Dirichlet series
| L(s) = 1 | + 64·2-s − 81·3-s + 1.53e3·4-s + 171·5-s − 5.18e3·6-s + 7.13e3·7-s + 7.74e3·9-s + 1.09e4·10-s − 2.61e4·11-s − 1.24e5·12-s − 4.16e3·13-s + 4.56e5·14-s − 1.38e4·15-s − 9.83e5·16-s + 1.09e6·17-s + 4.95e5·18-s − 4.36e5·19-s + 2.62e5·20-s − 5.77e5·21-s − 1.67e6·22-s − 2.89e5·23-s + 5.39e6·25-s − 2.66e5·26-s + 1.87e6·27-s + 1.09e7·28-s − 6.01e5·29-s − 8.86e5·30-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 0.577·3-s + 3·4-s + 0.122·5-s − 1.63·6-s + 1.12·7-s + 0.393·9-s + 0.346·10-s − 0.538·11-s − 1.73·12-s − 0.0404·13-s + 3.17·14-s − 0.0706·15-s − 3.75·16-s + 3.18·17-s + 1.11·18-s − 0.768·19-s + 0.367·20-s − 0.648·21-s − 1.52·22-s − 0.215·23-s + 2.76·25-s − 0.114·26-s + 0.679·27-s + 3.36·28-s − 0.157·29-s − 0.199·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.45606\times 10^{7}\) |
| Root analytic conductor: | \(3.04477\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{16} ,\ ( \ : [9/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(23.05115490\) |
| \(L(\frac12)\) | \(\approx\) | \(23.05115490\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{4} T + p^{8} T^{2} )^{4} \) |
| 3 | \( 1 + p^{4} T - 44 p^{3} T^{2} - 1189 p^{7} T^{3} - 7514 p^{10} T^{4} - 1189 p^{16} T^{5} - 44 p^{21} T^{6} + p^{31} T^{7} + p^{36} T^{8} \) | |
| good | 5 | \( 1 - 171 T - 5365598 T^{2} + 3021852159 T^{3} + 15190093296307 T^{4} - 9848225794363056 T^{5} - 28855275221226984284 T^{6} + \)\(19\!\cdots\!24\)\( p T^{7} + \)\(20\!\cdots\!56\)\( p^{2} T^{8} + \)\(19\!\cdots\!24\)\( p^{10} T^{9} - 28855275221226984284 p^{18} T^{10} - 9848225794363056 p^{27} T^{11} + 15190093296307 p^{36} T^{12} + 3021852159 p^{45} T^{13} - 5365598 p^{54} T^{14} - 171 p^{63} T^{15} + p^{72} T^{16} \) |
| 7 | \( 1 - 7135 T - 34982952 T^{2} + 486060236233 T^{3} - 488818768752817 T^{4} - 1865443274843300508 p T^{5} + \)\(17\!\cdots\!00\)\( p^{2} T^{6} + \)\(50\!\cdots\!36\)\( p^{3} T^{7} - \)\(19\!\cdots\!08\)\( p^{4} T^{8} + \)\(50\!\cdots\!36\)\( p^{12} T^{9} + \)\(17\!\cdots\!00\)\( p^{20} T^{10} - 1865443274843300508 p^{28} T^{11} - 488818768752817 p^{36} T^{12} + 486060236233 p^{45} T^{13} - 34982952 p^{54} T^{14} - 7135 p^{63} T^{15} + p^{72} T^{16} \) | |
| 11 | \( 1 + 26130 T - 7921887224 T^{2} - 120615622422744 T^{3} + 39842753827777544035 T^{4} + \)\(34\!\cdots\!20\)\( T^{5} - \)\(14\!\cdots\!92\)\( T^{6} - \)\(28\!\cdots\!34\)\( T^{7} + \)\(38\!\cdots\!60\)\( T^{8} - \)\(28\!\cdots\!34\)\( p^{9} T^{9} - \)\(14\!\cdots\!92\)\( p^{18} T^{10} + \)\(34\!\cdots\!20\)\( p^{27} T^{11} + 39842753827777544035 p^{36} T^{12} - 120615622422744 p^{45} T^{13} - 7921887224 p^{54} T^{14} + 26130 p^{63} T^{15} + p^{72} T^{16} \) | |
