Dirichlet series
| L(s) = 1 | + 2·2-s − 3·4-s + 8·5-s + 7·7-s − 7·8-s + 16·10-s + 3·11-s − 23·13-s + 14·14-s + 5·16-s − 3·17-s − 9·19-s − 24·20-s + 6·22-s − 12·23-s + 16·25-s − 46·26-s − 21·28-s + 9·29-s − 33·31-s + 5·32-s − 6·34-s + 56·35-s − 33·37-s − 18·38-s − 56·40-s + 3·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 3/2·4-s + 3.57·5-s + 2.64·7-s − 2.47·8-s + 5.05·10-s + 0.904·11-s − 6.37·13-s + 3.74·14-s + 5/4·16-s − 0.727·17-s − 2.06·19-s − 5.36·20-s + 1.27·22-s − 2.50·23-s + 16/5·25-s − 9.02·26-s − 3.96·28-s + 1.67·29-s − 5.92·31-s + 0.883·32-s − 1.02·34-s + 9.46·35-s − 5.42·37-s − 2.91·38-s − 8.85·40-s + 0.468·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 7^{7} \cdot 127^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 7^{7} \cdot 127^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{14} \cdot 7^{7} \cdot 127^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(4.34459\times 10^{12}\) |
| Root analytic conductor: | \(7.99301\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 3^{14} \cdot 7^{7} \cdot 127^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
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 | 3 | \( 1 \) |
| 7 | \( ( 1 - T )^{7} \) | |
| 127 | \( ( 1 + T )^{7} \) | |
| good | 2 | \( 1 - p T + 7 T^{2} - 13 T^{3} + 7 p^{2} T^{4} - 41 T^{5} + 73 T^{6} - 47 p T^{7} + 73 p T^{8} - 41 p^{2} T^{9} + 7 p^{5} T^{10} - 13 p^{4} T^{11} + 7 p^{5} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) |
| 5 | \( 1 - 8 T + 48 T^{2} - 198 T^{3} + 701 T^{4} - 2022 T^{5} + 5344 T^{6} - 2463 p T^{7} + 5344 p T^{8} - 2022 p^{2} T^{9} + 701 p^{3} T^{10} - 198 p^{4} T^{11} + 48 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - 3 T + 31 T^{2} - 93 T^{3} + 656 T^{4} - 1715 T^{5} + 9199 T^{6} - 22080 T^{7} + 9199 p T^{8} - 1715 p^{2} T^{9} + 656 p^{3} T^{10} - 93 p^{4} T^{11} + 31 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + 23 T + 289 T^{2} + 2532 T^{3} + 1316 p T^{4} + 93828 T^{5} + 430561 T^{6} + 1679255 T^{7} + 430561 p T^{8} + 93828 p^{2} T^{9} + 1316 p^{4} T^{10} + 2532 p^{4} T^{11} + 289 p^{5} T^{12} + 23 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 3 T + 50 T^{2} + 146 T^{3} + 1373 T^{4} + 4645 T^{5} + 30241 T^{6} + 99122 T^{7} + 30241 p T^{8} + 4645 p^{2} T^{9} + 1373 p^{3} T^{10} + 146 p^{4} T^{11} + 50 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 9 T + 127 T^{2} + 828 T^{3} + 6701 T^{4} + 34195 T^{5} + 201852 T^{6} + 825304 T^{7} + 201852 p T^{8} + 34195 p^{2} T^{9} + 6701 p^{3} T^{10} + 828 p^{4} T^{11} + 127 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 12 T + 150 T^{2} + 1067 T^{3} + 8328 T^{4} + 46995 T^{5} + 292164 T^{6} + 1356265 T^{7} + 292164 p T^{8} + 46995 p^{2} T^{9} + 8328 p^{3} T^{10} + 1067 p^{4} T^{11} + 150 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 9 T + 107 T^{2} - 652 T^{3} + 4764 T^{4} - 22810 T^{5} + 134487 T^{6} - 612145 T^{7} + 134487 p T^{8} - 22810 p^{2} T^{9} + 4764 p^{3} T^{10} - 652 p^{4} T^{11} + 107 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 33 T + 645 T^{2} + 8883 T^{3} + 95284 T^{4} + 826887 T^{5} + 5965621 T^{6} + 36144101 T^{7} + 5965621 p T^{8} + 826887 p^{2} T^{9} + 95284 p^{3} T^{10} + 8883 p^{4} T^{11} + 645 p^{5} T^{12} + 33 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 33 T + 653 T^{2} + 9315 T^{3} + 105122 T^{4} + 971473 T^{5} + 7548399 T^{6} + 49726081 T^{7} + 7548399 p T^{8} + 971473 p^{2} T^{9} + 105122 p^{3} T^{10} + 9315 p^{4} T^{11} + 653 p^{5} T^{12} + 33 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 3 T + 170 T^{2} - 50 T^{3} + 11626 T^{4} + 29278 T^{5} + 496448 T^{6} + 2150172 T^{7} + 496448 p T^{8} + 29278 p^{2} T^{9} + 11626 p^{3} T^{10} - 50 p^{4} T^{11} + 170 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 9 T + 217 T^{2} + 1559 T^{3} + 22058 T^{4} + 131173 T^{5} + 1390533 T^{6} + 6894030 T^{7} + 1390533 p T^{8} + 131173 p^{2} T^{9} + 22058 p^{3} T^{10} + 1559 p^{4} T^{11} + 217 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 11 T + 145 T^{2} + 1339 T^{3} + 14705 T^{4} + 119758 T^{5} + 953008 T^{6} + 6346454 T^{7} + 953008 p T^{8} + 119758 p^{2} T^{9} + 14705 p^{3} T^{10} + 1339 p^{4} T^{11} + 145 