Properties

Degree $20$
Conductor $1.123\times 10^{39}$
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  + 2·3-s + 4·5-s − 8·9-s + 6·13-s + 8·15-s + 12·17-s + 8·19-s + 4·23-s − 9·25-s − 18·27-s + 2·29-s + 8·31-s − 2·37-s + 12·39-s + 10·41-s − 2·43-s − 32·45-s − 14·47-s + 24·51-s + 8·53-s + 16·57-s + 24·59-s + 14·61-s + 24·65-s − 8·67-s + 8·69-s + 10·71-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s − 8/3·9-s + 1.66·13-s + 2.06·15-s + 2.91·17-s + 1.83·19-s + 0.834·23-s − 9/5·25-s − 3.46·27-s + 0.371·29-s + 1.43·31-s − 0.328·37-s + 1.92·39-s + 1.56·41-s − 0.304·43-s − 4.77·45-s − 2.04·47-s + 3.36·51-s + 1.09·53-s + 2.11·57-s + 3.12·59-s + 1.79·61-s + 2.97·65-s − 0.977·67-s + 0.963·69-s + 1.18·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{20} \cdot 41^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{20} \cdot 41^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{20} \cdot 7^{20} \cdot 41^{10}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{8036} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{20} \cdot 7^{20} \cdot 41^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(343.9221924\)
\(L(\frac12)\) \(\approx\) \(343.9221924\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( ( 1 - T )^{10} \)
good3 \( 1 - 2 T + 4 p T^{2} - 22 T^{3} + 83 T^{4} - 130 T^{5} + 134 p T^{6} - 542 T^{7} + 1513 T^{8} - 206 p^{2} T^{9} + 4817 T^{10} - 206 p^{3} T^{11} + 1513 p^{2} T^{12} - 542 p^{3} T^{13} + 134 p^{5} T^{14} - 130 p^{5} T^{15} + 83 p^{6} T^{16} - 22 p^{7} T^{17} + 4 p^{9} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \)
5 \( 1 - 4 T + p^{2} T^{2} - 88 T^{3} + 326 T^{4} - 994 T^{5} + 2938 T^{6} - 7978 T^{7} + 4149 p T^{8} - 50112 T^{9} + 116858 T^{10} - 50112 p T^{11} + 4149 p^{3} T^{12} - 7978 p^{3} T^{13} + 2938 p^{4} T^{14} - 994 p^{5} T^{15} + 326 p^{6} T^{16} - 88 p^{7} T^{17} + p^{10} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \)
11 \( 1 + 60 T^{2} - 48 T^{3} + 1750 T^{4} - 2512 T^{5} + 3186 p T^{6} - 60128 T^{7} + 548345 T^{8} - 910432 T^{9} + 6806204 T^{10} - 910432 p T^{11} + 548345 p^{2} T^{12} - 60128 p^{3} T^{13} + 3186 p^{5} T^{14} - 2512 p^{5} T^{15} + 1750 p^{6} T^{16} - 48 p^{7} T^{17} + 60 p^{8} T^{18} + p^{10} T^{20} \)
13 \( 1 - 6 T + 77 T^{2} - 378 T^{3} + 212 p T^{4} - 11378 T^{5} + 63254 T^{6} - 1338 p^{2} T^{7} + 1083085 T^{8} - 3478468 T^{9} + 15235973 T^{10} - 3478468 p T^{11} + 1083085 p^{2} T^{12} - 1338 p^{5} T^{13} + 63254 p^{4} T^{14} - 11378 p^{5} T^{15} + 212 p^{7} T^{16} - 378 p^{7} T^{17} + 77 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \)
17 \( 1 - 12 T + 9 p T^{2} - 1250 T^{3} + 9866 T^{4} - 64080 T^{5} + 390146 T^{6} - 2122674 T^{7} + 10664977 T^{8} - 49516210 T^{9} + 211311741 T^{10} - 49516210 p T^{11} + 10664977 p^{2} T^{12} - 2122674 p^{3} T^{13} + 390146 p^{4} T^{14} - 64080 p^{5} T^{15} + 9866 p^{6} T^{16} - 1250 p^{7} T^{17} + 9 p^{9} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \)
