Dirichlet series
| L(s) = 1 | + 3·2-s + 7·3-s + 3·4-s + 21·6-s + 8·7-s + 8-s + 28·9-s − 4·11-s + 21·12-s + 8·13-s + 24·14-s + 4·17-s + 84·18-s + 7·19-s + 56·21-s − 12·22-s + 10·23-s + 7·24-s + 24·26-s + 84·27-s + 24·28-s − 6·29-s + 4·31-s + 2·32-s − 28·33-s + 12·34-s + 84·36-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 4.04·3-s + 3/2·4-s + 8.57·6-s + 3.02·7-s + 0.353·8-s + 28/3·9-s − 1.20·11-s + 6.06·12-s + 2.21·13-s + 6.41·14-s + 0.970·17-s + 19.7·18-s + 1.60·19-s + 12.2·21-s − 2.55·22-s + 2.08·23-s + 1.42·24-s + 4.70·26-s + 16.1·27-s + 4.53·28-s − 1.11·29-s + 0.718·31-s + 0.353·32-s − 4.87·33-s + 2.05·34-s + 14·36-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 5^{14} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 5^{14} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{7} \cdot 5^{14} \cdot 19^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.46969\times 10^{7}\) |
| Root analytic conductor: | \(3.37323\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 3^{7} \cdot 5^{14} \cdot 19^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(571.3016363\) |
| \(L(\frac12)\) | \(\approx\) | \(571.3016363\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{7} \) |
| 5 | \( 1 \) | |
| 19 | \( ( 1 - T )^{7} \) | |
| good | 2 | \( 1 - 3 T + 3 p T^{2} - 5 p T^{3} + 15 T^{4} - 23 T^{5} + 19 p T^{6} - 7 p^{3} T^{7} + 19 p^{2} T^{8} - 23 p^{2} T^{9} + 15 p^{3} T^{10} - 5 p^{5} T^{11} + 3 p^{6} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) |
| 7 | \( 1 - 8 T + 48 T^{2} - 30 p T^{3} + 828 T^{4} - 2770 T^{5} + 8597 T^{6} - 23456 T^{7} + 8597 p T^{8} - 2770 p^{2} T^{9} + 828 p^{3} T^{10} - 30 p^{5} T^{11} + 48 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 4 T + 34 T^{2} + 152 T^{3} + 856 T^{4} + 2878 T^{5} + 12979 T^{6} + 40412 T^{7} + 12979 p T^{8} + 2878 p^{2} T^{9} + 856 p^{3} T^{10} + 152 p^{4} T^{11} + 34 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 8 T + 69 T^{2} - 358 T^{3} + 1959 T^{4} - 656 p T^{5} + 2863 p T^{6} - 136676 T^{7} + 2863 p^{2} T^{8} - 656 p^{3} T^{9} + 1959 p^{3} T^{10} - 358 p^{4} T^{11} + 69 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 4 T + 36 T^{2} - 60 T^{3} + 754 T^{4} - 1376 T^{5} + 12825 T^{6} - 11408 T^{7} + 12825 p T^{8} - 1376 p^{2} T^{9} + 754 p^{3} T^{10} - 60 p^{4} T^{11} + 36 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 10 T + 113 T^{2} - 888 T^{3} + 6821 T^{4} - 40006 T^{5} + 237093 T^{6} - 1158384 T^{7} + 237093 p T^{8} - 40006 p^{2} T^{9} + 6821 p^{3} T^{10} - 888 p^{4} T^{11} + 113 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 6 T + 115 T^{2} + 702 T^{3} + 7541 T^{4} + 39778 T^{5} + 316975 T^{6} + 1439028 T^{7} + 316975 p T^{8} + 39778 p^{2} T^{9} + 7541 p^{3} T^{10} + 702 p^{4} T^{11} + 115 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 4 T + 133 T^{2} - 668 T^{3} + 9417 T^{4} - 44348 T^{5} + 439237 T^{6} - 1717576 T^{7} + 439237 p T^{8} - 44348 p^{2} T^{9} + 9417 p^{3} T^{10} - 668 p^{4} T^{11} + 133 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 14 T + 217 T^{2} - 1950 T^{3} + 18843 T^{4} - 133762 T^{5} + 1017547 T^{6} - 5998948 T^{7} + 1017547 p T^{8} - 133762 p^{2} T^{9} + 18843 p^{3} T^{10} - 1950 p^{4} T^{11} + 217 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 2 T + 207 T^{2} - 610 T^{3} + 19693 T^{4} - 68974 T^{5} + 1162547 T^{6} - 3880604 T^{7} + 1162547 p T^{8} - 68974 p^{2} T^{9} + 19693 p^{3} T^{10} - 610 p^{4} T^{11} + 207 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 2 T + 176 T^{2} + 328 T^{3} + 15584 T^{4} + 23448 T^{5} + 921981 T^{6} + 1108836 T^{7} + 921981 p T^{8} + 23448 p^{2} T^{9} + 15584 p^{3} T^{10} + 328 p^{4} T^{11} + 176 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 30 T + 638 T^{2} - 9250 T^{3} + 111256 T^{4} - 22978 p T^{5} + 9205771 T^{6} - 66750740 T^{7} + 9205771 p T^{8} - 22978 p^{3} T^{9} + 111256 p^{3} T^{10} - 9250 p^{4} T^{11} + 638 p^{5} T^{12} - 30 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 307 T^{2} - 12 T^{3} + 42781 T^{4} - 2160 T^{5} + 3530455 T^{6} - 163400 T^{7} + 3530455 p T^{8} - 2160 p^{2} T^{9} + 42781 p^{3} T^{10} - 12 p^{4} T^{11} + 307 p^{5} T^{12} + p^{7} T^{14} \) | |