| 13 | \( 1 + 4163 T - 24491618040 T^{2} + 796144182200839 T^{3} + \)\(30\!\cdots\!61\)\( T^{4} - \)\(14\!\cdots\!40\)\( T^{5} - \)\(15\!\cdots\!10\)\( T^{6} + \)\(95\!\cdots\!50\)\( T^{7} + \)\(41\!\cdots\!92\)\( T^{8} + \)\(95\!\cdots\!50\)\( p^{9} T^{9} - \)\(15\!\cdots\!10\)\( p^{18} T^{10} - \)\(14\!\cdots\!40\)\( p^{27} T^{11} + \)\(30\!\cdots\!61\)\( p^{36} T^{12} + 796144182200839 p^{45} T^{13} - 24491618040 p^{54} T^{14} + 4163 p^{63} T^{15} + p^{72} T^{16} \) | |
| 17 | \( ( 1 - 549255 T + 288151451402 T^{2} - 122223914866033641 T^{3} + \)\(40\!\cdots\!46\)\( T^{4} - 122223914866033641 p^{9} T^{5} + 288151451402 p^{18} T^{6} - 549255 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 19 | \( ( 1 + 218191 T + 900262536256 T^{2} + 205400076594516871 T^{3} + \)\(37\!\cdots\!06\)\( T^{4} + 205400076594516871 p^{9} T^{5} + 900262536256 p^{18} T^{6} + 218191 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 23 | \( 1 + 289383 T - 2543066769524 T^{2} + 1846233949597537827 T^{3} + \)\(21\!\cdots\!11\)\( T^{4} - \)\(52\!\cdots\!24\)\( T^{5} + \)\(46\!\cdots\!16\)\( T^{6} + \)\(66\!\cdots\!64\)\( T^{7} - \)\(82\!\cdots\!44\)\( T^{8} + \)\(66\!\cdots\!64\)\( p^{9} T^{9} + \)\(46\!\cdots\!16\)\( p^{18} T^{10} - \)\(52\!\cdots\!24\)\( p^{27} T^{11} + \)\(21\!\cdots\!11\)\( p^{36} T^{12} + 1846233949597537827 p^{45} T^{13} - 2543066769524 p^{54} T^{14} + 289383 p^{63} T^{15} + p^{72} T^{16} \) | |
| 29 | \( 1 + 601707 T - 27669117066620 T^{2} + 78329737736173512699 T^{3} + \)\(39\!\cdots\!01\)\( T^{4} - \)\(18\!\cdots\!28\)\( T^{5} + \)\(14\!\cdots\!42\)\( T^{6} + \)\(17\!\cdots\!74\)\( T^{7} - \)\(51\!\cdots\!20\)\( T^{8} + \)\(17\!\cdots\!74\)\( p^{9} T^{9} + \)\(14\!\cdots\!42\)\( p^{18} T^{10} - \)\(18\!\cdots\!28\)\( p^{27} T^{11} + \)\(39\!\cdots\!01\)\( p^{36} T^{12} + 78329737736173512699 p^{45} T^{13} - 27669117066620 p^{54} T^{14} + 601707 p^{63} T^{15} + p^{72} T^{16} \) | |
| 31 | \( 1 - 5671315 T - 28081663789698 T^{2} + \)\(43\!\cdots\!63\)\( T^{3} - \)\(37\!\cdots\!55\)\( T^{4} - \)\(13\!\cdots\!56\)\( T^{5} + \)\(70\!\cdots\!90\)\( T^{6} + \)\(20\!\cdots\!58\)\( T^{7} - \)\(26\!\cdots\!80\)\( T^{8} + \)\(20\!\cdots\!58\)\( p^{9} T^{9} + \)\(70\!\cdots\!90\)\( p^{18} T^{10} - \)\(13\!\cdots\!56\)\( p^{27} T^{11} - \)\(37\!\cdots\!55\)\( p^{36} T^{12} + \)\(43\!\cdots\!63\)\( p^{45} T^{13} - 28081663789698 p^{54} T^{14} - 5671315 p^{63} T^{15} + p^{72} T^{16} \) | |