19 \( 1 - 8 T + 142 T^{2} - 854 T^{3} + 8779 T^{4} - 41722 T^{5} + 324726 T^{6} - 1272034 T^{7} + 8468743 T^{8} - 28827300 T^{9} + 175736485 T^{10} - 28827300 p T^{11} + 8468743 p^{2} T^{12} - 1272034 p^{3} T^{13} + 324726 p^{4} T^{14} - 41722 p^{5} T^{15} + 8779 p^{6} T^{16} - 854 p^{7} T^{17} + 142 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \)
23 \( 1 - 4 T + 150 T^{2} - 472 T^{3} + 10995 T^{4} - 29014 T^{5} + 530388 T^{6} - 1210146 T^{7} + 18532865 T^{8} - 36844898 T^{9} + 487837739 T^{10} - 36844898 p T^{11} + 18532865 p^{2} T^{12} - 1210146 p^{3} T^{13} + 530388 p^{4} T^{14} - 29014 p^{5} T^{15} + 10995 p^{6} T^{16} - 472 p^{7} T^{17} + 150 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \)
29 \( 1 - 2 T + 169 T^{2} - 200 T^{3} + 13298 T^{4} - 8180 T^{5} + 689398 T^{6} - 273872 T^{7} + 27573885 T^{8} - 10844658 T^{9} + 890329250 T^{10} - 10844658 p T^{11} + 27573885 p^{2} T^{12} - 273872 p^{3} T^{13} + 689398 p^{4} T^{14} - 8180 p^{5} T^{15} + 13298 p^{6} T^{16} - 200 p^{7} T^{17} + 169 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \)
31 \( 1 - 8 T + 173 T^{2} - 1172 T^{3} + 15518 T^{4} - 93450 T^{5} + 940146 T^{6} - 5050870 T^{7} + 42210833 T^{8} - 203386776 T^{9} + 1475895842 T^{10} - 203386776 p T^{11} + 42210833 p^{2} T^{12} - 5050870 p^{3} T^{13} + 940146 p^{4} T^{14} - 93450 p^{5} T^{15} + 15518 p^{6} T^{16} - 1172 p^{7} T^{17} + 173 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \)
37 \( 1 + 2 T + 207 T^{2} + 440 T^{3} + 23279 T^{4} + 49382 T^{5} + 1758529 T^{6} + 3562350 T^{7} + 97635520 T^{8} + 181110266 T^{9} + 4119614704 T^{10} + 181110266 p T^{11} + 97635520 p^{2} T^{12} + 3562350 p^{3} T^{13} + 1758529 p^{4} T^{14} + 49382 p^{5} T^{15} + 23279 p^{6} T^{16} + 440 p^{7} T^{17} + 207 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \)
43 \( 1 + 2 T + 221 T^{2} + 468 T^{3} + 25826 T^{4} + 55060 T^{5} + 2056468 T^{6} + 4339622 T^{7} + 123599993 T^{8} + 247838392 T^{9} + 5913403015 T^{10} + 247838392 p T^{11} + 123599993 p^{2} T^{12} + 4339622 p^{3} T^{13} + 2056468 p^{4} T^{14} + 55060 p^{5} T^{15} + 25826 p^{6} T^{16} + 468 p^{7} T^{17} + 221 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \)
47 \( 1 + 14 T + 347 T^{2} + 4286 T^{3} + 61213 T^{4} + 13504 p T^{5} + 6842191 T^{6} + 59787324 T^{7} + 525899550 T^{8} + 3920374812 T^{9} + 28988297588 T^{10} + 3920374812 p T^{11} + 525899550 p^{2} T^{12} + 59787324 p^{3} T^{13} + 6842191 p^{4} T^{14} + 13504 p^{6} T^{15} + 61213 p^{6} T^{16} + 4286 p^{7} T^{17} + 347 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \)
