Dirichlet series
| L(s) = 1 | − 2-s − 21·3-s − 17·4-s + 11·5-s + 21·6-s − 33·7-s + 5·8-s + 252·9-s − 11·10-s − 130·11-s + 357·12-s + 16·13-s + 33·14-s − 231·15-s + 134·16-s + 90·17-s − 252·18-s − 132·19-s − 187·20-s + 693·21-s + 130·22-s − 399·23-s − 105·24-s − 443·25-s − 16·26-s − 2.26e3·27-s + 561·28-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 4.04·3-s − 2.12·4-s + 0.983·5-s + 1.42·6-s − 1.78·7-s + 0.220·8-s + 28/3·9-s − 0.347·10-s − 3.56·11-s + 8.58·12-s + 0.341·13-s + 0.629·14-s − 3.97·15-s + 2.09·16-s + 1.28·17-s − 3.29·18-s − 1.59·19-s − 2.09·20-s + 7.20·21-s + 1.25·22-s − 3.61·23-s − 0.893·24-s − 3.54·25-s − 0.120·26-s − 16.1·27-s + 3.78·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 67^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 67^{7}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{7} \cdot 67^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(3.29939\times 10^{7}\) |
| Root analytic conductor: | \(3.44374\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 3^{7} \cdot 67^{7} ,\ ( \ : [3/2]^{7} ),\ -1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p T )^{7} \) |
| 67 | \( ( 1 - p T )^{7} \) | |
| good | 2 | \( 1 + T + 9 p T^{2} + 15 p T^{3} + 197 T^{4} + 535 T^{5} + 399 p^{2} T^{6} + 2623 p T^{7} + 399 p^{5} T^{8} + 535 p^{6} T^{9} + 197 p^{9} T^{10} + 15 p^{13} T^{11} + 9 p^{16} T^{12} + p^{18} T^{13} + p^{21} T^{14} \) |
| 5 | \( 1 - 11 T + 564 T^{2} - 4911 T^{3} + 151982 T^{4} - 1060709 T^{5} + 26199969 T^{6} - 153622402 T^{7} + 26199969 p^{3} T^{8} - 1060709 p^{6} T^{9} + 151982 p^{9} T^{10} - 4911 p^{12} T^{11} + 564 p^{15} T^{12} - 11 p^{18} T^{13} + p^{21} T^{14} \) | |
| 7 | \( 1 + 33 T + 1588 T^{2} + 42261 T^{3} + 1234294 T^{4} + 27110135 T^{5} + 630536859 T^{6} + 11313814502 T^{7} + 630536859 p^{3} T^{8} + 27110135 p^{6} T^{9} + 1234294 p^{9} T^{10} + 42261 p^{12} T^{11} + 1588 p^{15} T^{12} + 33 p^{18} T^{13} + p^{21} T^{14} \) | |
| 11 | \( 1 + 130 T + 12649 T^{2} + 795976 T^{3} + 43010377 T^{4} + 1823159982 T^{5} + 73998893113 T^{6} + 2649933947120 T^{7} + 73998893113 p^{3} T^{8} + 1823159982 p^{6} T^{9} + 43010377 p^{9} T^{10} + 795976 p^{12} T^{11} + 12649 p^{15} T^{12} + 130 p^{18} T^{13} + p^{21} T^{14} \) | |
| 13 | \( 1 - 16 T + 7959 T^{2} - 109108 T^{3} + 31715953 T^{4} - 346791968 T^{5} + 86867713007 T^{6} - 773800401336 T^{7} + 86867713007 p^{3} T^{8} - 346791968 p^{6} T^{9} + 31715953 p^{9} T^{10} - 109108 p^{12} T^{11} + 7959 p^{15} T^{12} - 16 p^{18} T^{13} + p^{21} T^{14} \) | |
| 17 | \( 1 - 90 T + 22744 T^{2} - 1911672 T^{3} + 225354330 T^{4} - 18527392242 T^{5} + 1406248452333 T^{6} - 111062566559448 T^{7} + 1406248452333 p^{3} T^{8} - 18527392242 p^{6} T^{9} + 225354330 p^{9} T^{10} - 1911672 p^{12} T^{11} + 22744 p^{15} T^{12} - 90 p^{18} T^{13} + p^{21} T^{14} \) | |
| 19 | \( 1 + 132 T + 27166 T^{2} + 2676198 T^{3} + 306425632 T^{4} + 27877812224 T^{5} + 2477154191679 T^{6} + 218161488799340 T^{7} + 2477154191679 p^{3} T^{8} + 27877812224 p^{6} T^{9} + 306425632 p^{9} T^{10} + 2676198 p^{12} T^{11} + 27166 p^{15} T^{12} + 132 p^{18} T^{13} + p^{21} T^{14} \) | |
| 23 | \( 1 + 399 T + 120929 T^{2} + 26998356 T^{3} + 5056877920 T^{4} + 789200793876 T^{5} + 107618659812076 T^{6} + 12648457670971446 T^{7} + 107618659812076 p^{3} T^{8} + 789200793876 p^{6} T^{9} + 5056877920 p^{9} T^{10} + 26998356 p^{12} T^{11} + 120929 p^{15} T^{12} + 399 p^{18} T^{13} + p^{21} T^{14} \) | |
