Dirichlet series
| L(s) = 1 | − 2·2-s − 4·3-s − 5·4-s − 4·5-s + 8·6-s − 12·7-s + 13·8-s − 3·9-s + 8·10-s − 7·11-s + 20·12-s − 7·13-s + 24·14-s + 16·15-s + 8·16-s − 4·17-s + 6·18-s − 20·19-s + 20·20-s + 48·21-s + 14·22-s + 23-s − 52·24-s − 6·25-s + 14·26-s + 39·27-s + 60·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s − 5/2·4-s − 1.78·5-s + 3.26·6-s − 4.53·7-s + 4.59·8-s − 9-s + 2.52·10-s − 2.11·11-s + 5.77·12-s − 1.94·13-s + 6.41·14-s + 4.13·15-s + 2·16-s − 0.970·17-s + 1.41·18-s − 4.58·19-s + 4.47·20-s + 10.4·21-s + 2.98·22-s + 0.208·23-s − 10.6·24-s − 6/5·25-s + 2.74·26-s + 7.50·27-s + 11.3·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(281^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(281^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(281^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(286.341\) |
| Root analytic conductor: | \(1.49793\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 281^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
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 | 281 | \( ( 1 + T )^{7} \) |
| good | 2 | \( 1 + p T + 9 T^{2} + 15 T^{3} + 41 T^{4} + 29 p T^{5} + 15 p^{3} T^{6} + 143 T^{7} + 15 p^{4} T^{8} + 29 p^{3} T^{9} + 41 p^{3} T^{10} + 15 p^{4} T^{11} + 9 p^{5} T^{12} + p^{7} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 + 4 T + 19 T^{2} + 49 T^{3} + 139 T^{4} + 10 p^{3} T^{5} + 589 T^{6} + 953 T^{7} + 589 p T^{8} + 10 p^{5} T^{9} + 139 p^{3} T^{10} + 49 p^{4} T^{11} + 19 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 5 | \( 1 + 4 T + 22 T^{2} + 62 T^{3} + 191 T^{4} + 416 T^{5} + 1041 T^{6} + 2069 T^{7} + 1041 p T^{8} + 416 p^{2} T^{9} + 191 p^{3} T^{10} + 62 p^{4} T^{11} + 22 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 + 12 T + 2 p^{2} T^{2} + 559 T^{3} + 53 p^{2} T^{4} + 1410 p T^{5} + 4632 p T^{6} + 91453 T^{7} + 4632 p^{2} T^{8} + 1410 p^{3} T^{9} + 53 p^{5} T^{10} + 559 p^{4} T^{11} + 2 p^{7} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 7 T + 5 p T^{2} + 17 p T^{3} + 7 p^{2} T^{4} + 1702 T^{5} + 8009 T^{6} + 13945 T^{7} + 8009 p T^{8} + 1702 p^{2} T^{9} + 7 p^{5} T^{10} + 17 p^{5} T^{11} + 5 p^{6} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + 7 T + 75 T^{2} + 379 T^{3} + 2363 T^{4} + 9356 T^{5} + 44537 T^{6} + 146019 T^{7} + 44537 p T^{8} + 9356 p^{2} T^{9} + 2363 p^{3} T^{10} + 379 p^{4} T^{11} + 75 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 4 T + 77 T^{2} + 276 T^{3} + 2872 T^{4} + 8810 T^{5} + 69214 T^{6} + 179139 T^{7} + 69214 p T^{8} + 8810 p^{2} T^{9} + 2872 p^{3} T^{10} + 276 p^{4} T^{11} + 77 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 20 T + 279 T^{2} + 2748 T^{3} + 22051 T^{4} + 143970 T^{5} + 801128 T^{6} + 3761121 T^{7} + 801128 p T^{8} + 143970 p^{2} T^{9} + 22051 p^{3} T^{10} + 2748 p^{4} T^{11} + 279 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - T + 71 T^{2} - 133 T^{3} + 3319 T^{4} - 5049 T^{5} + 103726 T^{6} - 154553 T^{7} + 103726 p T^{8} - 5049 p^{2} T^{9} + 3319 p^{3} T^{10} - 133 p^{4} T^{11} + 71 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 2 T + 91 T^{2} + 136 T^{3} + 3446 T^{4} + 256 T^{5} + 84068 T^{6} - 117975 T^{7} + 84068 p T^{8} + 256 p^{2} T^{9} + 3446 p^{3} T^{10} + 136 p^{4} T^{11} + 91 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 41 T + 861 T^{2} + 12168 T^{3} + 129951 T^{4} + 1115281 T^{5} + 7962097 T^{6} + 48079617 T^{7} + 7962097 p T^{8} + 1115281 p^{2} T^{9} + 129951 p^{3} T^{10} + 12168 p^{4} T^{11} + 861 p^{5} T^{12} + 41 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 2 T + 141 T^{2} + 361 T^{3} + 10784 T^{4} + 26861 T^{5} + 556172 T^{6} + 1243825 T^{7} + 556172 p T^{8} + 26861 p^{2} T^{9} + 10784 p^{3} T^{10} + 361 p^{4} T^{11} + 141 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 3 p T^{2} + 44 T^{3} + 5419 T^{4} + 8348 T^{5} + 92042 T^{6} + 515617 T^{7} + 92042 p T^{8} + 8348 p^{2} T^{9} + 5419 p^{3} T^{10} + 44 p^{4} T^{11} + 3 p^{6} T^{12} + p^{7} T^{14} \) | |
