Dirichlet series
| L(s) = 1 | − 16·2-s + 4·3-s + 144·4-s − 64·6-s + 27·7-s − 960·8-s − 62·9-s − 39·11-s + 576·12-s + 179·13-s − 432·14-s + 5.28e3·16-s + 247·17-s + 992·18-s − 25·19-s + 108·21-s + 624·22-s + 179·23-s − 3.84e3·24-s − 2.86e3·26-s − 102·27-s + 3.88e3·28-s − 565·29-s − 89·31-s − 2.53e4·32-s − 156·33-s − 3.95e3·34-s + ⋯ |
| L(s) = 1 | − 5.65·2-s + 0.769·3-s + 18·4-s − 4.35·6-s + 1.45·7-s − 42.4·8-s − 2.29·9-s − 1.06·11-s + 13.8·12-s + 3.81·13-s − 8.24·14-s + 82.5·16-s + 3.52·17-s + 12.9·18-s − 0.301·19-s + 1.12·21-s + 6.04·22-s + 1.62·23-s − 32.6·24-s − 21.6·26-s − 0.727·27-s + 26.2·28-s − 3.61·29-s − 0.515·31-s − 140.·32-s − 0.822·33-s − 19.9·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 5^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.75404\times 10^{14}\) |
| Root analytic conductor: | \(8.58792\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 5^{32} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(1.337726004\) |
| \(L(\frac12)\) | \(\approx\) | \(1.337726004\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p T )^{8} \) |
| 5 | \( 1 \) | |
| good | 3 | \( 1 - 4 T + 26 p T^{2} - 458 T^{3} + 3736 T^{4} - 21284 T^{5} + 47725 p T^{6} - 689974 T^{7} + 4284844 T^{8} - 689974 p^{3} T^{9} + 47725 p^{7} T^{10} - 21284 p^{9} T^{11} + 3736 p^{12} T^{12} - 458 p^{15} T^{13} + 26 p^{19} T^{14} - 4 p^{21} T^{15} + p^{24} T^{16} \) |
| 7 | \( 1 - 27 T + 1312 T^{2} - 21346 T^{3} + 754721 T^{4} - 1255896 p T^{5} + 303427660 T^{6} - 2702813353 T^{7} + 104452113604 T^{8} - 2702813353 p^{3} T^{9} + 303427660 p^{6} T^{10} - 1255896 p^{10} T^{11} + 754721 p^{12} T^{12} - 21346 p^{15} T^{13} + 1312 p^{18} T^{14} - 27 p^{21} T^{15} + p^{24} T^{16} \) | |
| 11 | \( 1 + 39 T + 3350 T^{2} + 111410 T^{3} + 774335 p T^{4} + 295217122 T^{5} + 17381158818 T^{6} + 489155464115 T^{7} + 23841275119380 T^{8} + 489155464115 p^{3} T^{9} + 17381158818 p^{6} T^{10} + 295217122 p^{9} T^{11} + 774335 p^{13} T^{12} + 111410 p^{15} T^{13} + 3350 p^{18} T^{14} + 39 p^{21} T^{15} + p^{24} T^{16} \) | |
| 13 | \( 1 - 179 T + 18963 T^{2} - 1425043 T^{3} + 83285016 T^{4} - 3916072579 T^{5} + 156156827690 T^{6} - 5745676940154 T^{7} + 237221890478099 T^{8} - 5745676940154 p^{3} T^{9} + 156156827690 p^{6} T^{10} - 3916072579 p^{9} T^{11} + 83285016 p^{12} T^{12} - 1425043 p^{15} T^{13} + 18963 p^{18} T^{14} - 179 p^{21} T^{15} + p^{24} T^{16} \) | |
| 17 | \( 1 - 247 T + 43092 T^{2} - 5416366 T^{3} + 35109328 p T^{4} - 56707178117 T^{5} + 5053326009255 T^{6} - 403779423876048 T^{7} + 29993946177008184 T^{8} - 403779423876048 p^{3} T^{9} + 5053326009255 p^{6} T^{10} - 56707178117 p^{9} T^{11} + 35109328 p^{13} T^{12} - 5416366 p^{15} T^{13} + 43092 p^{18} T^{14} - 247 p^{21} T^{15} + p^{24} T^{16} \) | |
| 19 | \( 1 + 25 T + 29022 T^{2} + 1048310 T^{3} + 24456062 p T^{4} + 16885403025 T^{5} + 5078356152059 T^{6} + 173645627039000 T^{7} + 40137236328736080 T^{8} + 173645627039000 p^{3} T^{9} + 5078356152059 p^{6} T^{10} + 16885403025 p^{9} T^{11} + 24456062 p^{13} T^{12} + 1048310 p^{15} T^{13} + 29022 p^{18} T^{14} + 25 p^{21} T^{15} + p^{24} T^{16} \) | |
