Dirichlet series
| L(s) = 1 | + 3·2-s − 10·3-s − 2·4-s − 30·6-s − 14·8-s + 55·9-s − 16·11-s + 20·12-s + 13-s − 6·16-s + 14·17-s + 165·18-s − 26·19-s − 48·22-s + 7·23-s + 140·24-s + 3·26-s − 220·27-s − 14·29-s − 22·31-s + 20·32-s + 160·33-s + 42·34-s − 110·36-s + 2·37-s − 78·38-s − 10·39-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 5.77·3-s − 4-s − 12.2·6-s − 4.94·8-s + 55/3·9-s − 4.82·11-s + 5.77·12-s + 0.277·13-s − 3/2·16-s + 3.39·17-s + 38.8·18-s − 5.96·19-s − 10.2·22-s + 1.45·23-s + 28.5·24-s + 0.588·26-s − 42.3·27-s − 2.59·29-s − 3.95·31-s + 3.53·32-s + 27.8·33-s + 7.20·34-s − 18.3·36-s + 0.328·37-s − 12.6·38-s − 1.60·39-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{20} \cdot 47^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{20} \cdot 47^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{10} \cdot 5^{20} \cdot 47^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.12149\times 10^{14}\) |
| Root analytic conductor: | \(5.30539\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(10\) |
| Selberg data: | \((20,\ 3^{10} \cdot 5^{20} \cdot 47^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T )^{10} \) |
| 5 | \( 1 \) | |
| 47 | \( ( 1 - T )^{10} \) | |
| good | 2 | \( 1 - 3 T + 11 T^{2} - 25 T^{3} + 61 T^{4} - 117 T^{5} + 231 T^{6} - 389 T^{7} + 333 p T^{8} - 31 p^{5} T^{9} + 375 p^{2} T^{10} - 31 p^{6} T^{11} + 333 p^{3} T^{12} - 389 p^{3} T^{13} + 231 p^{4} T^{14} - 117 p^{5} T^{15} + 61 p^{6} T^{16} - 25 p^{7} T^{17} + 11 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) |
| 7 | \( 1 + 34 T^{2} + 20 T^{3} + 571 T^{4} + 592 T^{5} + 976 p T^{6} + 8084 T^{7} + 66797 T^{8} + 1468 p^{2} T^{9} + 527642 T^{10} + 1468 p^{3} T^{11} + 66797 p^{2} T^{12} + 8084 p^{3} T^{13} + 976 p^{5} T^{14} + 592 p^{5} T^{15} + 571 p^{6} T^{16} + 20 p^{7} T^{17} + 34 p^{8} T^{18} + p^{10} T^{20} \) | |
| 11 | \( 1 + 16 T + 175 T^{2} + 1412 T^{3} + 9513 T^{4} + 54620 T^{5} + 25244 p T^{6} + 1261140 T^{7} + 5208066 T^{8} + 19609932 T^{9} + 67907474 T^{10} + 19609932 p T^{11} + 5208066 p^{2} T^{12} + 1261140 p^{3} T^{13} + 25244 p^{5} T^{14} + 54620 p^{5} T^{15} + 9513 p^{6} T^{16} + 1412 p^{7} T^{17} + 175 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - T + 57 T^{2} + 2 T^{3} + 1514 T^{4} + 1812 T^{5} + 27473 T^{6} + 59541 T^{7} + 427775 T^{8} + 1074676 T^{9} + 5975236 T^{10} + 1074676 p T^{11} + 427775 p^{2} T^{12} + 59541 p^{3} T^{13} + 27473 p^{4} T^{14} + 1812 p^{5} T^{15} + 1514 p^{6} T^{16} + 2 p^{7} T^{17} + 57 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 14 T + 185 T^{2} - 1582 T^{3} + 12748 T^{4} - 82164 T^{5} + 507371 T^{6} - 2690346 T^{7} + 13787821 T^{8} - 62318878 T^{9} + 272710356 T^{10} - 62318878 p T^{11} + 13787821 p^{2} T^{12} - 2690346 p^{3} T^{13} + 507371 p^{4} T^{14} - 82164 p^{5} T^{15} + 12748 p^{6} T^{16} - 1582 p^{7} T^{17} + 185 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 26 T + 391 T^{2} + 4176 T^{3} + 98 p^{2} T^{4} + 13174 p T^{5} + 1538657 T^{6} + 8436918 T^{7} + 42321217 T^{8} + 198331362 T^{9} + 883869960 T^{10} + 198331362 p T^{11} + 42321217 p^{2} T^{12} + 8436918 p^{3} T^{13} + 1538657 p^{4} T^{14} + 13174 p^{6} T^{15} + 98 p^{8} T^{16} + 4176 p^{7} T^{17} + 391 p^{8} T^{18} + 26 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 - 7 T + 152 T^{2} - 1043 T^{3} + 11649 T^{4} - 73882 T^{5} + 588002 T^{6} - 3296040 T^{7} + 21277889 T^{8} - 103239199 T^{9} + 567937530 T^{10} - 103239199 p T^{11} + 21277889 p^{2} T^{12} - 3296040 p^{3} T^{13} + 588002 p^{4} T^{14} - 73882 p^{5} T^{15} + 11649 p^{6} T^{16} - 1043 p^{7} T^{17} + 152 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 14 T + 126 T^{2} + 832 T^{3} + 5123 T^{4} + 25496 