Dirichlet series
| L(s) = 1 | + 16·4-s + 8·7-s − 4·13-s + 103·16-s + 20·19-s + 178·25-s + 128·28-s + 28·31-s − 148·37-s − 272·43-s − 186·49-s − 64·52-s − 224·61-s + 374·64-s + 24·67-s + 4·73-s + 320·76-s − 216·79-s − 32·91-s − 44·97-s + 2.84e3·100-s − 228·103-s − 340·109-s + 824·112-s + 448·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 4·4-s + 8/7·7-s − 0.307·13-s + 6.43·16-s + 1.05·19-s + 7.11·25-s + 32/7·28-s + 0.903·31-s − 4·37-s − 6.32·43-s − 3.79·49-s − 1.23·52-s − 3.67·61-s + 5.84·64-s + 0.358·67-s + 4/73·73-s + 4.21·76-s − 2.73·79-s − 0.351·91-s − 0.453·97-s + 28.4·100-s − 2.21·103-s − 3.11·109-s + 7.35·112-s + 3.61·124-s + 0.00787·127-s + 0.00763·131-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{32} \cdot 11^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.61249\times 10^{23}\) |
| Root analytic conductor: | \(5.44730\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{32} \cdot 11^{32} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(6.255149596\) |
| \(L(\frac12)\) | \(\approx\) | \(6.255149596\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) | |
| good | 2 | \( 1 - p^{4} T^{2} + 153 T^{4} - 587 p T^{6} + 1915 p^{2} T^{8} - 2701 p^{4} T^{10} + 217281 T^{12} - 497761 p T^{14} + 4170145 T^{16} - 497761 p^{5} T^{18} + 217281 p^{8} T^{20} - 2701 p^{16} T^{22} + 1915 p^{18} T^{24} - 587 p^{21} T^{26} + 153 p^{24} T^{28} - p^{32} T^{30} + p^{32} T^{32} \) |
| 5 | \( 1 - 178 T^{2} + 14778 T^{4} - 770368 T^{6} + 28639243 T^{8} - 818668258 T^{10} + 19271651352 T^{12} - 83084625136 p T^{14} + 9588673069861 T^{16} - 83084625136 p^{5} T^{18} + 19271651352 p^{8} T^{20} - 818668258 p^{12} T^{22} + 28639243 p^{16} T^{24} - 770368 p^{20} T^{26} + 14778 p^{24} T^{28} - 178 p^{28} T^{30} + p^{32} T^{32} \) | |
| 7 | \( ( 1 - 4 T + 117 T^{2} - 806 T^{3} + 10368 T^{4} - 11132 p T^{5} + 726296 T^{6} - 4835322 T^{7} + 41663687 T^{8} - 4835322 p^{2} T^{9} + 726296 p^{4} T^{10} - 11132 p^{7} T^{11} + 10368 p^{8} T^{12} - 806 p^{10} T^{13} + 117 p^{12} T^{14} - 4 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 13 | \( ( 1 + 2 T + 718 T^{2} + 1888 T^{3} + 260375 T^{4} + 546188 T^{5} + 64887908 T^{6} + 90279418 T^{7} + 12344502880 T^{8} + 90279418 p^{2} T^{9} + 64887908 p^{4} T^{10} + 546188 p^{6} T^{11} + 260375 p^{8} T^{12} + 1888 p^{10} T^{13} + 718 p^{12} T^{14} + 2 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 17 | \( 1 - 2034 T^{2} + 2137099 T^{4} - 1555825258 T^{6} + 885576421975 T^{8} - 418087252540946 T^{10} + 168801749233965397 T^{12} - 59268558645523686770 T^{14} + \)\(18\!\cdots\!16\)\( T^{16} - 59268558645523686770 p^{4} T^{18} + 168801749233965397 p^{8} T^{20} - 418087252540946 p^{12} T^{22} + 885576421975 p^{16} T^{24} - 1555825258 p^{20} T^{26} + 2137099 p^{24} T^{28} - 2034 p^{28} T^{30} + p^{32} T^{32} \) | |