| 23 | \( 1 - 179 T + 39323 T^{2} - 4116143 T^{3} + 845669571 T^{4} - 109920609624 T^{5} + 17395875133180 T^{6} - 1630691152015284 T^{7} + 205047824474561674 T^{8} - 1630691152015284 p^{3} T^{9} + 17395875133180 p^{6} T^{10} - 109920609624 p^{9} T^{11} + 845669571 p^{12} T^{12} - 4116143 p^{15} T^{13} + 39323 p^{18} T^{14} - 179 p^{21} T^{15} + p^{24} T^{16} \) | |
| 29 | \( 1 + 565 T + 265917 T^{2} + 82588765 T^{3} + 22616657268 T^{4} + 4944757781865 T^{5} + 998019534529924 T^{6} + 173320809150848750 T^{7} + 28784408280803237505 T^{8} + 173320809150848750 p^{3} T^{9} + 998019534529924 p^{6} T^{10} + 4944757781865 p^{9} T^{11} + 22616657268 p^{12} T^{12} + 82588765 p^{15} T^{13} + 265917 p^{18} T^{14} + 565 p^{21} T^{15} + p^{24} T^{16} \) | |
| 31 | \( 1 + 89 T + 204085 T^{2} + 522785 p T^{3} + 19028053945 T^{4} + 1325266015752 T^{5} + 1059716594028998 T^{6} + 62973270985320070 T^{7} + 38581577327933099030 T^{8} + 62973270985320070 p^{3} T^{9} + 1059716594028998 p^{6} T^{10} + 1325266015752 p^{9} T^{11} + 19028053945 p^{12} T^{12} + 522785 p^{16} T^{13} + 204085 p^{18} T^{14} + 89 p^{21} T^{15} + p^{24} T^{16} \) | |
| 37 | \( 1 - 287 T + 263247 T^{2} - 41739741 T^{3} + 25397450946 T^{4} - 1211311720107 T^{5} + 1241716482853020 T^{6} + 90589591335584382 T^{7} + 50710454906087085309 T^{8} + 90589591335584382 p^{3} T^{9} + 1241716482853020 p^{6} T^{10} - 1211311720107 p^{9} T^{11} + 25397450946 p^{12} T^{12} - 41739741 p^{15} T^{13} + 263247 p^{18} T^{14} - 287 p^{21} T^{15} + p^{24} T^{16} \) | |
| 41 | \( 1 + 394 T + 520380 T^{2} + 175897160 T^{3} + 120118591240 T^{4} + 34514935457472 T^{5} + 16109872456033553 T^{6} + 3861648373034550680 T^{7} + \)\(13\!\cdots\!80\)\( T^{8} + 3861648373034550680 p^{3} T^{9} + 16109872456033553 p^{6} T^{10} + 34514935457472 p^{9} T^{11} + 120118591240 p^{12} T^{12} + 175897160 p^{15} T^{13} + 520380 p^{18} T^{14} + 394 p^{21} T^{15} + p^{24} T^{16} \) | |
| 43 | \( 1 - 529 T + 414328 T^{2} - 139331268 T^{3} + 64713221751 T^{4} - 16515039487214 T^{5} + 6454461910629660 T^{6} - 1478695245459034599 T^{7} + \)\(54\!\cdots\!24\)\( T^{8} - 1478695245459034599 p^{3} T^{9} + 6454461910629660 p^{6} T^{10} - 16515039487214 p^{9} T^{11} + 64713221751 p^{12} T^{12} - 139331268 p^{15} T^{13} + 414328 p^{18} T^{14} - 529 p^{21} T^{15} + p^{24} T^{16} \) | |
| 47 | \( 1 + 758 T + 784692 T^{2} + 439041634 T^{3} + 271966134346 T^{4} + 2521337641614 p T^{5} + 54533132602498955 T^{6} + 19158219614764220602 T^{7} + \)\(69\!\cdots\!84\)\( T^{8} + 19158219614764220602 p^{3} T^{9} + 54533132602498955 p^{6} T^{10} + 2521337641614 p^{10} T^{11} + 271966134346 p^{12} T^{12} + 439041634 p^{15} T^{13} + 784692 p^{18} T^{14} + 758 p^{21} T^{15} + p^{24} T^{16} \) | |
