Properties

Degree 24
Conductor $ 2^{12} \cdot 3^{36} \cdot 7^{12} $
Sign $1$
Motivic weight 1
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 − 2·8-s + 9·9-s + 3·11-s + 12·13-s + 12·19-s + 12·23-s − 6·24-s − 12·25-s + 24·27-s − 3·29-s + 21·31-s + 9·33-s + 15·37-s + 36·39-s + 12·41-s + 21·43-s + 12·47-s − 42·53-s + 36·57-s + 18·59-s + 64-s − 33·67-s + 36·69-s + 24·71-s − 18·72-s + 9·73-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.707·8-s + 3·9-s + 0.904·11-s + 3.32·13-s + 2.75·19-s + 2.50·23-s − 1.22·24-s − 2.39·25-s + 4.61·27-s − 0.557·29-s + 3.77·31-s + 1.56·33-s + 2.46·37-s + 5.76·39-s + 1.87·41-s + 3.20·43-s + 1.75·47-s − 5.76·53-s + 4.76·57-s + 2.34·59-s + 1/8·64-s − 4.03·67-s + 4.33·69-s + 2.84·71-s − 2.12·72-s + 1.05·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(24\)
\( N \)  =  \(2^{12} \cdot 3^{36} \cdot 7^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{378} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(24,\ 2^{12} \cdot 3^{36} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )$
$L(1)$  $\approx$  $33.0783$
$L(\frac12)$  $\approx$  $33.0783$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 24. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 23.
$p$$F_p(T)$
bad2 \( ( 1 + T^{3} + T^{6} )^{2} \)
3 \( 1 - p T + p T^{3} + p^{2} T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} + p^{4} T^{9} - p^{6} T^{11} + p^{6} T^{12} \)
7 \( ( 1 + T^{3} + T^{6} )^{2} \)
good5 \( 1 + 12 T^{2} - 18 T^{3} + 78 T^{4} - 198 T^{5} + 467 T^{6} - 1233 T^{7} + 441 p T^{8} - 5832 T^{9} + 8847 T^{10} - 26739 T^{11} + 35469 T^{12} - 26739 p T^{13} + 8847 p^{2} T^{14} - 5832 p^{3} T^{15} + 441 p^{5} T^{16} - 1233 p^{5} T^{17} + 467 p^{6} T^{18} - 198 p^{7} T^{19} + 78 p^{8} T^{20} - 18 p^{9} T^{21} + 12 p^{10} T^{22} + p^{12} T^{24} \)
11 \( 1 - 3 T - 15 T^{2} + 126 T^{3} - 201 T^{4} - 1488 T^{5} + 7145 T^{6} - 1530 T^{7} - 5634 p T^{8} + 202716 T^{9} - 19692 T^{10} - 1304451 T^{11} + 4526883 T^{12} - 1304451 p T^{13} - 19692 p^{2} T^{14} + 202716 p^{3} T^{15} - 5634 p^{5} T^{16} - 1530 p^{5} T^{17} + 7145 p^{6} T^{18} - 1488 p^{7} T^{19} - 201 p^{8} T^{20} + 126 p^{9} T^{21} - 15 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \)
13 \( 1 - 12 T + 93 T^{2} - 461 T^{3} + 1830 T^{4} - 6510 T^{5} + 2113 p T^{6} - 9963 p T^{7} + 566109 T^{8} - 2155844 T^{9} + 552561 p T^{10} - 24098388 T^{11} + 81886711 T^{12} - 24098388 p T^{13} + 552561 p^{3} T^{14} - 2155844 p^{3} T^{15} + 566109 p^{4} T^{16} - 9963 p^{6} T^{17} + 2113 p^{7} T^{18} - 6510 p^{7} T^{19} + 1830 p^{8} T^{20} - 461 p^{9} T^{21} + 93 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \)
17 \( 1 - 45 T^{2} - 18 T^{3} + 810 T^{4} + 405 T^{5} - 4228 T^{6} + 10935 T^{7} - 8145 T^{8} - 541728 T^{9} - 2753640 T^{10} + 17820 p^{2} T^{11} + 87495483 T^{12} + 17820 p^{3} T^{13} - 2753640 p^{2} T^{14} - 541728 p^{3} T^{15} - 8145 p^{4} T^{16} + 10935 p^{5} T^{17} - 4228 p^{6} T^{18} + 405 p^{7} T^{19} + 810 p^{8} T^{20} - 18 p^{9} T^{21} - 45 p^{10} T^{22} + p^{12} T^{24} \)
