Dirichlet series
| L(s) = 1 | − 2·2-s − 10·5-s + 10·7-s + 2·8-s + 20·10-s − 9·11-s + 14·13-s − 20·14-s − 3·16-s − 8·17-s + 13·19-s + 18·22-s + 10·23-s + 55·25-s − 28·26-s − 10·29-s + 8·31-s + 9·32-s + 16·34-s − 100·35-s + 8·37-s − 26·38-s − 20·40-s + 5·41-s + 4·43-s − 20·46-s − 47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 4.47·5-s + 3.77·7-s + 0.707·8-s + 6.32·10-s − 2.71·11-s + 3.88·13-s − 5.34·14-s − 3/4·16-s − 1.94·17-s + 2.98·19-s + 3.83·22-s + 2.08·23-s + 11·25-s − 5.49·26-s − 1.85·29-s + 1.43·31-s + 1.59·32-s + 2.74·34-s − 16.9·35-s + 1.31·37-s − 4.21·38-s − 3.16·40-s + 0.780·41-s + 0.609·43-s − 2.94·46-s − 0.145·47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{10} \cdot 7^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{10} \cdot 7^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{20} \cdot 5^{10} \cdot 7^{10} \cdot 23^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.19908\times 10^{17}\) |
| Root analytic conductor: | \(7.60602\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{20} \cdot 5^{10} \cdot 7^{10} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(26.06785408\) |
| \(L(\frac12)\) | \(\approx\) | \(26.06785408\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 + T )^{10} \) | |
| 7 | \( ( 1 - T )^{10} \) | |
| 23 | \( ( 1 - T )^{10} \) | |
| good | 2 | \( 1 + p T + p^{2} T^{2} + 3 p T^{3} + 11 T^{4} + 11 T^{5} + p^{4} T^{6} + 5 p T^{7} + p^{2} T^{8} - 23 T^{9} - 3 p^{2} T^{10} - 23 p T^{11} + p^{4} T^{12} + 5 p^{4} T^{13} + p^{8} T^{14} + 11 p^{5} T^{15} + 11 p^{6} T^{16} + 3 p^{8} T^{17} + p^{10} T^{18} + p^{10} T^{19} + p^{10} T^{20} \) |
| 11 | \( 1 + 9 T + 79 T^{2} + 476 T^{3} + 2494 T^{4} + 11434 T^{5} + 45668 T^{6} + 169878 T^{7} + 587655 T^{8} + 1940263 T^{9} + 6537662 T^{10} + 1940263 p T^{11} + 587655 p^{2} T^{12} + 169878 p^{3} T^{13} + 45668 p^{4} T^{14} + 11434 p^{5} T^{15} + 2494 p^{6} T^{16} + 476 p^{7} T^{17} + 79 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - 14 T + 153 T^{2} - 1190 T^{3} + 8186 T^{4} - 47410 T^{5} + 252810 T^{6} - 1197928 T^{7} + 5312997 T^{8} - 21350378 T^{9} + 80685370 T^{10} - 21350378 p T^{11} + 5312997 p^{2} T^{12} - 1197928 p^{3} T^{13} + 252810 p^{4} T^{14} - 47410 p^{5} T^{15} + 8186 p^{6} T^{16} - 1190 p^{7} T^{17} + 153 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 8 T + 83 T^{2} + 504 T^{3} + 3150 T^{4} + 13908 T^{5} + 62918 T^{6} + 200324 T^{7} + 711201 T^{8} + 1576424 T^{9} + 7526894 T^{10} + 1576424 p T^{11} + 711201 p^{2} T^{12} + 200324 p^{3} T^{13} + 62918 p^{4} T^{14} + 13908 p^{5} T^{15} + 3150 p^{6} T^{16} + 504 p^{7} T^{17} + 83 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 13 T + 175 T^{2} - 1388 T^{3} + 10932 T^{4} - 62788 T^{5} + 365384 T^{6} - 1666610 T^{7} + 8163143 T^{8} - 33012821 T^{9} + 155689402 T^{10} - 33012821 p T^{11} + 8163143 p^{2} T^{12} - 1666610 p^{3} T^{13} + 365384 p^{4} T^{14} - 62788 p^{5} T^{15} + 10932 p^{6} T^{16} - 1388 p^{7} T^{17} + 175 p^{8} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 10 T + 114 T^{2} + 582 T^{3} + 5 p^{2} T^{4} + 19124 T^{5} + 137832 T^{6} + 700508 T^{7} + 4249242 T^{8} + 22444344 T^{9} + 124107660 T^{10} + 22444344 p T^{11} + 4249242 p^{2} T^{12} + 700508 p^{3} T^{13} + 137832 p^{4} T^{14} + 19124 p^{5} T^{15} + 5 p^{8} T^{16} + 582 p^{7} T^{17} + 114 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 8 T + 142 T^{2} - 950 T^{3} + 10373 T^{4} - 56814 T^{5} + 505400 T^{6} - 77418 p T^{7} + 19212346 T^{8} - 83120406 T^{9} + 632997492 T^{10} - 83120406 p T^{11} + 19212346 p^{2} T^{12} - 77418 