53 \( 1 - 8 T + 245 T^{2} - 1196 T^{3} + 23954 T^{4} - 58632 T^{5} + 1591642 T^{6} - 2732816 T^{7} + 116936509 T^{8} - 293998276 T^{9} + 7553425474 T^{10} - 293998276 p T^{11} + 116936509 p^{2} T^{12} - 2732816 p^{3} T^{13} + 1591642 p^{4} T^{14} - 58632 p^{5} T^{15} + 23954 p^{6} T^{16} - 1196 p^{7} T^{17} + 245 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \)
59 \( 1 - 24 T + 629 T^{2} - 8776 T^{3} + 126918 T^{4} - 1161538 T^{5} + 11387310 T^{6} - 65476022 T^{7} + 482347017 T^{8} - 1374681036 T^{9} + 15949341018 T^{10} - 1374681036 p T^{11} + 482347017 p^{2} T^{12} - 65476022 p^{3} T^{13} + 11387310 p^{4} T^{14} - 1161538 p^{5} T^{15} + 126918 p^{6} T^{16} - 8776 p^{7} T^{17} + 629 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \)
61 \( 1 - 14 T + 528 T^{2} - 6362 T^{3} + 130430 T^{4} - 1349402 T^{5} + 19744598 T^{6} - 175069242 T^{7} + 2021257345 T^{8} - 15296907412 T^{9} + 145932869524 T^{10} - 15296907412 p T^{11} + 2021257345 p^{2} T^{12} - 175069242 p^{3} T^{13} + 19744598 p^{4} T^{14} - 1349402 p^{5} T^{15} + 130430 p^{6} T^{16} - 6362 p^{7} T^{17} + 528 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \)
67 \( 1 + 8 T + 449 T^{2} + 2320 T^{3} + 87078 T^{4} + 265904 T^{5} + 10436462 T^{6} + 16581400 T^{7} + 935678857 T^{8} + 796918512 T^{9} + 68666864514 T^{10} + 796918512 p T^{11} + 935678857 p^{2} T^{12} + 16581400 p^{3} T^{13} + 10436462 p^{4} T^{14} + 265904 p^{5} T^{15} + 87078 p^{6} T^{16} + 2320 p^{7} T^{17} + 449 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \)
71 \( 1 - 10 T + 444 T^{2} - 4262 T^{3} + 86730 T^{4} - 810086 T^{5} + 9918442 T^{6} - 94008490 T^{7} + 789911765 T^{8} - 7996076064 T^{9} + 55295925748 T^{10} - 7996076064 p T^{11} + 789911765 p^{2} T^{12} - 94008490 p^{3} T^{13} + 9918442 p^{4} T^{14} - 810086 p^{5} T^{15} + 86730 p^{6} T^{16} - 4262 p^{7} T^{17} + 444 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \)
73 \( 1 - 44 T + 1220 T^{2} - 24508 T^{3} + 403542 T^{4} - 5667300 T^{5} + 70965058 T^{6} - 804708756 T^{7} + 8403572217 T^{8} - 80839769728 T^{9} + 719421847316 T^{10} - 80839769728 p T^{11} + 8403572217 p^{2} T^{12} - 804708756 p^{3} T^{13} + 70965058 p^{4} T^{14} - 5667300 p^{5} T^{15} + 403542 p^{6} T^{16} - 24508 p^{7} T^{17} + 1220 p^{8} T^{18} - 44 p^{9} T^{19} + p^{10} T^{20} \)
79 \( 1 - 10 T + 614 T^{2} - 5902 T^{3} + 179069 T^{4} - 1630352 T^{5} + 32727400 T^{6} - 275265280 T^{7} + 4153988066 T^{8} - 31232651988 T^{9} + 382732416868 T^{10} - 31232651988 p T^{11} + 4153988066 p^{2} T^{12} - 275265280 p^{3} T^{13} + 32727400 p^{4} T^{14} - 1630352 p^{5} T^{15} + 179069 p^{6} T^{16} - 5902 p^{7} T^{17} + 614 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \)