| 37 | \( ( 1 + 19368574 T + 236628706386520 T^{2} + \)\(34\!\cdots\!94\)\( T^{3} + \)\(53\!\cdots\!10\)\( T^{4} + \)\(34\!\cdots\!94\)\( p^{9} T^{5} + 236628706386520 p^{18} T^{6} + 19368574 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 41 | \( 1 + 18418410 T - 109735099241540 T^{2} + \)\(38\!\cdots\!48\)\( T^{3} + \)\(59\!\cdots\!55\)\( T^{4} - \)\(20\!\cdots\!24\)\( T^{5} + \)\(42\!\cdots\!96\)\( T^{6} + \)\(11\!\cdots\!90\)\( T^{7} - \)\(54\!\cdots\!52\)\( T^{8} + \)\(11\!\cdots\!90\)\( p^{9} T^{9} + \)\(42\!\cdots\!96\)\( p^{18} T^{10} - \)\(20\!\cdots\!24\)\( p^{27} T^{11} + \)\(59\!\cdots\!55\)\( p^{36} T^{12} + \)\(38\!\cdots\!48\)\( p^{45} T^{13} - 109735099241540 p^{54} T^{14} + 18418410 p^{63} T^{15} + p^{72} T^{16} \) | |
| 43 | \( 1 - 35096140 T + 538514563498182 T^{2} - \)\(45\!\cdots\!72\)\( T^{3} + \)\(12\!\cdots\!65\)\( T^{4} - \)\(14\!\cdots\!44\)\( T^{5} + \)\(92\!\cdots\!78\)\( T^{6} - \)\(22\!\cdots\!84\)\( T^{7} + \)\(22\!\cdots\!56\)\( T^{8} - \)\(22\!\cdots\!84\)\( p^{9} T^{9} + \)\(92\!\cdots\!78\)\( p^{18} T^{10} - \)\(14\!\cdots\!44\)\( p^{27} T^{11} + \)\(12\!\cdots\!65\)\( p^{36} T^{12} - \)\(45\!\cdots\!72\)\( p^{45} T^{13} + 538514563498182 p^{54} T^{14} - 35096140 p^{63} T^{15} + p^{72} T^{16} \) | |
| 47 | \( 1 + 79830825 T + 2518655718367696 T^{2} - \)\(36\!\cdots\!71\)\( T^{3} - \)\(53\!\cdots\!17\)\( T^{4} - \)\(17\!\cdots\!92\)\( T^{5} - \)\(10\!\cdots\!16\)\( T^{6} + \)\(12\!\cdots\!56\)\( T^{7} + \)\(64\!\cdots\!76\)\( T^{8} + \)\(12\!\cdots\!56\)\( p^{9} T^{9} - \)\(10\!\cdots\!16\)\( p^{18} T^{10} - \)\(17\!\cdots\!92\)\( p^{27} T^{11} - \)\(53\!\cdots\!17\)\( p^{36} T^{12} - \)\(36\!\cdots\!71\)\( p^{45} T^{13} + 2518655718367696 p^{54} T^{14} + 79830825 p^{63} T^{15} + p^{72} T^{16} \) | |
| 53 | \( ( 1 - 165348618 T + 20407779287030552 T^{2} - \)\(16\!\cdots\!46\)\( T^{3} + \)\(10\!\cdots\!90\)\( T^{4} - \)\(16\!\cdots\!46\)\( p^{9} T^{5} + 20407779287030552 p^{18} T^{6} - 165348618 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 59 | \( 1 + 90704166 T - 24060100201611800 T^{2} - \)\(16\!\cdots\!04\)\( T^{3} + \)\(42\!\cdots\!67\)\( T^{4} + \)\(20\!\cdots\!92\)\( T^{5} - \)\(48\!\cdots\!12\)\( T^{6} - \)\(65\!\cdots\!30\)\( T^{7} + \)\(48\!\cdots\!92\)\( T^{8} - \)\(65\!\cdots\!30\)\( p^{9} T^{9} - \)\(48\!\cdots\!12\)\( p^{18} T^{10} + \)\(20\!\cdots\!92\)\( p^{27} T^{11} + \)\(42\!\cdots\!67\)\( p^{36} T^{12} - \)\(16\!\cdots\!04\)\( p^{45} T^{13} - 24060100201611800 p^{54} T^{14} + 90704166 p^{63} T^{15} + p^{72} T^{16} \) | |