19 \( 1 - 12 T + 12 T^{2} + 52 T^{3} + 2649 T^{4} - 10209 T^{5} - 32138 T^{6} - 100341 T^{7} + 1360620 T^{8} + 1473505 T^{9} - 7164300 T^{10} - 30874461 T^{11} + 38067223 T^{12} - 30874461 p T^{13} - 7164300 p^{2} T^{14} + 1473505 p^{3} T^{15} + 1360620 p^{4} T^{16} - 100341 p^{5} T^{17} - 32138 p^{6} T^{18} - 10209 p^{7} T^{19} + 2649 p^{8} T^{20} + 52 p^{9} T^{21} + 12 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \)
23 \( 1 - 12 T + 57 T^{2} - 216 T^{3} + 870 T^{4} - 1974 T^{5} + 6398 T^{6} - 34596 T^{7} - 231615 T^{8} + 2798766 T^{9} - 5217057 T^{10} - 7068906 T^{11} - 3870063 T^{12} - 7068906 p T^{13} - 5217057 p^{2} T^{14} + 2798766 p^{3} T^{15} - 231615 p^{4} T^{16} - 34596 p^{5} T^{17} + 6398 p^{6} T^{18} - 1974 p^{7} T^{19} + 870 p^{8} T^{20} - 216 p^{9} T^{21} + 57 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \)
29 \( 1 + 3 T + 30 T^{2} + 207 T^{3} + 978 T^{4} + 4863 T^{5} + 23219 T^{6} + 31419 T^{7} + 849420 T^{8} - 4617 T^{9} - 2257902 T^{10} - 22744773 T^{11} - 94856679 T^{12} - 22744773 p T^{13} - 2257902 p^{2} T^{14} - 4617 p^{3} T^{15} + 849420 p^{4} T^{16} + 31419 p^{5} T^{17} + 23219 p^{6} T^{18} + 4863 p^{7} T^{19} + 978 p^{8} T^{20} + 207 p^{9} T^{21} + 30 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \)
31 \( 1 - 21 T + 255 T^{2} - 2414 T^{3} + 18624 T^{4} - 124887 T^{5} + 730225 T^{6} - 3464208 T^{7} + 11411442 T^{8} - 1810082 T^{9} - 324371289 T^{10} + 3091769544 T^{11} - 19862733551 T^{12} + 3091769544 p T^{13} - 324371289 p^{2} T^{14} - 1810082 p^{3} T^{15} + 11411442 p^{4} T^{16} - 3464208 p^{5} T^{17} + 730225 p^{6} T^{18} - 124887 p^{7} T^{19} + 18624 p^{8} T^{20} - 2414 p^{9} T^{21} + 255 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \)
37 \( 1 - 15 T + 69 T^{2} + 406 T^{3} - 7374 T^{4} + 38802 T^{5} + 15283 T^{6} - 1166850 T^{7} + 3628278 T^{8} + 22593355 T^{9} - 153725946 T^{10} - 917449914 T^{11} + 12989691655 T^{12} - 917449914 p T^{13} - 153725946 p^{2} T^{14} + 22593355 p^{3} T^{15} + 3628278 p^{4} T^{16} - 1166850 p^{5} T^{17} + 15283 p^{6} T^{18} + 38802 p^{7} T^{19} - 7374 p^{8} T^{20} + 406 p^{9} T^{21} + 69 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \)
41 \( 1 - 12 T + 39 T^{2} - 252 T^{3} + 6540 T^{4} - 62211 T^{5} + 254375 T^{6} - 1609425 T^{7} + 20028519 T^{8} - 143465580 T^{9} + 690787764 T^{10} - 4925423520 T^{11} + 40342142493 T^{12} - 4925423520 p T^{13} + 690787764 p^{2} T^{14} - 143465580 p^{3} T^{15} + 20028519 p^{4} T^{16} - 1609425 p^{5} T^{17} + 254375 p^{6} T^{18} - 62211 p^{7} T^{19} + 6540 p^{8} T^{20} - 252 p^{9} T^{21} + 39 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \)
43 \( 1 - 21 T + 192 T^{2} - 560 T^{3} - 8403 T^{4} + 123036 T^{5} - 639908 T^{6} - 1742472 T^{7} + 55902231 T^{8} - 419028878 T^{9} + 874994820 T^{10} + 12129804183 T^{11} - 134937795011 T^{12} + 12129804183 p T^{13} + 874994820 p^{2} T^{14} - 419028878 p^{3} T^{15} + 55902231 p^{4} T^{16} - 1742472 p^{5} T^{17} - 639908 p^{6} T^{18} + 123036 p^{7} T^{19} - 8403 p^{8} T^{20} - 560 p^{9} T^{21} + 192 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \)
47 \( 1 - 12 T - 72 T^{2} + 1215 T^{3} + 6273 T^{4} - 85908 T^{5} - 561724 T^{6} + 5249610 T^{7} + 35240229 T^{8} - 201170169 T^{9} - 2024333352 T^{10} + 3145093866 T^{11} + 108872443875 T^{12} + 3145093866 p T^{13} - 2024333352 p^{2} T^{14} - 201170169 p^{3} T^{15} + 35240229 p^{4} T^{16} + 5249610 p^{5} T^{17} - 561724 p^{6} T^{18} - 85908 p^{7} T^{19} + 6273 p^{8} T^{20} + 1215 p^{9} T^{21} - 72 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \)