T^{5} + 123156 T^{6} + 746962 T^{7} + 5683061 T^{8} + 36659224 T^{9} + 209999202 T^{10} + 36659224 p T^{11} + 5683061 p^{2} T^{12} + 746962 p^{3} T^{13} + 123156 p^{4} T^{14} + 25496 p^{5} T^{15} + 5123 p^{6} T^{16} + 832 p^{7} T^{17} + 126 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 22 T + 398 T^{2} + 4990 T^{3} + 55461 T^{4} + 510772 T^{5} + 4308236 T^{6} + 31928468 T^{7} + 220517218 T^{8} + 1371895096 T^{9} + 8019141820 T^{10} + 1371895096 p T^{11} + 220517218 p^{2} T^{12} + 31928468 p^{3} T^{13} + 4308236 p^{4} T^{14} + 510772 p^{5} T^{15} + 55461 p^{6} T^{16} + 4990 p^{7} T^{17} + 398 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 2 T + 155 T^{2} - 376 T^{3} + 11133 T^{4} - 34510 T^{5} + 527948 T^{6} - 2036746 T^{7} + 19987710 T^{8} - 89473678 T^{9} + 719073370 T^{10} - 89473678 p T^{11} + 19987710 p^{2} T^{12} - 2036746 p^{3} T^{13} + 527948 p^{4} T^{14} - 34510 p^{5} T^{15} + 11133 p^{6} T^{16} - 376 p^{7} T^{17} + 155 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 22 T + 501 T^{2} + 7326 T^{3} + 100624 T^{4} + 1111938 T^{5} + 11408051 T^{6} + 100977098 T^{7} + 830263515 T^{8} + 6049460352 T^{9} + 40998179640 T^{10} + 6049460352 p T^{11} + 830263515 p^{2} T^{12} + 100977098 p^{3} T^{13} + 11408051 p^{4} T^{14} + 1111938 p^{5} T^{15} + 100624 p^{6} T^{16} + 7326 p^{7} T^{17} + 501 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 11 T + 282 T^{2} + 1947 T^{3} + 719 p T^{4} + 129454 T^{5} + 1827912 T^{6} + 2707638 T^{7} + 68198418 T^{8} - 101851596 T^{9} + 2403900220 T^{10} - 101851596 p T^{11} + 68198418 p^{2} T^{12} + 2707638 p^{3} T^{13} + 1827912 p^{4} T^{14} + 129454 p^{5} T^{15} + 719 p^{7} T^{16} + 1947 p^{7} T^{17} + 282 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 - 22 T + 563 T^{2} - 8640 T^{3} + 130314 T^{4} - 1556998 T^{5} + 17454831 T^{6} - 171189974 T^{7} + 1548933415 T^{8} - 12791800922 T^{9} + 96878564336 T^{10} - 12791800922 p T^{11} + 1548933415 p^{2} T^{12} - 171189974 p^{3} T^{13} + 17454831 p^{4} T^{14} - 1556998 p^{5} T^{15} + 130314 p^{6} T^{16} - 8640 p^{7} T^{17} + 563 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 37 T + 879 T^{2} + 14390 T^{3} + 181782 T^{4} + 1724644 T^{5} + 11829823 T^{6} + 38200197 T^{7} - 301898285 T^{8} - 6583153976 T^{9} - 63179489668 T^{10} - 6583153976 p T^{11} - 301898285 p^{2} T^{12} + 38200197 p^{3} T^{13} + 11829823 p^{4} T^{14} + 1724644 p^{5} T^{15} + 181782 p^{6} T^{16} + 14390 p^{7} T^{17} + 879 p^{8} T^{18} + 37 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 25 T + 521 T^{2} + 6462 T^{3} + 75214 T^{4} + 643410 T^{5} + 6276399 T^{6} + 51901591 T^{7} + 531963431 T^{8} + 4363651120 T^{9} + 39088782668 T^{10} + 4363651120 p T^{11} + 531963431 p^{2} T^{12} + 51901591 p^{3} T^{13} + 6276399 p^{4} T^{14} + 643410 p^{5} T^{15} + 75214 p^{6} T^{16} + 6462 p^{7} T^{17} + 521 p^{8} T^{18} + 25 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 4 T + 330 T^{2} + 1800 T^{3} + 56785 T^{4} + 363820 T^{5} + 6897164 T^{6} + 46461796 T^{7} + 642113518 T^{8} + 4235655660 T^{9} + 47677530340 T^{10} + 4235655660 p T^{11} + 642113518 p^{2} T^{12} + 46461796 p^{3} T^{13} + 6897164 p^{4} T^{14} + 363820 p^{5} T^{15} + 56785 p^{6} T^{16} + 1800 p^{7} T^{17} + 330 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 27 T + 707 T^{2} + 11488 T^{3} + 177518 T^{4} + 2140228 T^{5} + 25006339 T^{6} + 249868629 T^{7} + 2470358887 T^{8} + 21864581174 T^{9} + 193406940432 T^{10} + 21864581174 p T^{11} + 2470358887 p^{2} T^{12} + 249868629 p^{3} T^{13} + 25006339 p^{4} T^{14} + 2140228 p^{5} T^{15} + 177518 p^{6} T^{16} + 11488 p^{7} T^{17} + 707 p^{8} T^{18} + 27 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - T + 368 