p^{4} T^{13} + 505400 p^{4} T^{14} - 56814 p^{5} T^{15} + 10373 p^{6} T^{16} - 950 p^{7} T^{17} + 142 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 8 T + 151 T^{2} - 524 T^{3} + 5198 T^{4} + 15842 T^{5} - 66576 T^{6} + 1349300 T^{7} - 1068939 T^{8} - 9374578 T^{9} + 206799786 T^{10} - 9374578 p T^{11} - 1068939 p^{2} T^{12} + 1349300 p^{3} T^{13} - 66576 p^{4} T^{14} + 15842 p^{5} T^{15} + 5198 p^{6} T^{16} - 524 p^{7} T^{17} + 151 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 5 T + 203 T^{2} - 1546 T^{3} + 22204 T^{4} - 196122 T^{5} + 1797520 T^{6} - 14846372 T^{7} + 113600075 T^{8} - 791597539 T^{9} + 5419362490 T^{10} - 791597539 p T^{11} + 113600075 p^{2} T^{12} - 14846372 p^{3} T^{13} + 1797520 p^{4} T^{14} - 196122 p^{5} T^{15} + 22204 p^{6} T^{16} - 1546 p^{7} T^{17} + 203 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 4 T + 141 T^{2} - 598 T^{3} + 12740 T^{4} - 56730 T^{5} + 855842 T^{6} - 3945194 T^{7} + 47903787 T^{8} - 216044866 T^{9} + 2255135202 T^{10} - 216044866 p T^{11} + 47903787 p^{2} T^{12} - 3945194 p^{3} T^{13} + 855842 p^{4} T^{14} - 56730 p^{5} T^{15} + 12740 p^{6} T^{16} - 598 p^{7} T^{17} + 141 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 + T + 150 T^{2} + 271 T^{3} + 11197 T^{4} + 22764 T^{5} + 687432 T^{6} + 1554580 T^{7} + 41388514 T^{8} + 111753414 T^{9} + 2156581188 T^{10} + 111753414 p T^{11} + 41388514 p^{2} T^{12} + 1554580 p^{3} T^{13} + 687432 p^{4} T^{14} + 22764 p^{5} T^{15} + 11197 p^{6} T^{16} + 271 p^{7} T^{17} + 150 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 9 T + 348 T^{2} + 2207 T^{3} + 51283 T^{4} + 229594 T^{5} + 4524684 T^{6} + 14154190 T^{7} + 292520500 T^{8} + 685048068 T^{9} + 16203221808 T^{10} + 685048068 p T^{11} + 292520500 p^{2} T^{12} + 14154190 p^{3} T^{13} + 4524684 p^{4} T^{14} + 229594 p^{5} T^{15} + 51283 p^{6} T^{16} + 2207 p^{7} T^{17} + 348 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 17 T + 459 T^{2} - 6050 T^{3} + 93072 T^{4} - 996748 T^{5} + 11375958 T^{6} - 103115138 T^{7} + 973715629 T^{8} - 7747102503 T^{9} + 64235219666 T^{10} - 7747102503 p T^{11} + 973715629 p^{2} T^{12} - 103115138 p^{3} T^{13} + 11375958 p^{4} T^{14} - 996748 p^{5} T^{15} + 93072 p^{6} T^{16} - 6050 p^{7} T^{17} + 459 p^{8} T^{18} - 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 19 T + 457 T^{2} - 5386 T^{3} + 75530 T^{4} - 591810 T^{5} + 5842398 T^{6} - 25230132 T^{7} + 200011785 T^{8} + 156295515 T^{9} + 4202354186 T^{10} + 156295515 p T^{11} + 200011785 p^{2} T^{12} - 25230132 p^{3} T^{13} + 5842398 p^{4} T^{14} - 591810 p^{5} T^{15} + 75530 p^{6} T^{16} - 5386 p^{7} T^{17} + 457 p^{8} T^{18} - 19 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 431 T^{2} + 622 T^{3} + 89760 T^{4} + 210488 T^{5} + 12325466 T^{6} + 32598276 T^{7} + 1245545215 T^{8} + 3136408834 T^{9} + 95569993870 T^{10} + 3136408834 p T^{11} + 1245545215 p^{2} T^{12} + 32598276 p^{3} T^{13} + 12325466 p^{4} T^{14} + 210488 p^{5} T^{15} + 89760 p^{6} T^{16} + 622 p^{7} T^{17} + 431 p^{8} T^{18} + p^{10} T^{20} \) | |
| 71 | \( 1 + 478 T^{2} + 628 T^{3} + 106293 T^{4} + 288820 T^{5} + 14782296 T^{6} + 58381828 T^{7} + 1478820474 T^{8} + 6728928388 T^{9} + 116359109396 T^{10} + 6728928388 p T^{11} + 1478820474 p^{2} T^{12} + 58381828 p^{3} T^{13} + 14782296 p^{4} T^{14} + 288820 p^{5} T^{15} + 106293 p^{6} T^{16} + 628 p^{7} T^{17} + 478 p^{8} T^{18} + p^{10} T^{20} \) | |
