Dirichlet series
| L(s) = 1 | − 5·2-s − 5·3-s + 10·4-s − 5·5-s + 25·6-s − 5·8-s + 10·9-s + 25·10-s − 2·11-s − 50·12-s + 13-s + 25·15-s − 20·16-s − 50·18-s − 2·19-s − 50·20-s + 10·22-s − 2·23-s + 25·24-s + 10·25-s − 5·26-s − 5·27-s + 18·29-s − 125·30-s − 14·31-s + 49·32-s + 10·33-s + ⋯ |
| L(s) = 1 | − 3.53·2-s − 2.88·3-s + 5·4-s − 2.23·5-s + 10.2·6-s − 1.76·8-s + 10/3·9-s + 7.90·10-s − 0.603·11-s − 14.4·12-s + 0.277·13-s + 6.45·15-s − 5·16-s − 11.7·18-s − 0.458·19-s − 11.1·20-s + 2.13·22-s − 0.417·23-s + 5.10·24-s + 2·25-s − 0.980·26-s − 0.962·27-s + 3.34·29-s − 22.8·30-s − 2.51·31-s + 8.66·32-s + 1.74·33-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 5^{10} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 5^{10} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 3^{10} \cdot 5^{10} \cdot 37^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.99227\times 10^{9}\) |
| Root analytic conductor: | \(2.97714\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 3^{10} \cdot 5^{10} \cdot 37^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.02218007218\) |
| \(L(\frac12)\) | \(\approx\) | \(0.02218007218\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} )^{5} \) |
| 3 | \( ( 1 + T + T^{2} )^{5} \) | |
| 5 | \( ( 1 + T + T^{2} )^{5} \) | |
| 37 | \( 1 + 25 T^{2} - 178 T^{3} + 1945 T^{4} + 1886 T^{5} + 1945 p T^{6} - 178 p^{2} T^{7} + 25 p^{3} T^{8} + p^{5} T^{10} \) | |
| good | 7 | \( 1 - 12 T^{2} - 24 T^{3} + 95 T^{4} + 328 T^{5} + 242 T^{6} - 3240 T^{7} - 8039 T^{8} + 160 p^{2} T^{9} + 91970 T^{10} + 160 p^{3} T^{11} - 8039 p^{2} T^{12} - 3240 p^{3} T^{13} + 242 p^{4} T^{14} + 328 p^{5} T^{15} + 95 p^{6} T^{16} - 24 p^{7} T^{17} - 12 p^{8} T^{18} + p^{10} T^{20} \) |
| 11 | \( ( 1 + T + 14 T^{2} + 43 T^{3} + 261 T^{4} + 212 T^{5} + 261 p T^{6} + 43 p^{2} T^{7} + 14 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 13 | \( 1 - T - 14 T^{2} + 21 T^{3} - 30 T^{4} - 23 T^{5} + 16 p^{2} T^{6} - 3579 T^{7} - 21003 T^{8} + 35654 T^{9} - 144628 T^{10} + 35654 p T^{11} - 21003 p^{2} T^{12} - 3579 p^{3} T^{13} + 16 p^{6} T^{14} - 23 p^{5} T^{15} - 30 p^{6} T^{16} + 21 p^{7} T^{17} - 14 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 32 T^{2} - 100 T^{3} + 449 T^{4} + 3228 T^{5} + 3650 T^{6} - 60992 T^{7} - 220791 T^{8} + 421000 T^{9} + 4949226 T^{10} + 421000 p T^{11} - 220791 p^{2} T^{12} - 60992 p^{3} T^{13} + 3650 p^{4} T^{14} + 3228 p^{5} T^{15} + 449 p^{6} T^{16} - 100 p^{7} T^{17} - 32 p^{8} T^{18} + p^{10} T^{20} \) | |
| 19 | \( 1 + 2 T - 44 T^{2} - 240 T^{3} + 801 T^{4} + 448 p T^{5} + 6322 T^{6} - 190086 T^{7} - 648543 T^{8} + 1568528 T^{9} + 18717590 T^{10} + 1568528 p T^{11} - 648543 p^{2} T^{12} - 190086 p^{3} T^{13} + 6322 p^{4} T^{14} + 448 p^{6} T^{15} + 801 p^{6} T^{16} - 240 p^{7} T^{17} - 44 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( ( 1 + T + 27 T^{2} - 104 T^{3} + 1170 T^{4} + 302 T^{5} + 1170 p T^{6} - 104 p^{2} T^{7} + 27 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 29 | \( ( 1 - 9 T + 108 