53 \( ( 1 + 21 T + 450 T^{2} + 5868 T^{3} + 70182 T^{4} + 637455 T^{5} + 5208409 T^{6} + 637455 p T^{7} + 70182 p^{2} T^{8} + 5868 p^{3} T^{9} + 450 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
59 \( 1 - 18 T + 72 T^{2} + 3006 T^{4} - 28638 T^{5} + 390494 T^{6} - 4658796 T^{7} + 22409190 T^{8} - 176439384 T^{9} + 1983739302 T^{10} - 9795803364 T^{11} + 37789076343 T^{12} - 9795803364 p T^{13} + 1983739302 p^{2} T^{14} - 176439384 p^{3} T^{15} + 22409190 p^{4} T^{16} - 4658796 p^{5} T^{17} + 390494 p^{6} T^{18} - 28638 p^{7} T^{19} + 3006 p^{8} T^{20} + 72 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \)
61 \( 1 + 189 T^{2} + 130 T^{3} + 25974 T^{4} - 17334 T^{5} + 2536246 T^{6} - 3252339 T^{7} + 204923790 T^{8} - 455484935 T^{9} + 14770394064 T^{10} - 36254574540 T^{11} + 930935536957 T^{12} - 36254574540 p T^{13} + 14770394064 p^{2} T^{14} - 455484935 p^{3} T^{15} + 204923790 p^{4} T^{16} - 3252339 p^{5} T^{17} + 2536246 p^{6} T^{18} - 17334 p^{7} T^{19} + 25974 p^{8} T^{20} + 130 p^{9} T^{21} + 189 p^{10} T^{22} + p^{12} T^{24} \)
67 \( 1 + 33 T + 642 T^{2} + 8368 T^{3} + 69393 T^{4} + 186288 T^{5} - 4701338 T^{6} - 91813068 T^{7} - 902003391 T^{8} - 4727097308 T^{9} + 9492444366 T^{10} + 478832565675 T^{11} + 5293151069269 T^{12} + 478832565675 p T^{13} + 9492444366 p^{2} T^{14} - 4727097308 p^{3} T^{15} - 902003391 p^{4} T^{16} - 91813068 p^{5} T^{17} - 4701338 p^{6} T^{18} + 186288 p^{7} T^{19} + 69393 p^{8} T^{20} + 8368 p^{9} T^{21} + 642 p^{10} T^{22} + 33 p^{11} T^{23} + p^{12} T^{24} \)
71 \( 1 - 24 T + 105 T^{2} + 2718 T^{3} - 33666 T^{4} - 35205 T^{5} + 2568722 T^{6} - 8972217 T^{7} - 55594071 T^{8} - 248099436 T^{9} + 7228158138 T^{10} + 28994592564 T^{11} - 876571769937 T^{12} + 28994592564 p T^{13} + 7228158138 p^{2} T^{14} - 248099436 p^{3} T^{15} - 55594071 p^{4} T^{16} - 8972217 p^{5} T^{17} + 2568722 p^{6} T^{18} - 35205 p^{7} T^{19} - 33666 p^{8} T^{20} + 2718 p^{9} T^{21} + 105 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \)
73 \( 1 - 9 T - 162 T^{2} + 1543 T^{3} + 6453 T^{4} - 30168 T^{5} - 1029257 T^{6} - 42849 p T^{7} + 217574586 T^{8} - 589643561 T^{9} - 12568109643 T^{10} + 46186633458 T^{11} + 321235971409 T^{12} + 46186633458 p T^{13} - 12568109643 p^{2} T^{14} - 589643561 p^{3} T^{15} + 217574586 p^{4} T^{16} - 42849 p^{6} T^{17} - 1029257 p^{6} T^{18} - 30168 p^{7} T^{19} + 6453 p^{8} T^{20} + 1543 p^{9} T^{21} - 162 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \)
79 \( 1 - 135 T^{2} + 1993 T^{3} - 2052 T^{4} - 232362 T^{5} + 3259333 T^{6} - 10002285 T^{7} - 260701254 T^{8} + 3944140684 T^{9} - 13669666056 T^{10} - 186122056770 T^{11} + 3171346188679 T^{12} - 186122056770 p T^{13} - 13669666056 p^{2} T^{14} + 3944140684 p^{3} T^{15} - 260701254 p^{4} T^{16} - 10002285 p^{5} T^{17} + 3259333 p^{6} T^{18} - 232362 p^{7} T^{19} - 2052 p^{8} T^{20} + 1993 p^{9} T^{21} - 135 p^{10} T^{22} + p^{12} T^{24} \)