| 53 | \( 1 + 56 T + 861328 T^{2} + 20068102 T^{3} + 354725727136 T^{4} + 1894129166506 T^{5} + 91932536311490645 T^{6} - 103035379136580124 T^{7} + \)\(16\!\cdots\!64\)\( T^{8} - 103035379136580124 p^{3} T^{9} + 91932536311490645 p^{6} T^{10} + 1894129166506 p^{9} T^{11} + 354725727136 p^{12} T^{12} + 20068102 p^{15} T^{13} + 861328 p^{18} T^{14} + 56 p^{21} T^{15} + p^{24} T^{16} \) | |
| 59 | \( 1 + 1220 T + 2036087 T^{2} + 1793989770 T^{3} + 1664695350853 T^{4} + 1121489731769670 T^{5} + 733224544342936169 T^{6} + \)\(38\!\cdots\!00\)\( T^{7} + \)\(19\!\cdots\!80\)\( T^{8} + \)\(38\!\cdots\!00\)\( p^{3} T^{9} + 733224544342936169 p^{6} T^{10} + 1121489731769670 p^{9} T^{11} + 1664695350853 p^{12} T^{12} + 1793989770 p^{15} T^{13} + 2036087 p^{18} T^{14} + 1220 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( 1 + 489 T + 1575055 T^{2} + 621459855 T^{3} + 1111083730090 T^{4} + 361578972829797 T^{5} + 467005356774259768 T^{6} + \)\(12\!\cdots\!30\)\( T^{7} + \)\(12\!\cdots\!65\)\( T^{8} + \)\(12\!\cdots\!30\)\( p^{3} T^{9} + 467005356774259768 p^{6} T^{10} + 361578972829797 p^{9} T^{11} + 1111083730090 p^{12} T^{12} + 621459855 p^{15} T^{13} + 1575055 p^{18} T^{14} + 489 p^{21} T^{15} + p^{24} T^{16} \) | |
| 67 | \( 1 - 2547 T + 4137912 T^{2} - 4723504026 T^{3} + 4326889325011 T^{4} - 3285937502563142 T^{5} + 2194176556517022280 T^{6} - \)\(13\!\cdots\!03\)\( T^{7} + \)\(74\!\cdots\!24\)\( T^{8} - \)\(13\!\cdots\!03\)\( p^{3} T^{9} + 2194176556517022280 p^{6} T^{10} - 3285937502563142 p^{9} T^{11} + 4326889325011 p^{12} T^{12} - 4723504026 p^{15} T^{13} + 4137912 p^{18} T^{14} - 2547 p^{21} T^{15} + p^{24} T^{16} \) | |
| 71 | \( 1 - 1576 T + 2062240 T^{2} - 1802050730 T^{3} + 1564601142700 T^{4} - 1075368517244678 T^{5} + 753772981185501113 T^{6} - \)\(44\!\cdots\!40\)\( T^{7} + 56118484608830905460 p^{2} T^{8} - \)\(44\!\cdots\!40\)\( p^{3} T^{9} + 753772981185501113 p^{6} T^{10} - 1075368517244678 p^{9} T^{11} + 1564601142700 p^{12} T^{12} - 1802050730 p^{15} T^{13} + 2062240 p^{18} T^{14} - 1576 p^{21} T^{15} + p^{24} T^{16} \) | |
| 73 | \( 1 + 166 T + 1843928 T^{2} - 4132708 T^{3} + 1618846353956 T^{4} - 128675140372744 T^{5} + 995478281705972145 T^{6} - 75680988855895892424 T^{7} + \)\(45\!\cdots\!44\)\( T^{8} - 75680988855895892424 p^{3} T^{9} + 995478281705972145 p^{6} T^{10} - 128675140372744 p^{9} T^{11} + 1618846353956 p^{12} T^{12} - 4132708 p^{15} T^{13} + 1843928 p^{18} T^{14} + 166 p^{21} T^{15} + p^{24} T^{16} \) | |
| 79 | \( 1 + 1540 T + 3069157 T^{2} + 3430256170 T^{3} + 54859141207 p T^{4} + 3921664767097740 T^{5} + 3798209682014350499 T^{6} + \)\(28\!\cdots\!50\)\( T^{7} + \)\(22\!\cdots\!80\)\( T^{8} + \)\(28\!\cdots\!50\)\( p^{3} T^{9} + 3798209682014350499 p^{6} T^{10} + 3921664767097740 p^{9} T^{11} + 54859141207 p^{13} T^{12} + 3430256170 p^{15} T^{13} + 3069157 p^{18} T^{14} + 1540 p^{21} T^{15} + p^{24} T^{16} \) | |