83 \( 1 - 20 T + 528 T^{2} - 7496 T^{3} + 121918 T^{4} - 1442276 T^{5} + 18880926 T^{6} - 198399444 T^{7} + 2251070337 T^{8} - 21080323924 T^{9} + 210699872260 T^{10} - 21080323924 p T^{11} + 2251070337 p^{2} T^{12} - 198399444 p^{3} T^{13} + 18880926 p^{4} T^{14} - 1442276 p^{5} T^{15} + 121918 p^{6} T^{16} - 7496 p^{7} T^{17} + 528 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} \)
89 \( 1 - 6 T + 2 p T^{2} - 40 T^{3} + 19025 T^{4} + 13312 T^{5} + 2644880 T^{6} - 1951728 T^{7} + 298984707 T^{8} + 249612878 T^{9} + 23616419385 T^{10} + 249612878 p T^{11} + 298984707 p^{2} T^{12} - 1951728 p^{3} T^{13} + 2644880 p^{4} T^{14} + 13312 p^{5} T^{15} + 19025 p^{6} T^{16} - 40 p^{7} T^{17} + 2 p^{9} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \)
97 \( 1 - 46 T + 1292 T^{2} - 26530 T^{3} + 430839 T^{4} - 5783946 T^{5} + 66062922 T^{6} - 659196450 T^{7} + 5979674797 T^{8} - 52593327006 T^{9} + 491620931417 T^{10} - 52593327006 p T^{11} + 5979674797 p^{2} T^{12} - 659196450 p^{3} T^{13} + 66062922 p^{4} T^{14} - 5783946 p^{5} T^{15} + 430839 p^{6} T^{16} - 26530 p^{7} T^{17} + 1292 p^{8} T^{18} - 46 p^{9} T^{19} + p^{10} T^{20} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.63739401406014886625076782712, −2.59092530608944814438035804628, −2.58093579966122292100818640842, −2.47637288781717771262369656558, −2.38678231195116075863502313380, −2.12993301204563500281882057720, −2.02988123158199272133510944566, −1.95351335419280275535001109835, −1.89766137736111020700337289517, −1.87372683082080326141060140420, −1.85501560562082013339845883393, −1.68725511538766507601537673522, −1.68116678599495515567825494601, −1.48255044269548766941303595874, −1.32570060818648344180165015796, −1.21009869364004268598471535734, −1.04276472495611986606309500530, −0.901786839796998459657247317865, −0.894181869907808049522342404079, −0.828744725805483586060587084912, −0.60292247404132829916732644767, −0.53637482350666364807205846753, −0.46694016863208501644468958117, −0.42589246309817497616644209910, −0.38822452387893386965980257193, 0.38822452387893386965980257193, 0.42589246309817497616644209910, 0.46694016863208501644468958117, 0.53637482350666364807205846753, 0.60292247404132829916732644767, 0.828744725805483586060587084912, 0.894181869907808049522342404079, 0.901786839796998459657247317865, 1.04276472495611986606309500530, 1.21009869364004268598471535734, 1.32570060818648344180165015796, 1.48255044269548766941303595874, 1.68116678599495515567825494601, 1.68725511538766507601537673522, 1.85501560562082013339845883393, 1.87372683082080326141060140420, 1.89766137736111020700337289517, 1.95351335419280275535001109835, 2.02988123158199272133510944566, 2.12993301204563500281882057720, 2.38678231195116075863502313380, 2.47637288781717771262369656558, 2.58093579966122292100818640842, 2.59092530608944814438035804628, 2.63739401406014886625076782712

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

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