Dirichlet series
| L(s) = 1 | + 6·2-s + 24·3-s + 3·4-s + 42·5-s + 144·6-s + 53·7-s − 53·8-s + 324·9-s + 252·10-s + 67·11-s + 72·12-s + 33·13-s + 318·14-s + 1.00e3·15-s − 173·16-s + 139·17-s + 1.94e3·18-s + 64·19-s + 126·20-s + 1.27e3·21-s + 402·22-s + 226·23-s − 1.27e3·24-s + 430·25-s + 198·26-s + 3.24e3·27-s + 159·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 4.61·3-s + 3/8·4-s + 3.75·5-s + 9.79·6-s + 2.86·7-s − 2.34·8-s + 12·9-s + 7.96·10-s + 1.83·11-s + 1.73·12-s + 0.704·13-s + 6.07·14-s + 17.3·15-s − 2.70·16-s + 1.98·17-s + 25.4·18-s + 0.772·19-s + 1.40·20-s + 13.2·21-s + 3.89·22-s + 2.04·23-s − 10.8·24-s + 3.43·25-s + 1.49·26-s + 23.0·27-s + 1.07·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 59^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 59^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 59^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.41486\times 10^{8}\) |
| Root analytic conductor: | \(3.23161\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 59^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(1407.461883\) |
| \(L(\frac12)\) | \(\approx\) | \(1407.461883\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - p T )^{8} \) |
| 59 | \( ( 1 - p T )^{8} \) | |
| good | 2 | \( 1 - 3 p T + 33 T^{2} - 127 T^{3} + 259 p T^{4} - 1719 T^{5} + 381 p^{4} T^{6} - 4631 p^{2} T^{7} + 445 p^{7} T^{8} - 4631 p^{5} T^{9} + 381 p^{10} T^{10} - 1719 p^{9} T^{11} + 259 p^{13} T^{12} - 127 p^{15} T^{13} + 33 p^{18} T^{14} - 3 p^{22} T^{15} + p^{24} T^{16} \) |
| 5 | \( 1 - 42 T + 1334 T^{2} - 30582 T^{3} + 123001 p T^{4} - 2058768 p T^{5} + 6206018 p^{2} T^{6} - 81290888 p^{2} T^{7} + 24260386244 T^{8} - 81290888 p^{5} T^{9} + 6206018 p^{8} T^{10} - 2058768 p^{10} T^{11} + 123001 p^{13} T^{12} - 30582 p^{15} T^{13} + 1334 p^{18} T^{14} - 42 p^{21} T^{15} + p^{24} T^{16} \) | |
| 7 | \( 1 - 53 T + 2691 T^{2} - 89868 T^{3} + 2781125 T^{4} - 70148347 T^{5} + 1659208393 T^{6} - 34379818384 T^{7} + 673155902428 T^{8} - 34379818384 p^{3} T^{9} + 1659208393 p^{6} T^{10} - 70148347 p^{9} T^{11} + 2781125 p^{12} T^{12} - 89868 p^{15} T^{13} + 2691 p^{18} T^{14} - 53 p^{21} T^{15} + p^{24} T^{16} \) | |
| 11 | \( 1 - 67 T + 7098 T^{2} - 254681 T^{3} + 14908169 T^{4} - 152345376 T^{5} + 6950223846 T^{6} + 551171670190 T^{7} - 7352982481060 T^{8} + 551171670190 p^{3} T^{9} + 6950223846 p^{6} T^{10} - 152345376 p^{9} T^{11} + 14908169 p^{12} T^{12} - 254681 p^{15} T^{13} + 7098 p^{18} T^{14} - 67 p^{21} T^{15} + p^{24} T^{16} \) | |
| 13 | \( 1 - 33 T + 10224 T^{2} - 232681 T^{3} + 43095629 T^{4} - 578041352 T^{5} + 105546706546 T^{6} - 48398968682 p T^{7} + 217989954715792 T^{8} - 48398968682 p^{4} T^{9} + 105546706546 p^{6} T^{10} - 578041352 p^{9} T^{11} + 43095629 p^{12} T^{12} - 232681 p^{15} T^{13} + 10224 p^{18} T^{14} - 33 p^{21} T^{15} + p^{24} T^{16} \) | |
