Dirichlet series
| L(s) = 1 | + 2·2-s + 2·3-s + 4·4-s − 2·5-s + 4·6-s + 8·7-s + 6·8-s + 7·9-s − 4·10-s + 6·11-s + 8·12-s + 4·13-s + 16·14-s − 4·15-s + 14·16-s + 3·17-s + 14·18-s − 7·19-s − 8·20-s + 16·21-s + 12·22-s + 12·24-s + 17·25-s + 8·26-s + 24·27-s + 32·28-s − 20·29-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 2·4-s − 0.894·5-s + 1.63·6-s + 3.02·7-s + 2.12·8-s + 7/3·9-s − 1.26·10-s + 1.80·11-s + 2.30·12-s + 1.10·13-s + 4.27·14-s − 1.03·15-s + 7/2·16-s + 0.727·17-s + 3.29·18-s − 1.60·19-s − 1.78·20-s + 3.49·21-s + 2.55·22-s + 2.44·24-s + 17/5·25-s + 1.56·26-s + 4.61·27-s + 6.04·28-s − 3.71·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 41^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 41^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(7^{10} \cdot 41^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3995.67\) |
| Root analytic conductor: | \(1.51383\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 7^{10} \cdot 41^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(49.29109784\) |
| \(L(\frac12)\) | \(\approx\) | \(49.29109784\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 - 8 T + 36 T^{2} - 115 T^{3} + 302 T^{4} - 783 T^{5} + 302 p T^{6} - 115 p^{2} T^{7} + 36 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \) |
| 41 | \( ( 1 + T )^{10} \) | |
| good | 2 | \( 1 - p T + p T^{3} - 3 p T^{4} + 11 T^{5} - 9 T^{6} + p^{3} T^{7} + 11 T^{8} - 11 p^{2} T^{9} + 61 T^{10} - 11 p^{3} T^{11} + 11 p^{2} T^{12} + p^{6} T^{13} - 9 p^{4} T^{14} + 11 p^{5} T^{15} - 3 p^{7} T^{16} + p^{8} T^{17} - p^{10} T^{19} + p^{10} T^{20} \) |
| 3 | \( 1 - 2 T - p T^{2} - 4 T^{3} + 23 T^{4} + 28 T^{5} + 7 T^{6} - 23 p^{2} T^{7} - 94 T^{8} + 95 T^{9} + 1087 T^{10} + 95 p T^{11} - 94 p^{2} T^{12} - 23 p^{5} T^{13} + 7 p^{4} T^{14} + 28 p^{5} T^{15} + 23 p^{6} T^{16} - 4 p^{7} T^{17} - p^{9} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 5 | \( 1 + 2 T - 13 T^{2} - 8 T^{3} + 119 T^{4} - 38 T^{5} - 689 T^{6} + 103 p T^{7} + 3022 T^{8} - 1229 T^{9} - 12631 T^{10} - 1229 p T^{11} + 3022 p^{2} T^{12} + 103 p^{4} T^{13} - 689 p^{4} T^{14} - 38 p^{5} T^{15} + 119 p^{6} T^{16} - 8 p^{7} T^{17} - 13 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - 6 T - 16 T^{2} + 126 T^{3} + 272 T^{4} - 1391 T^{5} - 6093 T^{6} + 10447 T^{7} + 110443 T^{8} - 53446 T^{9} - 1352963 T^{10} - 53446 p T^{11} + 110443 p^{2} T^{12} + 10447 p^{3} T^{13} - 6093 p^{4} T^{14} - 1391 p^{5} T^{15} + 272 p^{6} T^{16} + 126 p^{7} T^{17} - 16 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( ( 1 - 2 T + 57 T^{2} - 85 T^{3} + 1376 T^{4} - 1541 T^{5} + 1376 p T^{6} - 85 p^{2} T^{7} + 57 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 17 | \( 1 - 3 T - 3 p T^{2} + 126 T^{3} + 1418 T^{4} - 2471 T^{5} - 32276 T^{6} + 39771 T^{7} + 626554 T^{8} - 325955 T^{9} - 10861899 T^{10} - 325955 p T^{11} + 626554 p^{2} T^{12} + 39771 p^{3} T^{13} - 32276 p^{4} T^{14} - 2471 p^{5} T^{15} + 1418 p^{6} T^{16} + 126 p^{7} T^{17} - 3 p^{9} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 7 T - 30 T^{2} - 229 T^{3} + 765 T^{4} + 2984 T^{5} - 32505 T^{6} - 73291 T^{7} + 742704 T^{8} + 1021299 T^{9} - 11963649 T^{10} + 1021299 p T^{11} + 742704 p^{2} T^{12} - 73291 p^{3} T^{13} - 32505 p^{4} T^{14} + 2984 p^{5} T^{15} + 765 p^{6} T^{16} - 229 p^{7} T^{17} - 30 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 - 77 T^{2} + 38 T^{3} + 133 p T^{4} - 2146 T^{5} - 88609 T^{6} + 40601 T^{7} + 2204294 T^{8} - 225757 T^{9} - 51333237 T^{10} - 225757 p T^{11} + 2204294 p^{2} T^{12} + 40601 p^{3} T^{13} - 88609 p^{4} T^{14} - 2146 p^{5} T^{15} + 133 p^{7} T^{16} + 38 p^{7} T^{17} - 77 p^{8} T^{18} + p^{10} T^{20} \) | |