T^{2} - 21 p T^{3} + 4617 T^{4} - 19992 T^{5} + 4617 p T^{6} - 21 p^{3} T^{7} + 108 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 31 | \( ( 1 + 7 T + 105 T^{2} + 512 T^{3} + 5496 T^{4} + 22354 T^{5} + 5496 p T^{6} + 512 p^{2} T^{7} + 105 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 41 | \( 1 + 10 T - 55 T^{2} - 266 T^{3} + 5709 T^{4} - 1608 T^{5} - 273484 T^{6} + 564076 T^{7} + 7979345 T^{8} - 9431554 T^{9} - 174767387 T^{10} - 9431554 p T^{11} + 7979345 p^{2} T^{12} + 564076 p^{3} T^{13} - 273484 p^{4} T^{14} - 1608 p^{5} T^{15} + 5709 p^{6} T^{16} - 266 p^{7} T^{17} - 55 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( ( 1 - 3 T + 51 T^{2} + 180 T^{3} + 2614 T^{4} + 2510 T^{5} + 2614 p T^{6} + 180 p^{2} T^{7} + 51 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 47 | \( ( 1 - 15 T + 180 T^{2} - 1749 T^{3} + 14781 T^{4} - 104928 T^{5} + 14781 p T^{6} - 1749 p^{2} T^{7} + 180 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( 1 + 2 T - 187 T^{2} + 198 T^{3} + 20037 T^{4} - 48376 T^{5} - 1371820 T^{6} + 3824764 T^{7} + 72469361 T^{8} - 98662762 T^{9} - 3622887455 T^{10} - 98662762 p T^{11} + 72469361 p^{2} T^{12} + 3824764 p^{3} T^{13} - 1371820 p^{4} T^{14} - 48376 p^{5} T^{15} + 20037 p^{6} T^{16} + 198 p^{7} T^{17} - 187 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 7 T - 23 T^{2} - 1418 T^{3} - 10828 T^{4} - 14518 T^{5} + 1026671 T^{6} + 9438511 T^{7} + 34339551 T^{8} - 448465316 T^{9} - 5338705992 T^{10} - 448465316 p T^{11} + 34339551 p^{2} T^{12} + 9438511 p^{3} T^{13} + 1026671 p^{4} T^{14} - 14518 p^{5} T^{15} - 10828 p^{6} T^{16} - 1418 p^{7} T^{17} - 23 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 6 T + 81 T^{2} + 1242 T^{3} + 3729 T^{4} + 72896 T^{5} + 607320 T^{6} + 2639844 T^{7} + 47333457 T^{8} + 366694002 T^{9} + 2268886953 T^{10} + 366694002 p T^{11} + 47333457 p^{2} T^{12} + 2639844 p^{3} T^{13} + 607320 p^{4} T^{14} + 72896 p^{5} T^{15} + 3729 p^{6} T^{16} + 1242 p^{7} T^{17} + 81 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 5 T - 55 T^{2} - 674 T^{3} - 3208 T^{4} + 838 p T^{5} + 250131 T^{6} - 70733 p T^{7} - 33391853 T^{8} + 184042164 T^{9} + 4282600640 T^{10} + 184042164 p T^{11} - 33391853 p^{2} T^{12} - 70733 p^{4} T^{13} + 250131 p^{4} T^{14} + 838 p^{6} T^{15} - 3208 p^{6} T^{16} - 674 p^{7} T^{17} - 55 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 3 T - 254 T^{2} + 615 T^{3} + 34285 T^{4} - 57108 T^{5} - 3680784 T^{6} + 2842248 T^{7} + 340892489 T^{8} - 59777907 T^{9} - 26504952190 T^{10} - 59777907 p T^{11} + 340892489 p^{2} T^{12} + 2842248 p^{3} T^{13} - 3680784 p^{4} T^{14} - 57108 p^{5} T^{15} + 34285 p^{6} T^{16} + 615 p^{7} T^{17} - 254 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( ( 1 + 9 T + 173 T^{2} + 1188 T^{3} + 8170 T^{4} + 75478 T^{5} + 8170 p T^{6} + 1188 p^{2} T^{7} + 173 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 79 | \( 1 + 15 T - 100 T^{2} - 2621 T^{3} + 417 T^{4} + 153926 T^{5} - 215482 T^{6} + 1496172 T^{7} + 163871041 