Dirichlet series
| L(s) = 1 | − 4·2-s − 24·3-s − 12·4-s + 24·5-s + 96·6-s + 28·7-s + 64·8-s + 324·9-s − 96·10-s + 12·11-s + 288·12-s − 60·13-s − 112·14-s − 576·15-s − 16-s − 1.29e3·18-s − 140·19-s − 288·20-s − 672·21-s − 48·22-s − 96·23-s − 1.53e3·24-s − 102·25-s + 240·26-s − 3.24e3·27-s − 336·28-s + 380·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 4.61·3-s − 3/2·4-s + 2.14·5-s + 6.53·6-s + 1.51·7-s + 2.82·8-s + 12·9-s − 3.03·10-s + 0.328·11-s + 6.92·12-s − 1.28·13-s − 2.13·14-s − 9.91·15-s − 0.0156·16-s − 16.9·18-s − 1.69·19-s − 3.21·20-s − 6.98·21-s − 0.465·22-s − 0.870·23-s − 13.0·24-s − 0.815·25-s + 1.81·26-s − 23.0·27-s − 2.26·28-s + 2.43·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 17^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.68901\times 10^{13}\) |
| Root analytic conductor: | \(7.15224\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 17^{16} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(1.756195373\) |
| \(L(\frac12)\) | \(\approx\) | \(1.756195373\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p T )^{8} \) |
| 17 | \( 1 \) | |
| good | 2 | \( 1 + p^{2} T + 7 p^{2} T^{2} + 3 p^{5} T^{3} + 465 T^{4} + 369 p^{2} T^{5} + 2879 p T^{6} + 1937 p^{3} T^{7} + 13001 p^{2} T^{8} + 1937 p^{6} T^{9} + 2879 p^{7} T^{10} + 369 p^{11} T^{11} + 465 p^{12} T^{12} + 3 p^{20} T^{13} + 7 p^{20} T^{14} + p^{23} T^{15} + p^{24} T^{16} \) |
| 5 | \( 1 - 24 T + 678 T^{2} - 10108 T^{3} + 173557 T^{4} - 2165168 T^{5} + 31771686 T^{6} - 373058132 T^{7} + 4692450744 T^{8} - 373058132 p^{3} T^{9} + 31771686 p^{6} T^{10} - 2165168 p^{9} T^{11} + 173557 p^{12} T^{12} - 10108 p^{15} T^{13} + 678 p^{18} T^{14} - 24 p^{21} T^{15} + p^{24} T^{16} \) | |
| 7 | \( 1 - 4 p T + 2032 T^{2} - 41372 T^{3} + 1795200 T^{4} - 28843420 T^{5} + 980190864 T^{6} - 13179410588 T^{7} + 385910707006 T^{8} - 13179410588 p^{3} T^{9} + 980190864 p^{6} T^{10} - 28843420 p^{9} T^{11} + 1795200 p^{12} T^{12} - 41372 p^{15} T^{13} + 2032 p^{18} T^{14} - 4 p^{22} T^{15} + p^{24} T^{16} \) | |
| 11 | \( 1 - 12 T + 4878 T^{2} - 2760 T^{3} + 14205593 T^{4} - 736464 p T^{5} + 30313268526 T^{6} + 28060790172 T^{7} + 45116021369828 T^{8} + 28060790172 p^{3} T^{9} + 30313268526 p^{6} T^{10} - 736464 p^{10} T^{11} + 14205593 p^{12} T^{12} - 2760 p^{15} T^{13} + 4878 p^{18} T^{14} - 12 p^{21} T^{15} + p^{24} T^{16} \) | |
| 13 | \( 1 + 60 T + 11654 T^{2} + 501408 T^{3} + 56823505 T^{4} + 1698751432 T^{5} + 162984282454 T^{6} + 269911701660 p T^{7} + 2179688399908 p^{2} T^{8} + 269911701660 p^{4} T^{9} + 162984282454 p^{6} T^{10} + 1698751432 p^{9} T^{11} + 56823505 p^{12} T^{12} + 501408 p^{15} T^{13} + 11654 p^{18} T^{14} + 60 p^{21} T^{15} + p^{24} T^{16} \) | |