| 19 | \( ( 1 - 10 T + 1541 T^{2} - 12848 T^{3} + 1194081 T^{4} - 7050832 T^{5} + 617393333 T^{6} - 2562029742 T^{7} + 12972454372 p T^{8} - 2562029742 p^{2} T^{9} + 617393333 p^{4} T^{10} - 7050832 p^{6} T^{11} + 1194081 p^{8} T^{12} - 12848 p^{10} T^{13} + 1541 p^{12} T^{14} - 10 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 23 | \( 1 - 3898 T^{2} + 7470938 T^{4} - 9442830008 T^{6} + 8992724976405 T^{8} - 7035858281914756 T^{10} + 4797996886199722826 T^{12} - \)\(29\!\cdots\!26\)\( T^{14} + \)\(16\!\cdots\!40\)\( T^{16} - \)\(29\!\cdots\!26\)\( p^{4} T^{18} + 4797996886199722826 p^{8} T^{20} - 7035858281914756 p^{12} T^{22} + 8992724976405 p^{16} T^{24} - 9442830008 p^{20} T^{26} + 7470938 p^{24} T^{28} - 3898 p^{28} T^{30} + p^{32} T^{32} \) | |
| 29 | \( 1 - 7090 T^{2} + 24088979 T^{4} - 52860171052 T^{6} + 85979263288379 T^{8} - 113520722017723476 T^{10} + \)\(12\!\cdots\!45\)\( T^{12} - \)\(13\!\cdots\!54\)\( T^{14} + \)\(11\!\cdots\!32\)\( T^{16} - \)\(13\!\cdots\!54\)\( p^{4} T^{18} + \)\(12\!\cdots\!45\)\( p^{8} T^{20} - 113520722017723476 p^{12} T^{22} + 85979263288379 p^{16} T^{24} - 52860171052 p^{20} T^{26} + 24088979 p^{24} T^{28} - 7090 p^{28} T^{30} + p^{32} T^{32} \) | |
| 31 | \( ( 1 - 14 T + 3978 T^{2} - 62074 T^{3} + 6872087 T^{4} - 133270650 T^{5} + 7029590252 T^{6} - 182545665112 T^{7} + 6210443887425 T^{8} - 182545665112 p^{2} T^{9} + 7029590252 p^{4} T^{10} - 133270650 p^{6} T^{11} + 6872087 p^{8} T^{12} - 62074 p^{10} T^{13} + 3978 p^{12} T^{14} - 14 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 37 | \( ( 1 + 2 p T + 8842 T^{2} + 527868 T^{3} + 35589723 T^{4} + 1747694092 T^{5} + 86947356072 T^{6} + 3553006456782 T^{7} + 142971465181960 T^{8} + 3553006456782 p^{2} T^{9} + 86947356072 p^{4} T^{10} + 1747694092 p^{6} T^{11} + 35589723 p^{8} T^{12} + 527868 p^{10} T^{13} + 8842 p^{12} T^{14} + 2 p^{15} T^{15} + p^{16} T^{16} )^{2} \) | |
| 41 | \( 1 - 18838 T^{2} + 173358858 T^{4} - 1037317818784 T^{6} + 4528296295411213 T^{8} - 15321581034880516324 T^{10} + \)\(41\!\cdots\!18\)\( T^{12} - \)\(92\!\cdots\!66\)\( T^{14} + \)\(17\!\cdots\!40\)\( T^{16} - \)\(92\!\cdots\!66\)\( p^{4} T^{18} + \)\(41\!\cdots\!18\)\( p^{8} T^{20} - 15321581034880516324 p^{12} T^{22} + 4528296295411213 p^{16} T^{24} - 1037317818784 p^{20} T^{26} + 173358858 p^{24} T^{28} - 18838 p^{28} T^{30} + p^{32} T^{32} \) | |
| 43 | \( ( 1 + 136 T + 18164 T^{2} + 1483936 T^{3} + 115914165 T^{4} + 6877749720 T^{5} + 397184994126 T^{6} + 18797621494760 T^{7} + 882478921005452 T^{8} + 18797621494760 p^{2} T^{9} + 397184994126 p^{4} T^{10} + 6877749720 p^{6} T^{11} + 115914165 p^{8} T^{12} + 1483936 p^{10} T^{13} + 18164 p^{12} T^{14} + 136 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 47 | \( 1 - 14410 T^{2} + 112155039 T^{4} - 608867850142 T^{6} + 2578161157095895 T^{8} - 9021116111839284886 T^{10} + \)\(27\!\cdots\!77\)\( T^{12} - \)\(71\!\cdots\!70\)\( T^{14} + \)\(16\!\cdots\!56\)\( T^{16} - \)\(71\!\cdots\!70\)\( p^{4} T^{18} + \)\(27\!\cdots\!77\)\( p^{8} T^{20} - 9021116111839284886 p^{12} T^{22} + 2578161157095895 p^{16} T^{24} - 608867850142 p^{20} T^{26} + 112155039 p^{24} T^{28} - 14410 p^{28} T^{30} + p^{32} T^{32} \) | |