| 61 | \( 1 + 122811677 T - 16794869234119542 T^{2} - \)\(30\!\cdots\!17\)\( T^{3} + \)\(12\!\cdots\!67\)\( T^{4} + \)\(37\!\cdots\!44\)\( T^{5} + \)\(96\!\cdots\!32\)\( T^{6} - \)\(24\!\cdots\!00\)\( T^{7} - \)\(35\!\cdots\!88\)\( T^{8} - \)\(24\!\cdots\!00\)\( p^{9} T^{9} + \)\(96\!\cdots\!32\)\( p^{18} T^{10} + \)\(37\!\cdots\!44\)\( p^{27} T^{11} + \)\(12\!\cdots\!67\)\( p^{36} T^{12} - \)\(30\!\cdots\!17\)\( p^{45} T^{13} - 16794869234119542 p^{54} T^{14} + 122811677 p^{63} T^{15} + p^{72} T^{16} \) | |
| 67 | \( 1 - 221601736 T - 30022765389271134 T^{2} + \)\(34\!\cdots\!08\)\( T^{3} + \)\(14\!\cdots\!49\)\( T^{4} - \)\(16\!\cdots\!64\)\( T^{5} - \)\(32\!\cdots\!82\)\( T^{6} + \)\(17\!\cdots\!84\)\( T^{7} - \)\(45\!\cdots\!68\)\( T^{8} + \)\(17\!\cdots\!84\)\( p^{9} T^{9} - \)\(32\!\cdots\!82\)\( p^{18} T^{10} - \)\(16\!\cdots\!64\)\( p^{27} T^{11} + \)\(14\!\cdots\!49\)\( p^{36} T^{12} + \)\(34\!\cdots\!08\)\( p^{45} T^{13} - 30022765389271134 p^{54} T^{14} - 221601736 p^{63} T^{15} + p^{72} T^{16} \) | |
| 71 | \( ( 1 - 138204120 T + 168296185112945612 T^{2} - \)\(26\!\cdots\!40\)\( p T^{3} + \)\(11\!\cdots\!58\)\( T^{4} - \)\(26\!\cdots\!40\)\( p^{10} T^{5} + 168296185112945612 p^{18} T^{6} - 138204120 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 73 | \( ( 1 + 494458507 T + 225390744932020822 T^{2} + \)\(61\!\cdots\!37\)\( T^{3} + \)\(17\!\cdots\!82\)\( T^{4} + \)\(61\!\cdots\!37\)\( p^{9} T^{5} + 225390744932020822 p^{18} T^{6} + 494458507 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 79 | \( 1 - 7504315 p T + 23917252591792518 T^{2} - \)\(33\!\cdots\!07\)\( T^{3} + \)\(28\!\cdots\!33\)\( T^{4} + \)\(12\!\cdots\!76\)\( T^{5} - \)\(12\!\cdots\!02\)\( T^{6} - \)\(17\!\cdots\!50\)\( T^{7} - \)\(32\!\cdots\!68\)\( T^{8} - \)\(17\!\cdots\!50\)\( p^{9} T^{9} - \)\(12\!\cdots\!02\)\( p^{18} T^{10} + \)\(12\!\cdots\!76\)\( p^{27} T^{11} + \)\(28\!\cdots\!33\)\( p^{36} T^{12} - \)\(33\!\cdots\!07\)\( p^{45} T^{13} + 23917252591792518 p^{54} T^{14} - 7504315 p^{64} T^{15} + p^{72} T^{16} \) | |
| 83 | \( 1 + 478410747 T - 342935176722438074 T^{2} - \)\(30\!\cdots\!23\)\( T^{3} + \)\(12\!\cdots\!41\)\( T^{4} - \)\(85\!\cdots\!64\)\( T^{5} - \)\(24\!\cdots\!26\)\( T^{6} - \)\(12\!\cdots\!90\)\( T^{7} + \)\(28\!\cdots\!28\)\( T^{8} - \)\(12\!\cdots\!90\)\( p^{9} T^{9} - \)\(24\!\cdots\!26\)\( p^{18} T^{10} - \)\(85\!\cdots\!64\)\( p^{27} T^{11} + \)\(12\!\cdots\!41\)\( p^{36} T^{12} - \)\(30\!\cdots\!23\)\( p^{45} T^{13} - 342935176722438074 p^{54} T^{14} + 478410747 p^{63} T^{15} + p^{72} T^{16} \) | |