| 17 | \( 1 - 139 T + 41715 T^{2} - 4418424 T^{3} + 734411093 T^{4} - 61731375605 T^{5} + 7275749342311 T^{6} - 492047109900680 T^{7} + 44766333624762144 T^{8} - 492047109900680 p^{3} T^{9} + 7275749342311 p^{6} T^{10} - 61731375605 p^{9} T^{11} + 734411093 p^{12} T^{12} - 4418424 p^{15} T^{13} + 41715 p^{18} T^{14} - 139 p^{21} T^{15} + p^{24} T^{16} \) | |
| 19 | \( 1 - 64 T + 33039 T^{2} - 7626 p^{2} T^{3} + 558430853 T^{4} - 49452312066 T^{5} + 6340528997357 T^{6} - 519753872843152 T^{7} + 51291258925776956 T^{8} - 519753872843152 p^{3} T^{9} + 6340528997357 p^{6} T^{10} - 49452312066 p^{9} T^{11} + 558430853 p^{12} T^{12} - 7626 p^{17} T^{13} + 33039 p^{18} T^{14} - 64 p^{21} T^{15} + p^{24} T^{16} \) | |
| 23 | \( 1 - 226 T + 82865 T^{2} - 13806908 T^{3} + 2840253255 T^{4} - 375722632070 T^{5} + 56848929351555 T^{6} - 6344733945900084 T^{7} + 797111609500248712 T^{8} - 6344733945900084 p^{3} T^{9} + 56848929351555 p^{6} T^{10} - 375722632070 p^{9} T^{11} + 2840253255 p^{12} T^{12} - 13806908 p^{15} T^{13} + 82865 p^{18} T^{14} - 226 p^{21} T^{15} + p^{24} T^{16} \) | |
| 29 | \( 1 - 456 T + 212317 T^{2} - 61481552 T^{3} + 17399377291 T^{4} - 3806932662288 T^{5} + 810573027532895 T^{6} - 142182863227164408 T^{7} + 24271763595210245232 T^{8} - 142182863227164408 p^{3} T^{9} + 810573027532895 p^{6} T^{10} - 3806932662288 p^{9} T^{11} + 17399377291 p^{12} T^{12} - 61481552 p^{15} T^{13} + 212317 p^{18} T^{14} - 456 p^{21} T^{15} + p^{24} T^{16} \) | |
| 31 | \( 1 - 4 p T + 167203 T^{2} - 22121936 T^{3} + 13874741399 T^{4} - 1724599993662 T^{5} + 727652609739609 T^{6} - 79823327036454690 T^{7} + 25976144716079534040 T^{8} - 79823327036454690 p^{3} T^{9} + 727652609739609 p^{6} T^{10} - 1724599993662 p^{9} T^{11} + 13874741399 p^{12} T^{12} - 22121936 p^{15} T^{13} + 167203 p^{18} T^{14} - 4 p^{22} T^{15} + p^{24} T^{16} \) | |
| 37 | \( 1 - 127 T + 119817 T^{2} - 7358352 T^{3} + 8210621181 T^{4} - 847287196075 T^{5} + 523229065978253 T^{6} - 74519419704183566 T^{7} + 28449530756661339192 T^{8} - 74519419704183566 p^{3} T^{9} + 523229065978253 p^{6} T^{10} - 847287196075 p^{9} T^{11} + 8210621181 p^{12} T^{12} - 7358352 p^{15} T^{13} + 119817 p^{18} T^{14} - 127 p^{21} T^{15} + p^{24} T^{16} \) | |
| 41 | \( 1 - 425 T + 383575 T^{2} - 76310048 T^{3} + 39978997629 T^{4} + 2005731677997 T^{5} + 308831425021491 T^{6} + 1206213448425791772 T^{7} - \)\(12\!\cdots\!08\)\( T^{8} + 1206213448425791772 p^{3} T^{9} + 308831425021491 p^{6} T^{10} + 2005731677997 p^{9} T^{11} + 39978997629 p^{12} T^{12} - 76310048 p^{15} T^{13} + 383575 p^{18} T^{14} - 425 p^{21} T^{15} + p^{24} T^{16} \) | |
| 43 | \( 1 + 115 T + 387096 T^{2} + 23934401 T^{3} + 68236859569 T^{4} + 127527611968 T^{5} + 7543306640092286 T^{6} - 313813438035978102 T^{7} + \)\(64\!\cdots\!24\)\( T^{8} - 313813438035978102 p^{3} T^{9} + 7543306640092286 p^{6} T^{10} + 127527611968 p^{9} T^{11} + 68236859569 p^{12} T^{12} + 23934401 p^{15} T^{13} + 387096 p^{18} T^{14} + 115 p^{21} T^{15} + p^{24} T^{16} \) | |
