Dirichlet series
| L(s) = 1 | + 3·2-s + 10·3-s + 2·4-s + 5·5-s + 30·6-s − 2·8-s + 55·9-s + 15·10-s + 8·11-s + 20·12-s + 50·15-s + 11·17-s + 165·18-s − 19-s + 10·20-s + 24·22-s + 10·23-s − 20·24-s − 2·25-s + 220·27-s + 22·29-s + 150·30-s + 3·31-s + 6·32-s + 80·33-s + 33·34-s + 110·36-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 5.77·3-s + 4-s + 2.23·5-s + 12.2·6-s − 0.707·8-s + 55/3·9-s + 4.74·10-s + 2.41·11-s + 5.77·12-s + 12.9·15-s + 2.66·17-s + 38.8·18-s − 0.229·19-s + 2.23·20-s + 5.11·22-s + 2.08·23-s − 4.08·24-s − 2/5·25-s + 42.3·27-s + 4.08·29-s + 27.3·30-s + 0.538·31-s + 1.06·32-s + 13.9·33-s + 5.65·34-s + 55/3·36-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 7^{20} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 7^{20} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{10} \cdot 7^{20} \cdot 23^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.05694\times 10^{14}\) |
| Root analytic conductor: | \(5.19590\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{10} \cdot 7^{20} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(11518.79770\) |
| \(L(\frac12)\) | \(\approx\) | \(11518.79770\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{10} \) |
| 7 | \( 1 \) | |
| 23 | \( ( 1 - T )^{10} \) | |
| good | 2 | \( 1 - 3 T + 7 T^{2} - 13 T^{3} + 19 T^{4} - 23 T^{5} + 23 T^{6} - 15 T^{7} + p^{3} T^{8} - p T^{9} - 5 p T^{10} - p^{2} T^{11} + p^{5} T^{12} - 15 p^{3} T^{13} + 23 p^{4} T^{14} - 23 p^{5} T^{15} + 19 p^{6} T^{16} - 13 p^{7} T^{17} + 7 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) |
| 5 | \( 1 - p T + 27 T^{2} - 74 T^{3} + 226 T^{4} - 406 T^{5} + 961 T^{6} - 943 T^{7} + 1589 T^{8} + 3414 T^{9} - 3906 T^{10} + 3414 p T^{11} + 1589 p^{2} T^{12} - 943 p^{3} T^{13} + 961 p^{4} T^{14} - 406 p^{5} T^{15} + 226 p^{6} T^{16} - 74 p^{7} T^{17} + 27 p^{8} T^{18} - p^{10} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - 8 T + 86 T^{2} - 536 T^{3} + 3453 T^{4} - 17656 T^{5} + 87096 T^{6} - 373496 T^{7} + 1531194 T^{8} - 5591440 T^{9} + 1782764 p T^{10} - 5591440 p T^{11} + 1531194 p^{2} T^{12} - 373496 p^{3} T^{13} + 87096 p^{4} T^{14} - 17656 p^{5} T^{15} + 3453 p^{6} T^{16} - 536 p^{7} T^{17} + 86 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + 73 T^{2} - 32 T^{3} + 2827 T^{4} - 1656 T^{5} + 74194 T^{6} - 47428 T^{7} + 1433473 T^{8} - 67856 p T^{9} + 1632147 p T^{10} - 67856 p^{2} T^{11} + 1433473 p^{2} T^{12} - 47428 p^{3} T^{13} + 74194 p^{4} T^{14} - 1656 p^{5} T^{15} + 2827 p^{6} T^{16} - 32 p^{7} T^{17} + 73 p^{8} T^{18} + p^{10} T^{20} \) | |
| 17 | \( 1 - 11 T + 129 T^{2} - 896 T^{3} + 6638 T^{4} - 37500 T^{5} + 225939 T^{6} - 1109875 T^{7} + 5679407 T^{8} - 24338740 T^{9} + 6399594 p T^{10} - 24338740 p T^{11} + 5679407 p^{2} T^{12} - 1109875 p^{3} T^{13} + 225939 p^{4} T^{14} - 37500 p^{5} T^{15} + 6638 p^{6} T^{16} - 896 p^{7} T^{17} + 129 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + T + 87 T^{2} - 92 T^{3} + 3939 T^{4} - 7517 T^{5} + 143282 T^{6} - 261081 T^{7} + 3922252 T^{8} - 7292197 T^{9} + 81792730 T^{10} - 7292197 p T^{11} + 3922252 p^{2} T^{12} - 261081 p^{3} T^{13} + 143282 p^{4} T^{14} - 7517 p^{5} T^{15} + 3939 p^{6} T^{16} - 92 p^{7} T^{17} + 87 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - 22 T + 396 T^{2} - 5028 T^{3} + 56213 T^{4} - 523954 T^{5} + 4431344 T^{6} - 32918302 T^{7} + 225039538 T^{8} - 47551366 p T^{9} + 7821784840 T^{10} - 47551366 p^{2} T^{11} + 225039538 p^{2} T^{12} - 32918302 p^{3} T^{13} + 4431344 p^{4} T^{14} - 523954 p^{5} T^{15} + 56213 p^{6} T^{16} - 5028 p^{7} T^{17} + 396 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 3 T + 151 T^{2} - 498 T^{3} + 11467 