| 73 | \( 1 - 6 T + 409 T^{2} - 2818 T^{3} + 84910 T^{4} - 607222 T^{5} + 11973690 T^{6} - 82702900 T^{7} + 1264224853 T^{8} - 8064860894 T^{9} + 103943669138 T^{10} - 8064860894 p T^{11} + 1264224853 p^{2} T^{12} - 82702900 p^{3} T^{13} + 11973690 p^{4} T^{14} - 607222 p^{5} T^{15} + 84910 p^{6} T^{16} - 2818 p^{7} T^{17} + 409 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 32 T + 872 T^{2} - 16414 T^{3} + 282621 T^{4} - 4049510 T^{5} + 54037376 T^{6} - 634017246 T^{7} + 6989731362 T^{8} - 69309848118 T^{9} + 647861983920 T^{10} - 69309848118 p T^{11} + 6989731362 p^{2} T^{12} - 634017246 p^{3} T^{13} + 54037376 p^{4} T^{14} - 4049510 p^{5} T^{15} + 282621 p^{6} T^{16} - 16414 p^{7} T^{17} + 872 p^{8} T^{18} - 32 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 2 T + 345 T^{2} - 1144 T^{3} + 71706 T^{4} - 287782 T^{5} + 10441858 T^{6} - 46610930 T^{7} + 1183484661 T^{8} - 5292145946 T^{9} + 108460368554 T^{10} - 5292145946 p T^{11} + 1183484661 p^{2} T^{12} - 46610930 p^{3} T^{13} + 10441858 p^{4} T^{14} - 287782 p^{5} T^{15} + 71706 p^{6} T^{16} - 1144 p^{7} T^{17} + 345 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 10 T + 442 T^{2} + 4198 T^{3} + 109757 T^{4} + 1017532 T^{5} + 18853464 T^{6} + 164641476 T^{7} + 2428740914 T^{8} + 19538936320 T^{9} + 244094619996 T^{10} + 19538936320 p T^{11} + 2428740914 p^{2} T^{12} + 164641476 p^{3} T^{13} + 18853464 p^{4} T^{14} + 1017532 p^{5} T^{15} + 109757 p^{6} T^{16} + 4198 p^{7} T^{17} + 442 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 18 T + 450 T^{2} - 5998 T^{3} + 94293 T^{4} - 1055100 T^{5} + 13939208 T^{6} - 145410820 T^{7} + 1788217578 T^{8} - 17467758272 T^{9} + 194551404908 T^{10} - 17467758272 p T^{11} + 1788217578 p^{2} T^{12} - 145410820 p^{3} T^{13} + 13939208 p^{4} T^{14} - 1055100 p^{5} T^{15} + 94293 p^{6} T^{16} - 5998 p^{7} T^{17} + 450 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 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
−2.82365041532014148409962210617, −2.59001794959204712796517683416, −2.45195118606173719299496966482, −2.37441066961477263594425257723, −2.31242585207729911910147839369, −2.20244352299435158040804561276, −2.17882173302200824541150685733, −1.99962195929515961368911515568, −1.86847155803201193568173795472, −1.73613295433216404537305626726, −1.70195437636835559569697146870, −1.63735547218288848738425552127, −1.43749903475990619842336988150, −1.32385003995335249974383496825, −1.31436046476808226115444145952, −1.17201089413129127214519753449, −0.939222666250022427750556303901, −0.801726164730220970000361149343, −0.72054799008290791667783113600, −0.71855462920444387213485073510, −0.70606634863645320267851531312, −0.54013221939296004569194108611, −0.43005956756673183494651018582, −0.38844766845993852251686661413, −0.37632064324730146130386040967, 0.37632064324730146130386040967, 0.38844766845993852251686661413, 0.43005956756673183494651018582, 0.54013221939296004569194108611, 0.70606634863645320267851531312, 0.71855462920444387213485073510, 0.72054799008290791667783113600, 0.801726164730220970000361149343, 0.939222666250022427750556303901, 1.17201089413129127214519753449, 1.31436046476808226115444145952, 1.32385003995335249974383496825, 1.43749903475990619842336988150, 1.63735547218288848738425552127, 1.70195437636835559569697146870, 1.73613295433216404537305626726, 1.86847155803201193568173795472, 1.99962195929515961368911515568, 2.17882173302200824541150685733, 2.20244352299435158040804561276, 2.31242585207729911910147839369, 2.37441066961477263594425257723, 2.45195118606173719299496966482, 2.59001794959204712796517683416, 2.82365041532014148409962210617
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.