| 19 | \( 1 + 140 T + 28350 T^{2} + 3067424 T^{3} + 447319361 T^{4} + 40410789024 T^{5} + 4541319845574 T^{6} + 364362387697084 T^{7} + 36055952011785444 T^{8} + 364362387697084 p^{3} T^{9} + 4541319845574 p^{6} T^{10} + 40410789024 p^{9} T^{11} + 447319361 p^{12} T^{12} + 3067424 p^{15} T^{13} + 28350 p^{18} T^{14} + 140 p^{21} T^{15} + p^{24} T^{16} \) | |
| 23 | \( 1 + 96 T + 44642 T^{2} + 2214340 T^{3} + 1030515193 T^{4} + 35075740256 T^{5} + 17981499113214 T^{6} + 408366975428204 T^{7} + 238095860043999564 T^{8} + 408366975428204 p^{3} T^{9} + 17981499113214 p^{6} T^{10} + 35075740256 p^{9} T^{11} + 1030515193 p^{12} T^{12} + 2214340 p^{15} T^{13} + 44642 p^{18} T^{14} + 96 p^{21} T^{15} + p^{24} T^{16} \) | |
| 29 | \( 1 - 380 T + 211416 T^{2} - 58153660 T^{3} + 18532297880 T^{4} - 3943153474836 T^{5} + 908972326382232 T^{6} - 154121594946688308 T^{7} + 27717489561272166734 T^{8} - 154121594946688308 p^{3} T^{9} + 908972326382232 p^{6} T^{10} - 3943153474836 p^{9} T^{11} + 18532297880 p^{12} T^{12} - 58153660 p^{15} T^{13} + 211416 p^{18} T^{14} - 380 p^{21} T^{15} + p^{24} T^{16} \) | |
| 31 | \( 1 - 4 p T + 142772 T^{2} - 10613948 T^{3} + 9139927428 T^{4} - 275888914636 T^{5} + 366492234002124 T^{6} + 244260678408820 T^{7} + 11599249403978143990 T^{8} + 244260678408820 p^{3} T^{9} + 366492234002124 p^{6} T^{10} - 275888914636 p^{9} T^{11} + 9139927428 p^{12} T^{12} - 10613948 p^{15} T^{13} + 142772 p^{18} T^{14} - 4 p^{22} T^{15} + p^{24} T^{16} \) | |
| 37 | \( 1 - 1260 T + 988108 T^{2} - 554552572 T^{3} + 249776949076 T^{4} - 93080545637476 T^{5} + 29697260364358212 T^{6} - 8179528973080251284 T^{7} + \)\(19\!\cdots\!30\)\( T^{8} - 8179528973080251284 p^{3} T^{9} + 29697260364358212 p^{6} T^{10} - 93080545637476 p^{9} T^{11} + 249776949076 p^{12} T^{12} - 554552572 p^{15} T^{13} + 988108 p^{18} T^{14} - 1260 p^{21} T^{15} + p^{24} T^{16} \) | |
| 41 | \( 1 - 284 T + 406434 T^{2} - 99026896 T^{3} + 77769726413 T^{4} - 16158509250456 T^{5} + 9221257202053254 T^{6} - 1635427742979136812 T^{7} + \)\(75\!\cdots\!36\)\( T^{8} - 1635427742979136812 p^{3} T^{9} + 9221257202053254 p^{6} T^{10} - 16158509250456 p^{9} T^{11} + 77769726413 p^{12} T^{12} - 99026896 p^{15} T^{13} + 406434 p^{18} T^{14} - 284 p^{21} T^{15} + p^{24} T^{16} \) | |
| 43 | \( 1 + 76 T + 495622 T^{2} + 34228680 T^{3} + 115588023441 T^{4} + 7124267117728 T^{5} + 16561652904629302 T^{6} + 881953422001006420 T^{7} + \)\(15\!\cdots\!04\)\( T^{8} + 881953422001006420 p^{3} T^{9} + 16561652904629302 p^{6} T^{10} + 7124267117728 p^{9} T^{11} + 115588023441 p^{12} T^{12} + 34228680 p^{15} T^{13} + 495622 p^{18} T^{14} + 76 p^{21} T^{15} + p^{24} T^{16} \) | |