| 47 | \( 1 - 420 T + 510033 T^{2} - 212525604 T^{3} + 132434573859 T^{4} - 48849781353820 T^{5} + 22635655390360395 T^{6} - 7077924113842259132 T^{7} + \)\(27\!\cdots\!12\)\( T^{8} - 7077924113842259132 p^{3} T^{9} + 22635655390360395 p^{6} T^{10} - 48849781353820 p^{9} T^{11} + 132434573859 p^{12} T^{12} - 212525604 p^{15} T^{13} + 510033 p^{18} T^{14} - 420 p^{21} T^{15} + p^{24} T^{16} \) | |
| 53 | \( 1 - 98 T + 476858 T^{2} - 127872466 T^{3} + 154614099721 T^{4} - 36405470365648 T^{5} + 37170616678496342 T^{6} - 8270833252473074228 T^{7} + \)\(60\!\cdots\!72\)\( T^{8} - 8270833252473074228 p^{3} T^{9} + 37170616678496342 p^{6} T^{10} - 36405470365648 p^{9} T^{11} + 154614099721 p^{12} T^{12} - 127872466 p^{15} T^{13} + 476858 p^{18} T^{14} - 98 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( 1 + 1254 T + 1708513 T^{2} + 1389977778 T^{3} + 1129712821765 T^{4} + 723003867320286 T^{5} + 449668364866905475 T^{6} + \)\(24\!\cdots\!58\)\( T^{7} + \)\(12\!\cdots\!88\)\( T^{8} + \)\(24\!\cdots\!58\)\( p^{3} T^{9} + 449668364866905475 p^{6} T^{10} + 723003867320286 p^{9} T^{11} + 1129712821765 p^{12} T^{12} + 1389977778 p^{15} T^{13} + 1708513 p^{18} T^{14} + 1254 p^{21} T^{15} + p^{24} T^{16} \) | |
| 67 | \( 1 + 1010 T + 1543678 T^{2} + 15481098 p T^{3} + 975933393721 T^{4} + 481989678728880 T^{5} + 363479802337939222 T^{6} + \)\(14\!\cdots\!56\)\( T^{7} + \)\(11\!\cdots\!00\)\( T^{8} + \)\(14\!\cdots\!56\)\( p^{3} T^{9} + 363479802337939222 p^{6} T^{10} + 481989678728880 p^{9} T^{11} + 975933393721 p^{12} T^{12} + 15481098 p^{16} T^{13} + 1543678 p^{18} T^{14} + 1010 p^{21} T^{15} + p^{24} T^{16} \) | |
| 71 | \( 1 + 17 T + 959370 T^{2} + 36551791 T^{3} + 592934517177 T^{4} - 44411276389336 T^{5} + 284575980834926662 T^{6} - 24574896669827933770 T^{7} + \)\(10\!\cdots\!76\)\( T^{8} - 24574896669827933770 p^{3} T^{9} + 284575980834926662 p^{6} T^{10} - 44411276389336 p^{9} T^{11} + 592934517177 p^{12} T^{12} + 36551791 p^{15} T^{13} + 959370 p^{18} T^{14} + 17 p^{21} T^{15} + p^{24} T^{16} \) | |
| 73 | \( 1 + 1180 T + 1946205 T^{2} + 1332995726 T^{3} + 1247406522885 T^{4} + 529439079938304 T^{5} + 410685021899663427 T^{6} + \)\(11\!\cdots\!98\)\( T^{7} + \)\(12\!\cdots\!48\)\( T^{8} + \)\(11\!\cdots\!98\)\( p^{3} T^{9} + 410685021899663427 p^{6} T^{10} + 529439079938304 p^{9} T^{11} + 1247406522885 p^{12} T^{12} + 1332995726 p^{15} T^{13} + 1946205 p^{18} T^{14} + 1180 p^{21} T^{15} + p^{24} T^{16} \) | |
| 79 | \( 1 + 873 T + 3173490 T^{2} + 2401583303 T^{3} + 4657322517545 T^{4} + 3044522854888160 T^{5} + 4146909464738296566 T^{6} + \)\(23\!\cdots\!94\)\( T^{7} + \)\(24\!\cdots\!72\)\( T^{8} + \)\(23\!\cdots\!94\)\( p^{3} T^{9} + 4146909464738296566 p^{6} T^{10} + 3044522854888160 p^{9} T^{11} + 4657322517545 p^{12} T^{12} + 2401583303 p^{15} T^{13} + 3173490 p^{18} T^{14} + 873 p^{21} T^{15} + p^{24} T^{16} \) | |