T^{8} - 309716589 T^{9} - 20540238558 T^{10} - 309716589 p T^{11} + 163871041 p^{2} T^{12} + 1496172 p^{3} T^{13} - 215482 p^{4} T^{14} + 153926 p^{5} T^{15} + 417 p^{6} T^{16} - 2621 p^{7} T^{17} - 100 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 5 T - 192 T^{2} - 3431 T^{3} + 13939 T^{4} + 569812 T^{5} + 2491574 T^{6} - 55529742 T^{7} - 558230927 T^{8} + 1851171519 T^{9} + 62995980698 T^{10} + 1851171519 p T^{11} - 558230927 p^{2} T^{12} - 55529742 p^{3} T^{13} + 2491574 p^{4} T^{14} + 569812 p^{5} T^{15} + 13939 p^{6} T^{16} - 3431 p^{7} T^{17} - 192 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 25 T + 18 T^{2} - 2793 T^{3} + 22865 T^{4} + 604886 T^{5} - 1209830 T^{6} - 30496852 T^{7} + 443437409 T^{8} + 2561826437 T^{9} - 25418071668 T^{10} + 2561826437 p T^{11} + 443437409 p^{2} T^{12} - 30496852 p^{3} T^{13} - 1209830 p^{4} T^{14} + 604886 p^{5} T^{15} + 22865 p^{6} T^{16} - 2793 p^{7} T^{17} + 18 p^{8} T^{18} + 25 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( ( 1 - 3 T + 377 T^{2} - 1336 T^{3} + 63574 T^{4} - 202602 T^{5} + 63574 p T^{6} - 1336 p^{2} T^{7} + 377 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
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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
−3.74016857330862992747955734930, −3.32786053617755048151170323860, −3.31799258226891378768506310736, −3.27629351838599239376051851841, −3.03568498960165567414273994150, −3.00851107505951735999924962716, −2.75107929322078596192741094761, −2.71932402457632544111073779366, −2.70614686807533374372424312208, −2.56513083541861810257730969324, −2.44325174269202394435130641515, −2.06151730657264172038110859981, −1.95769296861225970550583339003, −1.90529415990135227318590774736, −1.77336913306248169021338905229, −1.63333789726635098541674284946, −1.38824747877735582973466204655, −1.28833881508080984965863859944, −0.987263270012034434811042085503, −0.971601163879615362915155230171, −0.863914775106318183986176893240, −0.57668769483507355944681932856, −0.45401652504799139742782852934, −0.23977084562860641030283364955, −0.20713576853572433885579660203, 0.20713576853572433885579660203, 0.23977084562860641030283364955, 0.45401652504799139742782852934, 0.57668769483507355944681932856, 0.863914775106318183986176893240, 0.971601163879615362915155230171, 0.987263270012034434811042085503, 1.28833881508080984965863859944, 1.38824747877735582973466204655, 1.63333789726635098541674284946, 1.77336913306248169021338905229, 1.90529415990135227318590774736, 1.95769296861225970550583339003, 2.06151730657264172038110859981, 2.44325174269202394435130641515, 2.56513083541861810257730969324, 2.70614686807533374372424312208, 2.71932402457632544111073779366, 2.75107929322078596192741094761, 3.00851107505951735999924962716, 3.03568498960165567414273994150, 3.27629351838599239376051851841, 3.31799258226891378768506310736, 3.32786053617755048151170323860, 3.74016857330862992747955734930
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.