Dirichlet series
| L(s) = 1 | − 2-s − 7·3-s − 2·4-s − 4·5-s + 7·6-s − 8-s + 17·9-s + 4·10-s + 14·12-s − 13-s + 28·15-s + 2·16-s + 11·17-s − 17·18-s − 9·19-s + 8·20-s + 3·23-s + 7·24-s + 6·25-s + 26-s − 5·27-s − 14·29-s − 28·30-s − 34·31-s + 8·32-s − 11·34-s − 34·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 4.04·3-s − 4-s − 1.78·5-s + 2.85·6-s − 0.353·8-s + 17/3·9-s + 1.26·10-s + 4.04·12-s − 0.277·13-s + 7.22·15-s + 1/2·16-s + 2.66·17-s − 4.00·18-s − 2.06·19-s + 1.78·20-s + 0.625·23-s + 1.42·24-s + 6/5·25-s + 0.196·26-s − 0.962·27-s − 2.59·29-s − 5.11·30-s − 6.10·31-s + 1.41·32-s − 1.88·34-s − 5.66·36-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{14} \cdot 41^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{14} \cdot 41^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(7^{14} \cdot 41^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(2.73400\times 10^{8}\) |
| Root analytic conductor: | \(4.00523\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 7^{14} \cdot 41^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 \) |
| 41 | \( ( 1 - T )^{7} \) | |
| good | 2 | \( 1 + T + 3 T^{2} + 3 p T^{3} + 11 T^{4} + p^{4} T^{5} + 7 p^{2} T^{6} + 43 T^{7} + 7 p^{3} T^{8} + p^{6} T^{9} + 11 p^{3} T^{10} + 3 p^{5} T^{11} + 3 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 + 7 T + 32 T^{2} + 110 T^{3} + 35 p^{2} T^{4} + 85 p^{2} T^{5} + 1616 T^{6} + 331 p^{2} T^{7} + 1616 p T^{8} + 85 p^{4} T^{9} + 35 p^{5} T^{10} + 110 p^{4} T^{11} + 32 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 5 | \( 1 + 4 T + 2 p T^{2} + 28 T^{3} + 109 T^{4} + 334 T^{5} + 28 p^{2} T^{6} + 1332 T^{7} + 28 p^{3} T^{8} + 334 p^{2} T^{9} + 109 p^{3} T^{10} + 28 p^{4} T^{11} + 2 p^{6} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 35 T^{2} + 32 T^{3} + 512 T^{4} + 1448 T^{5} + 4674 T^{6} + 23960 T^{7} + 4674 p T^{8} + 1448 p^{2} T^{9} + 512 p^{3} T^{10} + 32 p^{4} T^{11} + 35 p^{5} T^{12} + p^{7} T^{14} \) | |
| 13 | \( 1 + T + 59 T^{2} + 31 T^{3} + 1698 T^{4} + 434 T^{5} + 31576 T^{6} + 5143 T^{7} + 31576 p T^{8} + 434 p^{2} T^{9} + 1698 p^{3} T^{10} + 31 p^{4} T^{11} + 59 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 11 T + 123 T^{2} - 861 T^{3} + 5850 T^{4} - 30762 T^{5} + 157516 T^{6} - 657593 T^{7} + 157516 p T^{8} - 30762 p^{2} T^{9} + 5850 p^{3} T^{10} - 861 p^{4} T^{11} + 123 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 9 T + 72 T^{2} + 358 T^{3} + 2255 T^{4} + 10619 T^{5} + 60588 T^{6} + 249857 T^{7} + 60588 p T^{8} + 10619 p^{2} T^{9} + 2255 p^{3} T^{10} + 358 p^{4} T^{11} + 72 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 3 T + 106 T^{2} - 236 T^{3} + 5717 T^{4} - 10577 T^{5} + 194242 T^{6} - 292425 T^{7} + 194242 p T^{8} - 10577 p^{2} T^{9} + 5717 p^{3} T^{10} - 236 p^{4} T^{11} + 106 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 14 T + 162 T^{2} + 1246 T^{3} + 9813 T^{4} + 61032 T^{5} + 384012 T^{6} + 69032 p T^{7} + 384012 p T^{8} + 61032 p^{2} T^{9} + 9813 p^{3} T^{10} + 1246 p^{4} T^{11} + 162 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 34 T + 608 T^{2} + 7498 T^{3} + 2329 p T^{4} + 579438 T^{5} + 3996968 T^{6} + 23893956 T^{7} + 3996968 p T^{8} + 579438 p^{2} T^{9} + 2329 p^{4} T^{10} + 7498 p^{4} T^{11} + 608 p^{5} T^{12} + 34 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 7 T + 121 T^{2} - 651 T^{3} + 7671 T^{4} - 38793 T^{5} + 395023 T^{6} - 1773722 T^{7} + 395023 p T^{8} - 38793 p^{2} T^{9} + 7671 p^{3} T^{10} - 651 p^{4} T^{11} + 121 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 3 T + 91 T^{2} + 111 T^{3} + 6288 T^{4} - 1242 T^{5} + 282726 T^{6} - 294681 T^{7} + 282726 p T^{8} - 1242 p^{2} T^{9} + 6288 p^{3} T^{10} + 111 p^{4} T^{11} + 91 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 17 T + 289 T^{2} + 3311 T^{3} + 36705 T^{4} + 325523 T^{5} + 2728049 T^{6} + 19146138 T^{7} + 2728049 p T^{8} + 325523 p^{2} T^{9} + 36705 p^{3} T^{10} + 3311 p^{4} T^{11} + 289 p^{5} T^{12} + 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 24 T + 558 T^{2} - 8100 T^{3} + 108225 T^{4} - 1107964 T^{5} + 10410464 