| 89 | \( ( 1 - 437759976 T + 893458919580980108 T^{2} - \)\(16\!\cdots\!24\)\( T^{3} + \)\(34\!\cdots\!62\)\( T^{4} - \)\(16\!\cdots\!24\)\( p^{9} T^{5} + 893458919580980108 p^{18} T^{6} - 437759976 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 97 | \( 1 - 2679512242 T + 2581477857404996280 T^{2} - \)\(90\!\cdots\!04\)\( T^{3} + \)\(15\!\cdots\!03\)\( T^{4} - \)\(76\!\cdots\!96\)\( T^{5} + \)\(89\!\cdots\!12\)\( T^{6} - \)\(88\!\cdots\!54\)\( T^{7} + \)\(13\!\cdots\!12\)\( T^{8} - \)\(88\!\cdots\!54\)\( p^{9} T^{9} + \)\(89\!\cdots\!12\)\( p^{18} T^{10} - \)\(76\!\cdots\!96\)\( p^{27} T^{11} + \)\(15\!\cdots\!03\)\( p^{36} T^{12} - \)\(90\!\cdots\!04\)\( p^{45} T^{13} + 2581477857404996280 p^{54} T^{14} - 2679512242 p^{63} T^{15} + p^{72} T^{16} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.78796818261629734020816815472, −6.75924748201438295848413851656, −6.50223235517318009518860413943, −5.94562977871411555899744026850, −5.82107377140133322170471059545, −5.70672051620604176581149937067, −5.43428673700362557606761633352, −5.19670970745151676405986093024, −5.13134282575994964639025736593, −4.80951399571425990680523304513, −4.71608647256698820083084378942, −4.37470901851744675818618282856, −4.23268512626827424460640331377, −3.64682732177622174787504292979, −3.55120684282042239629106787078, −3.33754420547431095436530007832, −3.23923449864184731730828391069, −2.55550386570852146507245277467, −2.51146612933797516966187540559, −2.19592930952154904255705965039, −1.53557395962826439320122763786, −1.14302661342369246539331890697, −1.08558670819884071547486367257, −0.63635895315253094288590715611, −0.30439354790718382050638935453, 0.30439354790718382050638935453, 0.63635895315253094288590715611, 1.08558670819884071547486367257, 1.14302661342369246539331890697, 1.53557395962826439320122763786, 2.19592930952154904255705965039, 2.51146612933797516966187540559, 2.55550386570852146507245277467, 3.23923449864184731730828391069, 3.33754420547431095436530007832, 3.55120684282042239629106787078, 3.64682732177622174787504292979, 4.23268512626827424460640331377, 4.37470901851744675818618282856, 4.71608647256698820083084378942, 4.80951399571425990680523304513, 5.13134282575994964639025736593, 5.19670970745151676405986093024, 5.43428673700362557606761633352, 5.70672051620604176581149937067, 5.82107377140133322170471059545, 5.94562977871411555899744026850, 6.50223235517318009518860413943, 6.75924748201438295848413851656, 6.78796818261629734020816815472
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.