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + T + 216 T^{2} - 139 T^{3} + 22869 T^{4} - 30467 T^{5} + 1664224 T^{6} - 2175415 T^{7} + 1664224 p T^{8} - 30467 p^{2} T^{9} + 22869 p^{3} T^{10} - 139 p^{4} T^{11} + 216 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 30 T + 570 T^{2} - 7389 T^{3} + 75460 T^{4} - 617301 T^{5} + 4593588 T^{6} - 33613111 T^{7} + 4593588 p T^{8} - 617301 p^{2} T^{9} + 75460 p^{3} T^{10} - 7389 p^{4} T^{11} + 570 p^{5} T^{12} - 30 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 19 T + 7 p T^{2} + 5649 T^{3} + 75542 T^{4} + 765723 T^{5} + 7510227 T^{6} + 59966461 T^{7} + 7510227 p T^{8} + 765723 p^{2} T^{9} + 75542 p^{3} T^{10} + 5649 p^{4} T^{11} + 7 p^{6} T^{12} + 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 30 T + 675 T^{2} + 11360 T^{3} + 157485 T^{4} + 1838995 T^{5} + 18608402 T^{6} + 162480644 T^{7} + 18608402 p T^{8} + 1838995 p^{2} T^{9} + 157485 p^{3} T^{10} + 11360 p^{4} T^{11} + 675 p^{5} T^{12} + 30 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 8 T + 246 T^{2} + 1787 T^{3} + 34468 T^{4} + 225343 T^{5} + 3268094 T^{6} + 18726838 T^{7} + 3268094 p T^{8} + 225343 p^{2} T^{9} + 34468 p^{3} T^{10} + 1787 p^{4} T^{11} + 246 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 20 T + 510 T^{2} + 6961 T^{3} + 102634 T^{4} + 1069123 T^{5} + 11655826 T^{6} + 97527765 T^{7} + 11655826 p T^{8} + 1069123 p^{2} T^{9} + 102634 p^{3} T^{10} + 6961 p^{4} T^{11} + 510 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 8 T + 273 T^{2} - 2858 T^{3} + 45901 T^{4} - 424499 T^{5} + 5436714 T^{6} - 39832374 T^{7} + 5436714 p T^{8} - 424499 p^{2} T^{9} + 45901 p^{3} T^{10} - 2858 p^{4} T^{11} + 273 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 34 T + 962 T^{2} - 18636 T^{3} + 304656 T^{4} - 4073018 T^{5} + 46715214 T^{6} - 458242049 T^{7} + 46715214 p T^{8} - 4073018 p^{2} T^{9} + 304656 p^{3} T^{10} - 18636 p^{4} T^{11} + 962 p^{5} T^{12} - 34 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 12 T + 151 T^{2} - 2659 T^{3} + 35240 T^{4} - 317069 T^{5} + 3805221 T^{6} - 40673821 T^{7} + 3805221 p T^{8} - 317069 p^{2} T^{9} + 35240 p^{3} T^{10} - 2659 p^{4} T^{11} + 151 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 7 T + 326 T^{2} - 2842 T^{3} + 55405 T^{4} - 456619 T^{5} + 7188211 T^{6} - 47880078 T^{7} + 7188211 p T^{8} - 456619 p^{2} T^{9} + 55405 p^{3} T^{10} - 2842 p^{4} T^{11} + 326 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−4.00135965857059444806225029315, −3.96713110418069451655333286871, −3.93995324189305407984113147747, −3.76954604134175659866502237157, −3.50662786607944416646579748805, −3.50162832179824173452788866040, −3.09937233980481292456569367148, −3.08120065647249204041561772042, −3.03817039343586453247763620269, −3.00822341110264516281721200623, −2.48843884456712230385783681781, −2.38091953429631983289000196751, −2.36971574895449476526623964965, −2.28353828024938539006382076250, −2.15306947128086954361354234104, −2.00482167194477589797158999486, −1.94105546729738287151163311347, −1.92792248334769426393719457636, −1.91285762042880458483414421913, −1.89957056949398439560296119748, −1.55690183422270188191248524935, −1.38055266914188754748785379500, −1.35544145584558757428642471806, −1.07475042600521424257386195034, −1.03589665077963974341060406936, 0, 0, 0, 0, 0, 0, 0, 1.03589665077963974341060406936, 1.07475042600521424257386195034, 1.35544145584558757428642471806, 1.38055266914188754748785379500, 1.55690183422270188191248524935, 1.89957056949398439560296119748, 1.91285762042880458483414421913, 1.92792248334769426393719457636, 1.94105546729738287151163311347, 2.00482167194477589797158999486, 2.15306947128086954361354234104, 2.28353828024938539006382076250, 2.36971574895449476526623964965, 2.38091953429631983289000196751, 2.48843884456712230385783681781, 3.00822341110264516281721200623, 3.03817039343586453247763620269, 3.08120065647249204041561772042, 3.09937233980481292456569367148, 3.50162832179824173452788866040, 3.50662786607944416646579748805, 3.76954604134175659866502237157, 3.93995324189305407984113147747, 3.96713110418069451655333286871, 4.00135965857059444806225029315
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.