| 59 | \( 1 + 18 T + 313 T^{2} + 3004 T^{3} + 28433 T^{4} + 157182 T^{5} + 1104449 T^{6} + 5068232 T^{7} + 1104449 p T^{8} + 157182 p^{2} T^{9} + 28433 p^{3} T^{10} + 3004 p^{4} T^{11} + 313 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 12 T + 136 T^{2} - 1326 T^{3} + 16514 T^{4} - 156640 T^{5} + 1332393 T^{6} - 9048172 T^{7} + 1332393 p T^{8} - 156640 p^{2} T^{9} + 16514 p^{3} T^{10} - 1326 p^{4} T^{11} + 136 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 18 T + 401 T^{2} - 4124 T^{3} + 52353 T^{4} - 356766 T^{5} + 3695433 T^{6} - 21598600 T^{7} + 3695433 p T^{8} - 356766 p^{2} T^{9} + 52353 p^{3} T^{10} - 4124 p^{4} T^{11} + 401 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 18 T + 429 T^{2} + 4556 T^{3} + 62585 T^{4} + 456014 T^{5} + 5090621 T^{6} + 31694952 T^{7} + 5090621 p T^{8} + 456014 p^{2} T^{9} + 62585 p^{3} T^{10} + 4556 p^{4} T^{11} + 429 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 10 T + 384 T^{2} - 3526 T^{3} + 72286 T^{4} - 562294 T^{5} + 8204377 T^{6} - 52431988 T^{7} + 8204377 p T^{8} - 562294 p^{2} T^{9} + 72286 p^{3} T^{10} - 3526 p^{4} T^{11} + 384 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 4 T + 225 T^{2} + 600 T^{3} + 17885 T^{4} - 516 T^{5} + 443613 T^{6} - 3708336 T^{7} + 443613 p T^{8} - 516 p^{2} T^{9} + 17885 p^{3} T^{10} + 600 p^{4} T^{11} + 225 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 18 T + 497 T^{2} - 5040 T^{3} + 76161 T^{4} - 429934 T^{5} + 5624649 T^{6} - 22366400 T^{7} + 5624649 p T^{8} - 429934 p^{2} T^{9} + 76161 p^{3} T^{10} - 5040 p^{4} T^{11} + 497 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 8 T + 295 T^{2} + 4354 T^{3} + 54701 T^{4} + 726512 T^{5} + 8547675 T^{6} + 70970572 T^{7} + 8547675 p T^{8} + 726512 p^{2} T^{9} + 54701 p^{3} T^{10} + 4354 p^{4} T^{11} + 295 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 20 T + 661 T^{2} - 9862 T^{3} + 187003 T^{4} - 2169732 T^{5} + 29663215 T^{6} - 271833540 T^{7} + 29663215 p T^{8} - 2169732 p^{2} T^{9} + 187003 p^{3} T^{10} - 9862 p^{4} T^{11} + 661 p^{5} T^{12} - 20 p^{6} T^{13} + p^{7} T^{14} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.42100362547314653520049320774, −4.21484762534657526675284618157, −4.21146477499088793574630610540, −4.15366271845764524040492103983, −3.95198485131790894192183191184, −3.61395822346537152470895871587, −3.57540018032704838220580838964, −3.47614486751650999411694429390, −3.41800268268785571511205179455, −3.26873600058186081136913099736, −3.07747372627636594366744220936, −2.92190288206840948444097653772, −2.76509303078047834291891773967, −2.66883015917272507888968962437, −2.48201975974414638158174047922, −2.32000829280833842498351382562, −2.03233052475643602027243928755, −2.01239289406869129022996892180, −1.96669181962175537554896908957, −1.58617856632162597819679640825, −1.32811574402730015808210028949, −1.20264728710957525567220177707, −0.987322599412682405163559365636, −0.974895974358141710390128569315, −0.78757292590302080830929777821, 0.78757292590302080830929777821, 0.974895974358141710390128569315, 0.987322599412682405163559365636, 1.20264728710957525567220177707, 1.32811574402730015808210028949, 1.58617856632162597819679640825, 1.96669181962175537554896908957, 2.01239289406869129022996892180, 2.03233052475643602027243928755, 2.32000829280833842498351382562, 2.48201975974414638158174047922, 2.66883015917272507888968962437, 2.76509303078047834291891773967, 2.92190288206840948444097653772, 3.07747372627636594366744220936, 3.26873600058186081136913099736, 3.41800268268785571511205179455, 3.47614486751650999411694429390, 3.57540018032704838220580838964, 3.61395822346537152470895871587, 3.95198485131790894192183191184, 4.15366271845764524040492103983, 4.21146477499088793574630610540, 4.21484762534657526675284618157, 4.42100362547314653520049320774
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.