| 53 | \( 1 - 24436 T^{2} + 309860538 T^{4} - 2662731812674 T^{6} + 17249750715319795 T^{8} - 89091997635889422436 T^{10} + \)\(37\!\cdots\!16\)\( T^{12} - \)\(13\!\cdots\!92\)\( T^{14} + \)\(41\!\cdots\!85\)\( T^{16} - \)\(13\!\cdots\!92\)\( p^{4} T^{18} + \)\(37\!\cdots\!16\)\( p^{8} T^{20} - 89091997635889422436 p^{12} T^{22} + 17249750715319795 p^{16} T^{24} - 2662731812674 p^{20} T^{26} + 309860538 p^{24} T^{28} - 24436 p^{28} T^{30} + p^{32} T^{32} \) | |
| 59 | \( 1 - 20682 T^{2} + 198130262 T^{4} - 1153138487844 T^{6} + 4326529642531863 T^{8} - 8745369823524440418 T^{10} - \)\(95\!\cdots\!32\)\( T^{12} + \)\(16\!\cdots\!64\)\( T^{14} - \)\(21\!\cdots\!87\)\( p^{2} T^{16} + \)\(16\!\cdots\!64\)\( p^{4} T^{18} - \)\(95\!\cdots\!32\)\( p^{8} T^{20} - 8745369823524440418 p^{12} T^{22} + 4326529642531863 p^{16} T^{24} - 1153138487844 p^{20} T^{26} + 198130262 p^{24} T^{28} - 20682 p^{28} T^{30} + p^{32} T^{32} \) | |
| 61 | \( ( 1 + 112 T + 21493 T^{2} + 1326876 T^{3} + 134785133 T^{4} + 2241406816 T^{5} + 159075236043 T^{6} - 26423445075596 T^{7} - 717448526447500 T^{8} - 26423445075596 p^{2} T^{9} + 159075236043 p^{4} T^{10} + 2241406816 p^{6} T^{11} + 134785133 p^{8} T^{12} + 1326876 p^{10} T^{13} + 21493 p^{12} T^{14} + 112 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 12 T + 13613 T^{2} - 121480 T^{3} + 120754069 T^{4} - 1956334700 T^{5} + 777459351015 T^{6} - 12985935239032 T^{7} + 3804482475877080 T^{8} - 12985935239032 p^{2} T^{9} + 777459351015 p^{4} T^{10} - 1956334700 p^{6} T^{11} + 120754069 p^{8} T^{12} - 121480 p^{10} T^{13} + 13613 p^{12} T^{14} - 12 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 71 | \( 1 - 32664 T^{2} + 582373471 T^{4} - 103852571306 p T^{6} + 73712493122078731 T^{8} - \)\(61\!\cdots\!26\)\( T^{10} + \)\(43\!\cdots\!53\)\( T^{12} - \)\(26\!\cdots\!76\)\( T^{14} + \)\(14\!\cdots\!88\)\( T^{16} - \)\(26\!\cdots\!76\)\( p^{4} T^{18} + \)\(43\!\cdots\!53\)\( p^{8} T^{20} - \)\(61\!\cdots\!26\)\( p^{12} T^{22} + 73712493122078731 p^{16} T^{24} - 103852571306 p^{21} T^{26} + 582373471 p^{24} T^{28} - 32664 p^{28} T^{30} + p^{32} T^{32} \) | |
| 73 | \( ( 1 - 2 T + 28881 T^{2} + 79288 T^{3} + 398736745 T^{4} + 2373174904 T^{5} + 3498841626669 T^{6} + 25082690682734 T^{7} + 21777004472176804 T^{8} + 25082690682734 p^{2} T^{9} + 3498841626669 p^{4} T^{10} + 2373174904 p^{6} T^{11} + 398736745 p^{8} T^{12} + 79288 p^{10} T^{13} + 28881 p^{12} T^{14} - 2 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 108 T + 33145 T^{2} + 3235546 T^{3} + 561354580 T^{4} + 47468168324 T^{5} + 6091852673746 T^{6} + 437688075575774 T^{7} + 45380145174506261 T^{8} + 437688075575774 p^{2} T^{9} + 6091852673746 p^{4} T^{10} + 47468168324 p^{6} T^{11} + 561354580 p^{8} T^{12} + 3235546 p^{10} T^{13} + 33145 p^{12} T^{14} + 108 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 83 | \( 1 - 56836 T^{2} + 1507726937 T^{4} - 24632124674530 T^{6} + 277176756528405302 T^{8} - \)\(23\!\cdots\!24\)\( T^{10} + \)\(15\!\cdots\!72\)\( T^{12} - \)\(89\!\cdots\!78\)\( T^{14} + \)\(56\!\cdots\!61\)\( T^{16} - \)\(89\!\cdots\!78\)\( p^{4} T^{18} + \)\(15\!\cdots\!72\)\( p^{8} T^{20} - \)\(23\!\cdots\!24\)\( p^{12} T^{22} + 277176756528405302 p^{16} T^{24} - 24632124674530 p^{20} T^{26} + 1507726937 p^{24} T^{28} - 56836 p^{28} T^{30} + p^{32} T^{32} \) | |