83 \( 1 - 15 T + 147 T^{2} - 1854 T^{3} + 25071 T^{4} - 149289 T^{5} + 1163042 T^{6} - 9519381 T^{7} + 3954789 T^{8} + 847137582 T^{9} - 4335015195 T^{10} + 98252949789 T^{11} - 1318582794483 T^{12} + 98252949789 p T^{13} - 4335015195 p^{2} T^{14} + 847137582 p^{3} T^{15} + 3954789 p^{4} T^{16} - 9519381 p^{5} T^{17} + 1163042 p^{6} T^{18} - 149289 p^{7} T^{19} + 25071 p^{8} T^{20} - 1854 p^{9} T^{21} + 147 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \)
89 \( 1 + 51 T + 1197 T^{2} + 18234 T^{3} + 216342 T^{4} + 2052636 T^{5} + 13370033 T^{6} + 34362828 T^{7} - 4394628 p T^{8} - 7793030061 T^{9} - 89642793852 T^{10} - 757259381556 T^{11} - 6184239975177 T^{12} - 757259381556 p T^{13} - 89642793852 p^{2} T^{14} - 7793030061 p^{3} T^{15} - 4394628 p^{5} T^{16} + 34362828 p^{5} T^{17} + 13370033 p^{6} T^{18} + 2052636 p^{7} T^{19} + 216342 p^{8} T^{20} + 18234 p^{9} T^{21} + 1197 p^{10} T^{22} + 51 p^{11} T^{23} + p^{12} T^{24} \)
97 \( 1 - 48 T + 1326 T^{2} - 27506 T^{3} + 457320 T^{4} - 6561246 T^{5} + 84857617 T^{6} - 1022695011 T^{7} + 11895196383 T^{8} - 134109734294 T^{9} + 1470887073393 T^{10} - 15554728611663 T^{11} + 156712969708897 T^{12} - 15554728611663 p T^{13} + 1470887073393 p^{2} T^{14} - 134109734294 p^{3} T^{15} + 11895196383 p^{4} T^{16} - 1022695011 p^{5} T^{17} + 84857617 p^{6} T^{18} - 6561246 p^{7} T^{19} + 457320 p^{8} T^{20} - 27506 p^{9} T^{21} + 1326 p^{10} T^{22} - 48 p^{11} T^{23} + p^{12} T^{24} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.69053266882218623481876545239, −3.65349180541093577790132769535, −3.62192364191525133828972228113, −3.59479794811347143620665616144, −3.51221263816703722229489801130, −3.31700702797424029257380303677, −3.18837348496177616236648340716, −3.17072233000105987731955856039, −2.73065897231044341020928123777, −2.71664299695834535069410245037, −2.70370745430852939930921421962, −2.65760509903550317201465595517, −2.64712349851005546416659957992, −2.36365321909676315775877685512, −2.29588860622682684348711465207, −2.24806431147703686224251074466, −1.93202047470728197306742524262, −1.50713166035009154506417130709, −1.43556569145556678055849144973, −1.35973275009077066035019972678, −1.17125434997677926028711906738, −1.12287684533827572201688538925, −1.06425874539303324001085827736, −1.03555920838297663525483149733, −0.53240959774674558491690338395, 0.53240959774674558491690338395, 1.03555920838297663525483149733, 1.06425874539303324001085827736, 1.12287684533827572201688538925, 1.17125434997677926028711906738, 1.35973275009077066035019972678, 1.43556569145556678055849144973, 1.50713166035009154506417130709, 1.93202047470728197306742524262, 2.24806431147703686224251074466, 2.29588860622682684348711465207, 2.36365321909676315775877685512, 2.64712349851005546416659957992, 2.65760509903550317201465595517, 2.70370745430852939930921421962, 2.71664299695834535069410245037, 2.73065897231044341020928123777, 3.17072233000105987731955856039, 3.18837348496177616236648340716, 3.31700702797424029257380303677, 3.51221263816703722229489801130, 3.59479794811347143620665616144, 3.62192364191525133828972228113, 3.65349180541093577790132769535, 3.69053266882218623481876545239

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

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