T^{4} - 45009 T^{5} + 576396 T^{6} - 86747 p T^{7} + 22159156 T^{8} - 114169875 T^{9} + 725780426 T^{10} - 114169875 p T^{11} + 22159156 p^{2} T^{12} - 86747 p^{4} T^{13} + 576396 p^{4} T^{14} - 45009 p^{5} T^{15} + 11467 p^{6} T^{16} - 498 p^{7} T^{17} + 151 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 3 T + 161 T^{2} + 278 T^{3} + 14485 T^{4} + 20569 T^{5} + 956036 T^{6} + 1132127 T^{7} + 48764866 T^{8} + 51284227 T^{9} + 2012183766 T^{10} + 51284227 p T^{11} + 48764866 p^{2} T^{12} + 1132127 p^{3} T^{13} + 956036 p^{4} T^{14} + 20569 p^{5} T^{15} + 14485 p^{6} T^{16} + 278 p^{7} T^{17} + 161 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 26 T + 512 T^{2} - 6720 T^{3} + 76013 T^{4} - 685282 T^{5} + 5667904 T^{6} - 40070790 T^{7} + 276322178 T^{8} - 1731175150 T^{9} + 11388208416 T^{10} - 1731175150 p T^{11} + 276322178 p^{2} T^{12} - 40070790 p^{3} T^{13} + 5667904 p^{4} T^{14} - 685282 p^{5} T^{15} + 76013 p^{6} T^{16} - 6720 p^{7} T^{17} + 512 p^{8} T^{18} - 26 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 27 T + 581 T^{2} - 9052 T^{3} + 120253 T^{4} - 1359081 T^{5} + 13645946 T^{6} - 122446777 T^{7} + 999237334 T^{8} - 7437857957 T^{9} + 50915042966 T^{10} - 7437857957 p T^{11} + 999237334 p^{2} T^{12} - 122446777 p^{3} T^{13} + 13645946 p^{4} T^{14} - 1359081 p^{5} T^{15} + 120253 p^{6} T^{16} - 9052 p^{7} T^{17} + 581 p^{8} T^{18} - 27 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 + 11 T + 339 T^{2} + 4008 T^{3} + 58016 T^{4} + 644720 T^{5} + 6567981 T^{6} + 61877523 T^{7} + 522453455 T^{8} + 4033587116 T^{9} + 29290339072 T^{10} + 4033587116 p T^{11} + 522453455 p^{2} T^{12} + 61877523 p^{3} T^{13} + 6567981 p^{4} T^{14} + 644720 p^{5} T^{15} + 58016 p^{6} T^{16} + 4008 p^{7} T^{17} + 339 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 - 5 T + 227 T^{2} - 246 T^{3} + 24762 T^{4} + 17790 T^{5} + 2290301 T^{6} + 2254795 T^{7} + 165363027 T^{8} + 249573438 T^{9} + 9291376356 T^{10} + 249573438 p T^{11} + 165363027 p^{2} T^{12} + 2254795 p^{3} T^{13} + 2290301 p^{4} T^{14} + 17790 p^{5} T^{15} + 24762 p^{6} T^{16} - 246 p^{7} T^{17} + 227 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 10 T + 424 T^{2} - 3184 T^{3} + 80053 T^{4} - 463650 T^{5} + 158136 p T^{6} - 43041298 T^{7} + 783568482 T^{8} - 3035519534 T^{9} + 51542536320 T^{10} - 3035519534 p T^{11} + 783568482 p^{2} T^{12} - 43041298 p^{3} T^{13} + 158136 p^{5} T^{14} - 463650 p^{5} T^{15} + 80053 p^{6} T^{16} - 3184 p^{7} T^{17} + 424 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 22 T + 714 T^{2} + 11302 T^{3} + 210533 T^{4} + 2605624 T^{5} + 35344744 T^{6} + 356013880 T^{7} + 3820584514 T^{8} + 31888404540 T^{9} + 280566644636 T^{10} + 31888404540 p T^{11} + 3820584514 p^{2} T^{12} + 356013880 p^{3} T^{13} + 35344744 p^{4} T^{14} + 2605624 p^{5} T^{15} + 210533 p^{6} T^{16} + 11302 p^{7} T^{17} + 714 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 2 T + 483 T^{2} + 862 T^{3} + 111131 T^{4} + 167714 T^{5} + 16179750 T^{6} + 20077692 T^{7} + 1667956063 T^{8} + 1723051688 T^{9} + 128324914779 T^{10} + 1723051688 p T^{11} + 1667956063 p^{2} T^{12} + 20077692 p^{3} T^{13} + 16179750 p^{4} T^{14} + 167714 p^{5} T^{15} + 111131 p^{6} T^{16} + 862 p^{7} T^{17} + 483 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 27 T + 637 T^{2} - 11032 T^{3} + 170394 T^{4} - 2275812 T^{5} + 27918611 T^{6} - 308350531 T^{7} + 3166296871 T^{8} - 29776947596 T^{9} + 261349838452 T^{10} - 29776947596 p T^{11} + 3166296871 p^{2} T^{12} - 308350531 p^{3} T^{13} + 27918611 p^{4} T^{14} - 2275812 p^{5} T^{15} + 170394 p^{6} T^{16} - 11032 p^{7} T^{17} + 637 p^{8} T^{18} - 27 