| 29 | \( 1 + 302 T + 118692 T^{2} + 24523836 T^{3} + 211525510 p T^{4} + 976849670402 T^{5} + 6762459808413 p T^{6} + 27082052279613256 T^{7} + 6762459808413 p^{4} T^{8} + 976849670402 p^{6} T^{9} + 211525510 p^{10} T^{10} + 24523836 p^{12} T^{11} + 118692 p^{15} T^{12} + 302 p^{18} T^{13} + p^{21} T^{14} \) | |
| 31 | \( 1 + 555 T + 271396 T^{2} + 88527427 T^{3} + 25510960062 T^{4} + 5959786423933 T^{5} + 1257751884350603 T^{6} + 227313677390195178 T^{7} + 1257751884350603 p^{3} T^{8} + 5959786423933 p^{6} T^{9} + 25510960062 p^{9} T^{10} + 88527427 p^{12} T^{11} + 271396 p^{15} T^{12} + 555 p^{18} T^{13} + p^{21} T^{14} \) | |
| 37 | \( 1 - 297 T + 244257 T^{2} - 53082932 T^{3} + 27333286436 T^{4} - 4757108863596 T^{5} + 1945271636872882 T^{6} - 284859005095362286 T^{7} + 1945271636872882 p^{3} T^{8} - 4757108863596 p^{6} T^{9} + 27333286436 p^{9} T^{10} - 53082932 p^{12} T^{11} + 244257 p^{15} T^{12} - 297 p^{18} T^{13} + p^{21} T^{14} \) | |
| 41 | \( 1 + 717 T + 524798 T^{2} + 217402755 T^{3} + 92867110408 T^{4} + 28152925652091 T^{5} + 9212861250096577 T^{6} + 2301992224371063114 T^{7} + 9212861250096577 p^{3} T^{8} + 28152925652091 p^{6} T^{9} + 92867110408 p^{9} T^{10} + 217402755 p^{12} T^{11} + 524798 p^{15} T^{12} + 717 p^{18} T^{13} + p^{21} T^{14} \) | |
| 43 | \( 1 + 245 T + 359696 T^{2} + 76388477 T^{3} + 62401534594 T^{4} + 11109706720611 T^{5} + 6942528586006991 T^{6} + 1048659332073804406 T^{7} + 6942528586006991 p^{3} T^{8} + 11109706720611 p^{6} T^{9} + 62401534594 p^{9} T^{10} + 76388477 p^{12} T^{11} + 359696 p^{15} T^{12} + 245 p^{18} T^{13} + p^{21} T^{14} \) | |
| 47 | \( 1 + 1072 T + 701142 T^{2} + 361323318 T^{3} + 171310164116 T^{4} + 71686508192356 T^{5} + 26832050950548543 T^{6} + 8985939150556985372 T^{7} + 26832050950548543 p^{3} T^{8} + 71686508192356 p^{6} T^{9} + 171310164116 p^{9} T^{10} + 361323318 p^{12} T^{11} + 701142 p^{15} T^{12} + 1072 p^{18} T^{13} + p^{21} T^{14} \) | |
| 53 | \( 1 - 5 p T + 755634 T^{2} - 182171295 T^{3} + 266533136548 T^{4} - 56856210837867 T^{5} + 58221326573118101 T^{6} - 10598351489083918522 T^{7} + 58221326573118101 p^{3} T^{8} - 56856210837867 p^{6} T^{9} + 266533136548 p^{9} T^{10} - 182171295 p^{12} T^{11} + 755634 p^{15} T^{12} - 5 p^{19} T^{13} + p^{21} T^{14} \) | |
| 59 | \( 1 + 255 T + 969971 T^{2} + 205350210 T^{3} + 446647333452 T^{4} + 77109073022682 T^{5} + 130446123069888346 T^{6} + 318413138268648246 p T^{7} + 130446123069888346 p^{3} T^{8} + 77109073022682 p^{6} T^{9} + 446647333452 p^{9} T^{10} + 205350210 p^{12} T^{11} + 969971 p^{15} T^{12} + 255 p^{18} T^{13} + p^{21} T^{14} \) | |
| 61 | \( 1 - 418 T + 533547 T^{2} - 69919496 T^{3} + 165816862261 T^{4} - 21570768811310 T^{5} + 56110297038126535 T^{6} - 8894859365737545968 T^{7} + 56110297038126535 p^{3} T^{8} - 21570768811310 p^{6} T^{9} + 165816862261 p^{9} T^{10} - 69919496 p^{12} T^{11} + 533547 p^{15} T^{12} - 418 p^{18} T^{13} + p^{21} T^{14} \) | |
| 71 | \( 1 + 1194 T + 2119729 T^{2} + 2030552208 T^{3} + 2030237493621 T^{4} + 1591046209096086 T^{5} + 1143282411189789429 T^{6} + \)\(72\!\cdots\!92\)\( T^{7} + 1143282411189789429 p^{3} T^{8} + 1591046209096086 p^{6} T^{9} + 2030237493621 p^{9} T^{10} + 2030552208 p^{12} T^{11} + 2119729 p^{15} T^{12} + 1194 p^{18} T^{13} + p^{21} T^{14} \) | |