| 43 | \( 1 + 3 T + 171 T^{2} + 276 T^{3} + 14303 T^{4} + 6609 T^{5} + 792239 T^{6} + 40999 T^{7} + 792239 p T^{8} + 6609 p^{2} T^{9} + 14303 p^{3} T^{10} + 276 p^{4} T^{11} + 171 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - T + 106 T^{2} - 206 T^{3} + 7960 T^{4} - 16018 T^{5} + 491801 T^{6} - 545769 T^{7} + 491801 p T^{8} - 16018 p^{2} T^{9} + 7960 p^{3} T^{10} - 206 p^{4} T^{11} + 106 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 17 T + 444 T^{2} - 5264 T^{3} + 77100 T^{4} - 686226 T^{5} + 7069481 T^{6} - 48225581 T^{7} + 7069481 p T^{8} - 686226 p^{2} T^{9} + 77100 p^{3} T^{10} - 5264 p^{4} T^{11} + 444 p^{5} T^{12} - 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 13 T + 258 T^{2} + 2931 T^{3} + 35939 T^{4} + 334078 T^{5} + 3145973 T^{6} + 414733 p T^{7} + 3145973 p T^{8} + 334078 p^{2} T^{9} + 35939 p^{3} T^{10} + 2931 p^{4} T^{11} + 258 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 9 T + 382 T^{2} + 2909 T^{3} + 64906 T^{4} + 413109 T^{5} + 6359219 T^{6} + 32822579 T^{7} + 6359219 p T^{8} + 413109 p^{2} T^{9} + 64906 p^{3} T^{10} + 2909 p^{4} T^{11} + 382 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 3 T + 246 T^{2} + 521 T^{3} + 32407 T^{4} + 60602 T^{5} + 3012993 T^{6} + 5250245 T^{7} + 3012993 p T^{8} + 60602 p^{2} T^{9} + 32407 p^{3} T^{10} + 521 p^{4} T^{11} + 246 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 8 T + 325 T^{2} + 2000 T^{3} + 50554 T^{4} + 266464 T^{5} + 5172662 T^{6} + 23317851 T^{7} + 5172662 p T^{8} + 266464 p^{2} T^{9} + 50554 p^{3} T^{10} + 2000 p^{4} T^{11} + 325 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 9 T + 420 T^{2} + 3562 T^{3} + 80447 T^{4} + 613504 T^{5} + 9140707 T^{6} + 58589551 T^{7} + 9140707 p T^{8} + 613504 p^{2} T^{9} + 80447 p^{3} T^{10} + 3562 p^{4} T^{11} + 420 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 32 T + 768 T^{2} + 12200 T^{3} + 167427 T^{4} + 1837526 T^{5} + 18911187 T^{6} + 169703141 T^{7} + 18911187 p T^{8} + 1837526 p^{2} T^{9} + 167427 p^{3} T^{10} + 12200 p^{4} T^{11} + 768 p^{5} T^{12} + 32 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 19 T + 447 T^{2} - 5708 T^{3} + 83720 T^{4} - 879567 T^{5} + 10195499 T^{6} - 89964333 T^{7} + 10195499 p T^{8} - 879567 p^{2} T^{9} + 83720 p^{3} T^{10} - 5708 p^{4} T^{11} + 447 p^{5} T^{12} - 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 19 T + 354 T^{2} + 4378 T^{3} + 66723 T^{4} + 734988 T^{5} + 8319611 T^{6} + 72907301 T^{7} + 8319611 p T^{8} + 734988 p^{2} T^{9} + 66723 p^{3} T^{10} + 4378 p^{4} T^{11} + 354 p^{5} T^{12} + 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 8 T + 549 T^{2} - 4536 T^{3} + 138610 T^{4} - 1072854 T^{5} + 20894882 T^{6} - 137393457 T^{7} + 20894882 p T^{8} - 1072854 p^{2} T^{9} + 138610 p^{3} T^{10} - 4536 p^{4} T^{11} + 549 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.34678676820549694068553010751, −6.15035498107428970780097652319, −6.11543720791650024207928329237, −5.89616094323065318068734685319, −5.89141678177081728494653911979, −5.58164819039525448648736694880, −5.32605891816178610784152666915, −5.29613423736032361184031428580, −5.23991723306277044140734668799, −4.97529442749967332070860116476, −4.93054380911553532545501505634, −4.45932304979261015420349870331, −4.39251485821711677277954777063, −4.25928897303008177980220111457, −3.97230515968157272645198769952, −3.88061524495113906451478558810, −3.71988865505446053413988485439, −3.52285728720877864562146960071, −3.29829961265860336091332816138, −2.98944869240213258032792052921, −2.96221525236079866860070015242, −2.66707970025067239422157328007, −2.24021784305655155594558329869, −2.16549391209167219496028651966, −1.84960585360784252074799543769, 0, 0, 0, 0, 0, 0, 0, 1.84960585360784252074799543769, 2.16549391209167219496028651966, 2.24021784305655155594558329869, 2.66707970025067239422157328007, 2.96221525236079866860070015242, 2.98944869240213258032792052921, 3.29829961265860336091332816138, 3.52285728720877864562146960071, 3.71988865505446053413988485439, 3.88061524495113906451478558810, 3.97230515968157272645198769952, 4.25928897303008177980220111457, 4.39251485821711677277954777063, 4.45932304979261015420349870331, 4.93054380911553532545501505634, 4.97529442749967332070860116476, 5.23991723306277044140734668799, 5.29613423736032361184031428580, 5.32605891816178610784152666915, 5.58164819039525448648736694880, 5.89141678177081728494653911979, 5.89616094323065318068734685319, 6.11543720791650024207928329237, 6.15035498107428970780097652319, 6.34678676820549694068553010751
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.