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 8 T + 345 T^{2} - 2300 T^{3} + 54515 T^{4} - 330976 T^{5} + 84146 p T^{6} - 35986516 T^{7} + 604629557 T^{8} - 3281408828 T^{9} + 49781106667 T^{10} - 3281408828 p T^{11} + 604629557 p^{2} T^{12} - 35986516 p^{3} T^{13} + 84146 p^{5} T^{14} - 330976 p^{5} T^{15} + 54515 p^{6} T^{16} - 2300 p^{7} T^{17} + 345 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 21 T + 525 T^{2} - 8126 T^{3} + 128643 T^{4} - 1617253 T^{5} + 20204082 T^{6} - 216467389 T^{7} + 2311417652 T^{8} - 21734709489 T^{9} + 205078494702 T^{10} - 21734709489 p T^{11} + 2311417652 p^{2} T^{12} - 216467389 p^{3} T^{13} + 20204082 p^{4} T^{14} - 1617253 p^{5} T^{15} + 128643 p^{6} T^{16} - 8126 p^{7} T^{17} + 525 p^{8} T^{18} - 21 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 12 T + 362 T^{2} - 2682 T^{3} + 66305 T^{4} - 380966 T^{5} + 8925952 T^{6} - 41824294 T^{7} + 974731974 T^{8} - 3873777678 T^{9} + 87218795372 T^{10} - 3873777678 p T^{11} + 974731974 p^{2} T^{12} - 41824294 p^{3} T^{13} + 8925952 p^{4} T^{14} - 380966 p^{5} T^{15} + 66305 p^{6} T^{16} - 2682 p^{7} T^{17} + 362 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 6 T + 590 T^{2} + 4154 T^{3} + 170593 T^{4} + 1273912 T^{5} + 32349800 T^{6} + 233618900 T^{7} + 4443951646 T^{8} + 29034364316 T^{9} + 456195269916 T^{10} + 29034364316 p T^{11} + 4443951646 p^{2} T^{12} + 233618900 p^{3} T^{13} + 32349800 p^{4} T^{14} + 1273912 p^{5} T^{15} + 170593 p^{6} T^{16} + 4154 p^{7} T^{17} + 590 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 + 6 T + 286 T^{2} + 2290 T^{3} + 42685 T^{4} + 304468 T^{5} + 6024104 T^{6} + 33297868 T^{7} + 750467106 T^{8} + 5053993384 T^{9} + 76658721332 T^{10} + 5053993384 p T^{11} + 750467106 p^{2} T^{12} + 33297868 p^{3} T^{13} + 6024104 p^{4} T^{14} + 304468 p^{5} T^{15} + 42685 p^{6} T^{16} + 2290 p^{7} T^{17} + 286 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−2.98365872291062848153340621852, −2.92880458969754344015886424454, −2.90839811030633634314932236828, −2.68161540090911009798529812564, −2.67885461736113436404943327548, −2.52457171812087519856918938749, −2.40827566755106785074787401300, −2.38804536613833213541134765177, −2.34341960124748917091175001033, −2.24394479268618871659796885667, −2.13875619152704609144295115741, −2.09397542592944505439351929654, −1.81430862883218587120395211211, −1.68364154378331664636394314968, −1.64224581593017860357102926229, −1.60061756968054563383923493376, −1.42918714266862521344608951973, −1.26512562295588653213288921293, −1.13098758380533523393978876197, −1.07850548943618944747679333832, −1.01957132746078541012230013810, −0.956129374099559479662995589806, −0.65907157551676310506464901960, −0.63849332819860667444542434517, −0.50535875408761507247755398874, 0.50535875408761507247755398874, 0.63849332819860667444542434517, 0.65907157551676310506464901960, 0.956129374099559479662995589806, 1.01957132746078541012230013810, 1.07850548943618944747679333832, 1.13098758380533523393978876197, 1.26512562295588653213288921293, 1.42918714266862521344608951973, 1.60061756968054563383923493376, 1.64224581593017860357102926229, 1.68364154378331664636394314968, 1.81430862883218587120395211211, 2.09397542592944505439351929654, 2.13875619152704609144295115741, 2.24394479268618871659796885667, 2.34341960124748917091175001033, 2.38804536613833213541134765177, 2.40827566755106785074787401300, 2.52457171812087519856918938749, 2.67885461736113436404943327548, 2.68161540090911009798529812564, 2.90839811030633634314932236828, 2.92880458969754344015886424454, 2.98365872291062848153340621852
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.