| 89 | \( 1 - 50138 T^{2} + 1370519842 T^{4} - 26422369652440 T^{6} + 399704691200648693 T^{8} - \)\(50\!\cdots\!84\)\( T^{10} + \)\(54\!\cdots\!74\)\( T^{12} - \)\(51\!\cdots\!10\)\( T^{14} + \)\(43\!\cdots\!20\)\( T^{16} - \)\(51\!\cdots\!10\)\( p^{4} T^{18} + \)\(54\!\cdots\!74\)\( p^{8} T^{20} - \)\(50\!\cdots\!84\)\( p^{12} T^{22} + 399704691200648693 p^{16} T^{24} - 26422369652440 p^{20} T^{26} + 1370519842 p^{24} T^{28} - 50138 p^{28} T^{30} + p^{32} T^{32} \) | |
| 97 | \( ( 1 + 22 T + 49408 T^{2} + 115060 T^{3} + 1140430559 T^{4} - 13214673230 T^{5} + 16943162093090 T^{6} - 298782127218388 T^{7} + 183519623479606515 T^{8} - 298782127218388 p^{2} T^{9} + 16943162093090 p^{4} T^{10} - 13214673230 p^{6} T^{11} + 1140430559 p^{8} T^{12} + 115060 p^{10} T^{13} + 49408 p^{12} T^{14} + 22 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.30216204237992959822375597223, −2.28622003426101014839589859493, −2.21582238204903596349852820941, −1.90764861437370235471974654036, −1.84690931381655541386114578609, −1.78036059341324792364733088959, −1.73019590720618898092452401230, −1.68397407304568711660063459454, −1.66039979811011384558957396947, −1.59495915895024368899752192017, −1.51986815764111421908597372045, −1.49720553241703831401654629329, −1.45346080303393718782037971788, −1.42154885705058961842055415158, −1.41280971491162689703113427857, −1.08172393379005662337233322463, −1.04772798670041969001002807062, −0.884757922641631687962485860169, −0.69138690564720384534901657845, −0.68445452829530084082143923318, −0.50267943142042643862178788653, −0.42230269678795086168434843816, −0.40029535427700379262812180900, −0.17297057741204928159720311492, −0.04113872671198303554959569150, 0.04113872671198303554959569150, 0.17297057741204928159720311492, 0.40029535427700379262812180900, 0.42230269678795086168434843816, 0.50267943142042643862178788653, 0.68445452829530084082143923318, 0.69138690564720384534901657845, 0.884757922641631687962485860169, 1.04772798670041969001002807062, 1.08172393379005662337233322463, 1.41280971491162689703113427857, 1.42154885705058961842055415158, 1.45346080303393718782037971788, 1.49720553241703831401654629329, 1.51986815764111421908597372045, 1.59495915895024368899752192017, 1.66039979811011384558957396947, 1.68397407304568711660063459454, 1.73019590720618898092452401230, 1.78036059341324792364733088959, 1.84690931381655541386114578609, 1.90764861437370235471974654036, 2.21582238204903596349852820941, 2.28622003426101014839589859493, 2.30216204237992959822375597223
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.