T^{2} + 735 T^{3} + 64477 T^{4} + 267268 T^{5} + 8036612 T^{6} + 37838088 T^{7} + 827563034 T^{8} + 3372613510 T^{9} + 68683017944 T^{10} + 3372613510 p T^{11} + 827563034 p^{2} T^{12} + 37838088 p^{3} T^{13} + 8036612 p^{4} T^{14} + 267268 p^{5} T^{15} + 64477 p^{6} T^{16} + 735 p^{7} T^{17} + 368 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 5 T + 289 T^{2} - 98 T^{3} + 38597 T^{4} + 52709 T^{5} + 4867508 T^{6} + 1778561 T^{7} + 479144622 T^{8} + 518462451 T^{9} + 36466510990 T^{10} + 518462451 p T^{11} + 479144622 p^{2} T^{12} + 1778561 p^{3} T^{13} + 4867508 p^{4} T^{14} + 52709 p^{5} T^{15} + 38597 p^{6} T^{16} - 98 p^{7} T^{17} + 289 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 2 T + 468 T^{2} - 1662 T^{3} + 108301 T^{4} - 453712 T^{5} + 16895008 T^{6} - 71032448 T^{7} + 1969107386 T^{8} - 7873529892 T^{9} + 181398106456 T^{10} - 7873529892 p T^{11} + 1969107386 p^{2} T^{12} - 71032448 p^{3} T^{13} + 16895008 p^{4} T^{14} - 453712 p^{5} T^{15} + 108301 p^{6} T^{16} - 1662 p^{7} T^{17} + 468 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 9 T + 760 T^{2} - 6831 T^{3} + 268985 T^{4} - 2330694 T^{5} + 58424276 T^{6} - 470084782 T^{7} + 8627280510 T^{8} - 61832790316 T^{9} + 905150624408 T^{10} - 61832790316 p T^{11} + 8627280510 p^{2} T^{12} - 470084782 p^{3} T^{13} + 58424276 p^{4} T^{14} - 2330694 p^{5} T^{15} + 268985 p^{6} T^{16} - 6831 p^{7} T^{17} + 760 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 + 40 T + 1203 T^{2} + 26060 T^{3} + 495909 T^{4} + 8013764 T^{5} + 118300260 T^{6} + 1554179036 T^{7} + 18940428138 T^{8} + 209344682284 T^{9} + 2159616884338 T^{10} + 209344682284 p T^{11} + 18940428138 p^{2} T^{12} + 1554179036 p^{3} T^{13} + 118300260 p^{4} T^{14} + 8013764 p^{5} T^{15} + 495909 p^{6} T^{16} + 26060 p^{7} T^{17} + 1203 p^{8} T^{18} + 40 p^{9} T^{19} + p^{10} T^{20} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.50602768883336844838348511593, −3.49635718241382694524558806585, −3.49373666371956576995951423043, −3.47133596235339948933378791940, −3.14511984262275692621296484328, −3.01108706317753123568515569384, −2.87601961382559525812062471964, −2.87206771632797146647419503620, −2.84971835674577985769433866448, −2.54483973336813209396775839275, −2.40022823262089500577828197956, −2.39388657199994343600461266506, −2.27242840159609778459365468046, −2.18556626009350893630432512441, −1.92919746222043525426397491976, −1.91156799650231233859437154185, −1.89562189248047912932650678234, −1.69204557649914068247513798883, −1.49096638385107611570910252512, −1.27635961934406556473192303466, −1.27498432073636273691718216550, −1.25089707033932039898319058397, −1.20445986569006262176566468361, −1.09762008334160691404717546246, −0.963483021507229044700754029748, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.963483021507229044700754029748, 1.09762008334160691404717546246, 1.20445986569006262176566468361, 1.25089707033932039898319058397, 1.27498432073636273691718216550, 1.27635961934406556473192303466, 1.49096638385107611570910252512, 1.69204557649914068247513798883, 1.89562189248047912932650678234, 1.91156799650231233859437154185, 1.92919746222043525426397491976, 2.18556626009350893630432512441, 2.27242840159609778459365468046, 2.39388657199994343600461266506, 2.40022823262089500577828197956, 2.54483973336813209396775839275, 2.84971835674577985769433866448, 2.87206771632797146647419503620, 2.87601961382559525812062471964, 3.01108706317753123568515569384, 3.14511984262275692621296484328, 3.47133596235339948933378791940, 3.49373666371956576995951423043, 3.49635718241382694524558806585, 3.50602768883336844838348511593
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.