3} T^{8} + 702 p^{3} T^{9} + 144 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 126 T^{2} - 133 T^{3} + 7065 T^{4} - 19557 T^{5} + 301413 T^{6} - 19557 p T^{7} + 7065 p^{2} T^{8} - 133 p^{3} T^{9} + 126 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 + 9 T + 153 T^{2} + 1071 T^{3} + 9720 T^{4} + 55890 T^{5} + 447481 T^{6} + 55890 p T^{7} + 9720 p^{2} T^{8} + 1071 p^{3} T^{9} + 153 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 6 T + 243 T^{2} - 1044 T^{3} + 25866 T^{4} - 84831 T^{5} + 1678915 T^{6} - 84831 p T^{7} + 25866 p^{2} T^{8} - 1044 p^{3} T^{9} + 243 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 12 T + 126 T^{2} - 450 T^{3} - 243 T^{4} + 66381 T^{5} - 549449 T^{6} + 66381 p T^{7} - 243 p^{2} T^{8} - 450 p^{3} T^{9} + 126 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 33 T + 738 T^{2} + 11432 T^{3} + 144438 T^{4} + 1463667 T^{5} + 12600519 T^{6} + 1463667 p T^{7} + 144438 p^{2} T^{8} + 11432 p^{3} T^{9} + 738 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 12 T + 267 T^{2} - 1223 T^{3} + 15453 T^{4} + 58248 T^{5} + 249381 T^{6} + 58248 p T^{7} + 15453 p^{2} T^{8} - 1223 p^{3} T^{9} + 267 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 6 T + 219 T^{2} - 1323 T^{3} + 26097 T^{4} - 126330 T^{5} + 2207167 T^{6} - 126330 p T^{7} + 26097 p^{2} T^{8} - 1323 p^{3} T^{9} + 219 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 21 T + 462 T^{2} + 5846 T^{3} + 75516 T^{4} + 703383 T^{5} + 6872013 T^{6} + 703383 p T^{7} + 75516 p^{2} T^{8} + 5846 p^{3} T^{9} + 462 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 6 T + 318 T^{2} + 1660 T^{3} + 51063 T^{4} + 223077 T^{5} + 4986927 T^{6} + 223077 p T^{7} + 51063 p^{2} T^{8} + 1660 p^{3} T^{9} + 318 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 3 T + 261 T^{2} - 1377 T^{3} + 39231 T^{4} - 180093 T^{5} + 4112461 T^{6} - 180093 p T^{7} + 39231 p^{2} T^{8} - 1377 p^{3} T^{9} + 261 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 6 T + 360 T^{2} + 2052 T^{3} + 63306 T^{4} + 316770 T^{5} + 6931009 T^{6} + 316770 p T^{7} + 63306 p^{2} T^{8} + 2052 p^{3} T^{9} + 360 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 12 T + 372 T^{2} + 3604 T^{3} + 71208 T^{4} + 568656 T^{5} + 8425329 T^{6} + 568656 p T^{7} + 71208 p^{2} T^{8} + 3604 p^{3} T^{9} + 372 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.77154489185933841219313503799, −4.50091350067846839315070346272, −4.34762462645922113531938842883, −4.34691701994515359049942819752, −4.34296458866556445409855883928, −4.18341278016561173778867875189, −4.13543216406025299753444412576, −3.52591302076254071498477493294, −3.38013871282655209487157027826, −3.36240757774389183468953615502, −3.35933794917938146560678304852, −3.25699285326967078097017165137, −2.85245457092288354857092176815, −2.61976697153725802287487446576, −2.54760059904369443849354523269, −2.54347012208775947851408110337, −2.51523026988855656672538224039, −2.28937102866571329042545064976, −1.95705933967124778086860231442, −1.86437530047505787225708673720, −1.72844701227893290591534731136, −1.21945887005585865471658314893, −1.20826386128722804190988212048, −1.03171791460913506055429851023, −0.903250219460902122904670640883, 0, 0, 0, 0, 0, 0, 0.903250219460902122904670640883, 1.03171791460913506055429851023, 1.20826386128722804190988212048, 1.21945887005585865471658314893, 1.72844701227893290591534731136, 1.86437530047505787225708673720, 1.95705933967124778086860231442, 2.28937102866571329042545064976, 2.51523026988855656672538224039, 2.54347012208775947851408110337, 2.54760059904369443849354523269, 2.61976697153725802287487446576, 2.85245457092288354857092176815, 3.25699285326967078097017165137, 3.35933794917938146560678304852, 3.36240757774389183468953615502, 3.38013871282655209487157027826, 3.52591302076254071498477493294, 4.13543216406025299753444412576, 4.18341278016561173778867875189, 4.34296458866556445409855883928, 4.34691701994515359049942819752, 4.34762462645922113531938842883, 4.50091350067846839315070346272, 4.77154489185933841219313503799

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

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