| 83 | \( 1 - 759 T + 4074169 T^{2} - 2541429334 T^{3} + 7402782870155 T^{4} - 3845087154866879 T^{5} + 7948194156208147511 T^{6} - \)\(34\!\cdots\!40\)\( T^{7} + \)\(55\!\cdots\!72\)\( T^{8} - \)\(34\!\cdots\!40\)\( p^{3} T^{9} + 7948194156208147511 p^{6} T^{10} - 3845087154866879 p^{9} T^{11} + 7402782870155 p^{12} T^{12} - 2541429334 p^{15} T^{13} + 4074169 p^{18} T^{14} - 759 p^{21} T^{15} + p^{24} T^{16} \) | |
| 89 | \( 1 - 988 T + 2203135 T^{2} - 2366626630 T^{3} + 3321595962725 T^{4} - 2662106139728074 T^{5} + 3373694952184310825 T^{6} - \)\(24\!\cdots\!64\)\( T^{7} + \)\(24\!\cdots\!60\)\( T^{8} - \)\(24\!\cdots\!64\)\( p^{3} T^{9} + 3373694952184310825 p^{6} T^{10} - 2662106139728074 p^{9} T^{11} + 3321595962725 p^{12} T^{12} - 2366626630 p^{15} T^{13} + 2203135 p^{18} T^{14} - 988 p^{21} T^{15} + p^{24} T^{16} \) | |
| 97 | \( 1 + 668 T + 4219340 T^{2} + 2402993618 T^{3} + 8641245317515 T^{4} + 4088976800053608 T^{5} + 11577670635533381980 T^{6} + \)\(46\!\cdots\!22\)\( T^{7} + \)\(11\!\cdots\!60\)\( T^{8} + \)\(46\!\cdots\!22\)\( p^{3} T^{9} + 11577670635533381980 p^{6} T^{10} + 4088976800053608 p^{9} T^{11} + 8641245317515 p^{12} T^{12} + 2402993618 p^{15} T^{13} + 4219340 p^{18} T^{14} + 668 p^{21} T^{15} + p^{24} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.35860817900245858538110485146, −4.86128461949867782747229504820, −4.78175984910134312931435393127, −4.53098844311359287533723092040, −4.44347615582221479879858583209, −4.43475181838014569196791164359, −4.33758419198079730335547234243, −4.24570465544780806227149436518, −4.22159345571315775540981695953, −3.52697469299822199448925613870, −3.42890203885916183776586359926, −3.36664135883380148032726680006, −3.25581763744218795927894628597, −3.05264810605709453813499907702, −2.81465768932797624550635757335, −2.75514719633032800620745044377, −2.22294934577190646855413797929, −2.14835795249272276993995709581, −2.07608455963124982518416107762, −1.82446042836825657889543150674, −1.59192764700765547128510977036, −1.35041683845663168362340337737, −1.29915326717631840793766491340, −1.08595591761035775567808517584, −0.828348935450798559131087068459, 0.828348935450798559131087068459, 1.08595591761035775567808517584, 1.29915326717631840793766491340, 1.35041683845663168362340337737, 1.59192764700765547128510977036, 1.82446042836825657889543150674, 2.07608455963124982518416107762, 2.14835795249272276993995709581, 2.22294934577190646855413797929, 2.75514719633032800620745044377, 2.81465768932797624550635757335, 3.05264810605709453813499907702, 3.25581763744218795927894628597, 3.36664135883380148032726680006, 3.42890203885916183776586359926, 3.52697469299822199448925613870, 4.22159345571315775540981695953, 4.24570465544780806227149436518, 4.33758419198079730335547234243, 4.43475181838014569196791164359, 4.44347615582221479879858583209, 4.53098844311359287533723092040, 4.78175984910134312931435393127, 4.86128461949867782747229504820, 5.35860817900245858538110485146
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.