| 29 | \( ( 1 + 10 T + 154 T^{2} + 1024 T^{3} + 8899 T^{4} + 42621 T^{5} + 8899 p T^{6} + 1024 p^{2} T^{7} + 154 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 31 | \( 1 - 6 T - 57 T^{2} + 472 T^{3} + 757 T^{4} - 10368 T^{5} - 4219 T^{6} - 123075 T^{7} + 2033500 T^{8} + 5253845 T^{9} - 112378329 T^{10} + 5253845 p T^{11} + 2033500 p^{2} T^{12} - 123075 p^{3} T^{13} - 4219 p^{4} T^{14} - 10368 p^{5} T^{15} + 757 p^{6} T^{16} + 472 p^{7} T^{17} - 57 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 18 T + 2 p T^{2} + 384 T^{3} - 1647 T^{4} - 14801 T^{5} + 48562 T^{6} - 508226 T^{7} + 8263775 T^{8} + 3837217 T^{9} - 385416905 T^{10} + 3837217 p T^{11} + 8263775 p^{2} T^{12} - 508226 p^{3} T^{13} + 48562 p^{4} T^{14} - 14801 p^{5} T^{15} - 1647 p^{6} T^{16} + 384 p^{7} T^{17} + 2 p^{9} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( ( 1 + 14 T + 162 T^{2} + 1230 T^{3} + 7725 T^{4} + 51435 T^{5} + 7725 p T^{6} + 1230 p^{2} T^{7} + 162 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 47 | \( 1 + 3 T - 139 T^{2} - 574 T^{3} + 9696 T^{4} + 47035 T^{5} - 425060 T^{6} - 2526513 T^{7} + 12950606 T^{8} + 54446897 T^{9} - 378378327 T^{10} + 54446897 p T^{11} + 12950606 p^{2} T^{12} - 2526513 p^{3} T^{13} - 425060 p^{4} T^{14} + 47035 p^{5} T^{15} + 9696 p^{6} T^{16} - 574 p^{7} T^{17} - 139 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 - 9 T - 64 T^{2} - 235 T^{3} + 10113 T^{4} + 27650 T^{5} + 7753 T^{6} - 6361889 T^{7} - 6637266 T^{8} + 48032331 T^{9} + 2449196135 T^{10} + 48032331 p T^{11} - 6637266 p^{2} T^{12} - 6361889 p^{3} T^{13} + 7753 p^{4} T^{14} + 27650 p^{5} T^{15} + 10113 p^{6} T^{16} - 235 p^{7} T^{17} - 64 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 19 T + 69 T^{2} + 994 T^{3} - 10320 T^{4} + 43687 T^{5} - 280758 T^{6} + 446353 T^{7} + 25825034 T^{8} - 183280321 T^{9} + 559637617 T^{10} - 183280321 p T^{11} + 25825034 p^{2} T^{12} + 446353 p^{3} T^{13} - 280758 p^{4} T^{14} + 43687 p^{5} T^{15} - 10320 p^{6} T^{16} + 994 p^{7} T^{17} + 69 p^{8} T^{18} - 19 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 23 T + 208 T^{2} + 739 T^{3} - 31393 T^{4} + 246920 T^{5} + 577277 T^{6} - 22998515 T^{7} + 134106750 T^{8} + 614594089 T^{9} - 11935706585 T^{10} + 614594089 p T^{11} + 134106750 p^{2} T^{12} - 22998515 p^{3} T^{13} + 577277 p^{4} T^{14} + 246920 p^{5} T^{15} - 31393 p^{6} T^{16} + 739 p^{7} T^{17} + 208 p^{8} T^{18} - 23 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 11 T - 79 T^{2} + 1010 T^{3} + 2242 T^{4} - 14343 T^{5} - 242472 T^{6} - 42471 T^{7} + 15451954 T^{8} - 61918455 T^{9} + 21903105 T^{10} - 61918455 p T^{11} + 15451954 p^{2} T^{12} - 42471 p^{3} T^{13} - 242472 p^{4} T^{14} - 14343 p^{5} T^{15} + 2242 p^{6} T^{16} + 1010 p^{7} T^{17} - 79 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( ( 1 + 205 T^{2} + 61 T^{3} + 23652 T^{4} + 1421 T^{5} + 23652 p T^{6} + 61 p^{2} T^{7} + 205 p^{3} T^{8} + p^{5} T^{10} )^{2} \) | |