| 83 | \( 1 - 929 T + 3192013 T^{2} - 2114311123 T^{3} + 4453876755901 T^{4} - 2243553413996524 T^{5} + 3873090378397321250 T^{6} - \)\(15\!\cdots\!54\)\( T^{7} + \)\(24\!\cdots\!54\)\( T^{8} - \)\(15\!\cdots\!54\)\( p^{3} T^{9} + 3873090378397321250 p^{6} T^{10} - 2243553413996524 p^{9} T^{11} + 4453876755901 p^{12} T^{12} - 2114311123 p^{15} T^{13} + 3192013 p^{18} T^{14} - 929 p^{21} T^{15} + p^{24} T^{16} \) | |
| 89 | \( 1 + 1150 T + 3775147 T^{2} + 2886348600 T^{3} + 6042067745463 T^{4} + 3499900267843900 T^{5} + 6365375889650535029 T^{6} + \)\(31\!\cdots\!50\)\( T^{7} + \)\(51\!\cdots\!20\)\( T^{8} + \)\(31\!\cdots\!50\)\( p^{3} T^{9} + 6365375889650535029 p^{6} T^{10} + 3499900267843900 p^{9} T^{11} + 6042067745463 p^{12} T^{12} + 2886348600 p^{15} T^{13} + 3775147 p^{18} T^{14} + 1150 p^{21} T^{15} + p^{24} T^{16} \) | |
| 97 | \( 1 - 632 T + 4051692 T^{2} - 1487361826 T^{3} + 7677909587811 T^{4} - 1053131834623492 T^{5} + 9327233457339635510 T^{6} + \)\(22\!\cdots\!52\)\( T^{7} + \)\(90\!\cdots\!39\)\( T^{8} + \)\(22\!\cdots\!52\)\( p^{3} T^{9} + 9327233457339635510 p^{6} T^{10} - 1053131834623492 p^{9} T^{11} + 7677909587811 p^{12} T^{12} - 1487361826 p^{15} T^{13} + 4051692 p^{18} T^{14} - 632 p^{21} T^{15} + p^{24} T^{16} \) | |
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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
−3.66144241289903067333037111553, −3.30163130689450921631176645128, −3.17331508518770915275983494402, −3.05697198066675454806759565248, −2.96507106992486802284538267408, −2.94169301659605556175930482926, −2.89505184765455542519566561686, −2.86381884765811122199979320967, −2.67314695962071250172624552724, −2.23555965677225500777846211872, −2.23051924417237627269898298389, −1.91104870107562838974905869903, −1.74736175494083124537429285420, −1.71294857450523396937078706039, −1.69941050166545581097406100181, −1.56735006731892549212051452426, −1.47519469060589493827059673935, −1.21844710552070063501257751982, −1.20702059904232830410970580587, −0.67574071779948984728446194366, −0.63890998982087346687774393905, −0.62538239499700763597809751549, −0.60638455703089458103342337627, −0.45541027852408461052568422885, −0.14532312328804863226173401609, 0.14532312328804863226173401609, 0.45541027852408461052568422885, 0.60638455703089458103342337627, 0.62538239499700763597809751549, 0.63890998982087346687774393905, 0.67574071779948984728446194366, 1.20702059904232830410970580587, 1.21844710552070063501257751982, 1.47519469060589493827059673935, 1.56735006731892549212051452426, 1.69941050166545581097406100181, 1.71294857450523396937078706039, 1.74736175494083124537429285420, 1.91104870107562838974905869903, 2.23051924417237627269898298389, 2.23555965677225500777846211872, 2.67314695962071250172624552724, 2.86381884765811122199979320967, 2.89505184765455542519566561686, 2.94169301659605556175930482926, 2.96507106992486802284538267408, 3.05697198066675454806759565248, 3.17331508518770915275983494402, 3.30163130689450921631176645128, 3.66144241289903067333037111553
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.