| 73 | \( 1 - 995 T + 1995565 T^{2} - 1405191688 T^{3} + 1682422298588 T^{4} - 898691201756752 T^{5} + 862620396675994770 T^{6} - \)\(39\!\cdots\!22\)\( T^{7} + 862620396675994770 p^{3} T^{8} - 898691201756752 p^{6} T^{9} + 1682422298588 p^{9} T^{10} - 1405191688 p^{12} T^{11} + 1995565 p^{15} T^{12} - 995 p^{18} T^{13} + p^{21} T^{14} \) | |
| 79 | \( 1 + 2640 T + 4147213 T^{2} + 4352043552 T^{3} + 3750155025025 T^{4} + 2811865160939504 T^{5} + 2095389973016555157 T^{6} + \)\(14\!\cdots\!32\)\( T^{7} + 2095389973016555157 p^{3} T^{8} + 2811865160939504 p^{6} T^{9} + 3750155025025 p^{9} T^{10} + 4352043552 p^{12} T^{11} + 4147213 p^{15} T^{12} + 2640 p^{18} T^{13} + p^{21} T^{14} \) | |
| 83 | \( 1 + 2579 T + 5794900 T^{2} + 8508860117 T^{3} + 11216042833030 T^{4} + 11632488029207865 T^{5} + 11056776765634490023 T^{6} + \)\(87\!\cdots\!70\)\( T^{7} + 11056776765634490023 p^{3} T^{8} + 11632488029207865 p^{6} T^{9} + 11216042833030 p^{9} T^{10} + 8508860117 p^{12} T^{11} + 5794900 p^{15} T^{12} + 2579 p^{18} T^{13} + p^{21} T^{14} \) | |
| 89 | \( 1 + 1604 T + 3567844 T^{2} + 3909328178 T^{3} + 5592539105542 T^{4} + 56161006918752 p T^{5} + 5685233180415121117 T^{6} + \)\(42\!\cdots\!28\)\( T^{7} + 5685233180415121117 p^{3} T^{8} + 56161006918752 p^{7} T^{9} + 5592539105542 p^{9} T^{10} + 3909328178 p^{12} T^{11} + 3567844 p^{15} T^{12} + 1604 p^{18} T^{13} + p^{21} T^{14} \) | |
| 97 | \( 1 + 808 T + 2914363 T^{2} + 2834641908 T^{3} + 5362881471945 T^{4} + 4736280886443000 T^{5} + 7063874922316465019 T^{6} + \)\(50\!\cdots\!92\)\( T^{7} + 7063874922316465019 p^{3} T^{8} + 4736280886443000 p^{6} T^{9} + 5362881471945 p^{9} T^{10} + 2834641908 p^{12} T^{11} + 2914363 p^{15} T^{12} + 808 p^{18} T^{13} + p^{21} T^{14} \) | |
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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
−6.16564977061532937386079742212, −6.10066470819373576559670832206, −5.85951561873273689613271451889, −5.78824429574468743777258371203, −5.55222917138779569494868050327, −5.53383123461648469749152591037, −5.18122565366868498781456821423, −5.15408313951818526000535237729, −5.05309081581990871273255910736, −4.95028748439116911709406261047, −4.93646751948018415383376413498, −4.16910835378809975419789832129, −4.03832076092577603264777613661, −3.97235633188766596751258606339, −3.95680369860517672465497797938, −3.86457209866958767910738699770, −3.53755083455186998000745540419, −3.22936243609164713926677344935, −2.81835184548955140560614178987, −2.59190208003117246628181082690, −2.25727955595924697580377979475, −1.84421275672066076522139511220, −1.59973788613426455576143405669, −1.59414625185452997779919612534, −1.39009802536701110048075548297, 0, 0, 0, 0, 0, 0, 0, 1.39009802536701110048075548297, 1.59414625185452997779919612534, 1.59973788613426455576143405669, 1.84421275672066076522139511220, 2.25727955595924697580377979475, 2.59190208003117246628181082690, 2.81835184548955140560614178987, 3.22936243609164713926677344935, 3.53755083455186998000745540419, 3.86457209866958767910738699770, 3.95680369860517672465497797938, 3.97235633188766596751258606339, 4.03832076092577603264777613661, 4.16910835378809975419789832129, 4.93646751948018415383376413498, 4.95028748439116911709406261047, 5.05309081581990871273255910736, 5.15408313951818526000535237729, 5.18122565366868498781456821423, 5.53383123461648469749152591037, 5.55222917138779569494868050327, 5.78824429574468743777258371203, 5.85951561873273689613271451889, 6.10066470819373576559670832206, 6.16564977061532937386079742212
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.