| 73 | \( 1 + 13 T - 86 T^{2} - 905 T^{3} + 13406 T^{4} + 75611 T^{5} - 348676 T^{6} + 336091 T^{7} - 23283159 T^{8} - 6294026 T^{9} + 4995107620 T^{10} - 6294026 p T^{11} - 23283159 p^{2} T^{12} + 336091 p^{3} T^{13} - 348676 p^{4} T^{14} + 75611 p^{5} T^{15} + 13406 p^{6} T^{16} - 905 p^{7} T^{17} - 86 p^{8} T^{18} + 13 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 41 T + 733 T^{2} - 8706 T^{3} + 1208 p T^{4} - 946091 T^{5} + 6898586 T^{6} - 40642481 T^{7} + 229690774 T^{8} - 233690255 T^{9} - 8874765303 T^{10} - 233690255 p T^{11} + 229690774 p^{2} T^{12} - 40642481 p^{3} T^{13} + 6898586 p^{4} T^{14} - 946091 p^{5} T^{15} + 1208 p^{7} T^{16} - 8706 p^{7} T^{17} + 733 p^{8} T^{18} - 41 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( ( 1 + 2 T + 338 T^{2} + 520 T^{3} + 50961 T^{4} + 61081 T^{5} + 50961 p T^{6} + 520 p^{2} T^{7} + 338 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 89 | \( 1 + 14 T + 2 T^{2} + 32 T^{3} + 6355 T^{4} - 26749 T^{5} + 723178 T^{6} + 18631550 T^{7} + 17093627 T^{8} + 19250247 T^{9} + 10958940527 T^{10} + 19250247 p T^{11} + 17093627 p^{2} T^{12} + 18631550 p^{3} T^{13} + 723178 p^{4} T^{14} - 26749 p^{5} T^{15} + 6355 p^{6} T^{16} + 32 p^{7} T^{17} + 2 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( ( 1 + 27 T + 448 T^{2} + 5083 T^{3} + 45427 T^{4} + 421253 T^{5} + 45427 p T^{6} + 5083 p^{2} T^{7} + 448 p^{3} T^{8} + 27 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 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
−4.36556414199632414482172147239, −4.35337305251035312706967162320, −4.32846428742201685593973427285, −4.06018229922405820352259572088, −4.01172856471926842034693949428, −3.94230794677447970302021714193, −3.88448479431798720993026874451, −3.76854643482262177768681129187, −3.65820300316052625142935505530, −3.50922053139028650305250851601, −3.31828554185243684447927770490, −3.28278340169382457419401472611, −2.83305577190805113889428931608, −2.65008240419087173523914137655, −2.64021840685113120661501660660, −2.57140395967832264579634936611, −2.52428238375729260954924751300, −2.08591924021796490576646812447, −2.01553587732130678935907064399, −1.67884865473270156429444266859, −1.40404897157678454948839910104, −1.37888708210959600307973133890, −1.27443961252480879036797547552, −1.18760131452415854738669706936, −1.02332690452219514892186144934, 1.02332690452219514892186144934, 1.18760131452415854738669706936, 1.27443961252480879036797547552, 1.37888708210959600307973133890, 1.40404897157678454948839910104, 1.67884865473270156429444266859, 2.01553587732130678935907064399, 2.08591924021796490576646812447, 2.52428238375729260954924751300, 2.57140395967832264579634936611, 2.64021840685113120661501660660, 2.65008240419087173523914137655, 2.83305577190805113889428931608, 3.28278340169382457419401472611, 3.31828554185243684447927770490, 3.50922053139028650305250851601, 3.65820300316052625142935505530, 3.76854643482262177768681129187, 3.88448479431798720993026874451, 3.94230794677447970302021714193, 4.01172856471926842034693949428, 4.06018229922405820352259572088, 4.32846428742201685593973427285, 4.35337305251035312706967162320, 4.36556414199632414482172147239
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.