T^{6} - 79084328 T^{7} + 10410464 p T^{8} - 1107964 p^{2} T^{9} + 108225 p^{3} T^{10} - 8100 p^{4} T^{11} + 558 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 4 T + 320 T^{2} + 1216 T^{3} + 47827 T^{4} + 165810 T^{5} + 4320864 T^{6} + 12717596 T^{7} + 4320864 p T^{8} + 165810 p^{2} T^{9} + 47827 p^{3} T^{10} + 1216 p^{4} T^{11} + 320 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 16 T + 437 T^{2} + 4848 T^{3} + 75368 T^{4} + 637128 T^{5} + 7247662 T^{6} + 48913976 T^{7} + 7247662 p T^{8} + 637128 p^{2} T^{9} + 75368 p^{3} T^{10} + 4848 p^{4} T^{11} + 437 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 24 T + 472 T^{2} + 6040 T^{3} + 72643 T^{4} + 683720 T^{5} + 6437456 T^{6} + 766576 p T^{7} + 6437456 p T^{8} + 683720 p^{2} T^{9} + 72643 p^{3} T^{10} + 6040 p^{4} T^{11} + 472 p^{5} T^{12} + 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 12 T + 291 T^{2} + 4116 T^{3} + 49600 T^{4} + 569260 T^{5} + 5785078 T^{6} + 47941040 T^{7} + 5785078 p T^{8} + 569260 p^{2} T^{9} + 49600 p^{3} T^{10} + 4116 p^{4} T^{11} + 291 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 14 T + 381 T^{2} + 5104 T^{3} + 72100 T^{4} + 815200 T^{5} + 8333298 T^{6} + 75539380 T^{7} + 8333298 p T^{8} + 815200 p^{2} T^{9} + 72100 p^{3} T^{10} + 5104 p^{4} T^{11} + 381 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 8 T + 273 T^{2} + 3016 T^{3} + 44397 T^{4} + 494520 T^{5} + 4907901 T^{6} + 49239664 T^{7} + 4907901 p T^{8} + 494520 p^{2} T^{9} + 44397 p^{3} T^{10} + 3016 p^{4} T^{11} + 273 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 14 T + 291 T^{2} + 1240 T^{3} + 22896 T^{4} + 53760 T^{5} + 3234166 T^{6} + 15310380 T^{7} + 3234166 p T^{8} + 53760 p^{2} T^{9} + 22896 p^{3} T^{10} + 1240 p^{4} T^{11} + 291 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 19 T + 536 T^{2} - 8224 T^{3} + 130645 T^{4} - 1612251 T^{5} + 18578340 T^{6} - 183326933 T^{7} + 18578340 p T^{8} - 1612251 p^{2} T^{9} + 130645 p^{3} T^{10} - 8224 p^{4} T^{11} + 536 p^{5} T^{12} - 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 23 T + 696 T^{2} + 10604 T^{3} + 188081 T^{4} + 2171679 T^{5} + 28535268 T^{6} + 263605149 T^{7} + 28535268 p T^{8} + 2171679 p^{2} T^{9} + 188081 p^{3} T^{10} + 10604 p^{4} T^{11} + 696 p^{5} T^{12} + 23 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−4.74642877096576466171246336354, −4.66270318051160749699495183219, −4.57023394331127434209420550300, −4.33642527024137610328358059111, −4.32444439249848317337922087158, −3.96846392079338160439739818891, −3.94349438788921793800265487092, −3.81555527645217820408896656731, −3.61679529244145795524241479224, −3.61308239254329280610334111029, −3.46073297715374862476610365993, −3.30052190791738997535648038683, −3.29378123648091932868940690369, −2.91783273354181528370222676999, −2.78877283043525520604486342700, −2.74905300835946266266699731535, −2.30835334666381218080859059773, −2.21291116233899769301232040099, −2.15093388882201481634125928672, −2.06368885107870880755614556910, −1.43654786343104594193318061165, −1.35811667322077932279876778572, −1.29470927201546675427199286336, −1.09842022797683956139000250102, −0.993989100658042100958235072894, 0, 0, 0, 0, 0, 0, 0, 0.993989100658042100958235072894, 1.09842022797683956139000250102, 1.29470927201546675427199286336, 1.35811667322077932279876778572, 1.43654786343104594193318061165, 2.06368885107870880755614556910, 2.15093388882201481634125928672, 2.21291116233899769301232040099, 2.30835334666381218080859059773, 2.74905300835946266266699731535, 2.78877283043525520604486342700, 2.91783273354181528370222676999, 3.29378123648091932868940690369, 3.30052190791738997535648038683, 3.46073297715374862476610365993, 3.61308239254329280610334111029, 3.61679529244145795524241479224, 3.81555527645217820408896656731, 3.94349438788921793800265487092, 3.96846392079338160439739818891, 4.32444439249848317337922087158, 4.33642527024137610328358059111, 4.57023394331127434209420550300, 4.66270318051160749699495183219, 4.74642877096576466171246336354
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.