| 47 | \( 1 + 184 T + 637080 T^{2} + 103089112 T^{3} + 191321055276 T^{4} + 27342155842616 T^{5} + 35434978313886824 T^{6} + 4375714231389295128 T^{7} + \)\(44\!\cdots\!38\)\( T^{8} + 4375714231389295128 p^{3} T^{9} + 35434978313886824 p^{6} T^{10} + 27342155842616 p^{9} T^{11} + 191321055276 p^{12} T^{12} + 103089112 p^{15} T^{13} + 637080 p^{18} T^{14} + 184 p^{21} T^{15} + p^{24} T^{16} \) | |
| 53 | \( 1 - 1064 T + 1192552 T^{2} - 820460376 T^{3} + 540017129676 T^{4} - 276937305839784 T^{5} + 136574961148603480 T^{6} - 57341078404040908568 T^{7} + \)\(23\!\cdots\!26\)\( T^{8} - 57341078404040908568 p^{3} T^{9} + 136574961148603480 p^{6} T^{10} - 276937305839784 p^{9} T^{11} + 540017129676 p^{12} T^{12} - 820460376 p^{15} T^{13} + 1192552 p^{18} T^{14} - 1064 p^{21} T^{15} + p^{24} T^{16} \) | |
| 59 | \( 1 - 1224 T + 1144860 T^{2} - 801494536 T^{3} + 548993519848 T^{4} - 325649280533928 T^{5} + 182418607886863700 T^{6} - 91855288272457229096 T^{7} + \)\(43\!\cdots\!70\)\( T^{8} - 91855288272457229096 p^{3} T^{9} + 182418607886863700 p^{6} T^{10} - 325649280533928 p^{9} T^{11} + 548993519848 p^{12} T^{12} - 801494536 p^{15} T^{13} + 1144860 p^{18} T^{14} - 1224 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( 1 - 324 T + 490604 T^{2} + 71271404 T^{3} + 70992511124 T^{4} + 61392876344084 T^{5} + 20146278923309412 T^{6} + 14049089065153701028 T^{7} + \)\(59\!\cdots\!30\)\( T^{8} + 14049089065153701028 p^{3} T^{9} + 20146278923309412 p^{6} T^{10} + 61392876344084 p^{9} T^{11} + 70992511124 p^{12} T^{12} + 71271404 p^{15} T^{13} + 490604 p^{18} T^{14} - 324 p^{21} T^{15} + p^{24} T^{16} \) | |
| 67 | \( 1 + 624 T + 1529832 T^{2} + 898150096 T^{3} + 1192036459564 T^{4} + 619766929411856 T^{5} + 599543238997230680 T^{6} + \)\(27\!\cdots\!56\)\( T^{7} + \)\(21\!\cdots\!10\)\( T^{8} + \)\(27\!\cdots\!56\)\( p^{3} T^{9} + 599543238997230680 p^{6} T^{10} + 619766929411856 p^{9} T^{11} + 1192036459564 p^{12} T^{12} + 898150096 p^{15} T^{13} + 1529832 p^{18} T^{14} + 624 p^{21} T^{15} + p^{24} T^{16} \) | |
| 71 | \( 1 - 2636 T + 4732372 T^{2} - 5813943532 T^{3} + 5776655507652 T^{4} - 4624040246445852 T^{5} + 3260437087462743660 T^{6} - \)\(20\!\cdots\!64\)\( T^{7} + \)\(12\!\cdots\!74\)\( T^{8} - \)\(20\!\cdots\!64\)\( p^{3} T^{9} + 3260437087462743660 p^{6} T^{10} - 4624040246445852 p^{9} T^{11} + 5776655507652 p^{12} T^{12} - 5813943532 p^{15} T^{13} + 4732372 p^{18} T^{14} - 2636 p^{21} T^{15} + p^{24} T^{16} \) | |
| 73 | \( 1 - 1640 T + 1963372 T^{2} - 1693750680 T^{3} + 1407763939144 T^{4} - 1072289764835752 T^{5} + 787320946600626756 T^{6} - \)\(51\!\cdots\!92\)\( T^{7} + \)\(32\!\cdots\!78\)\( T^{8} - \)\(51\!\cdots\!92\)\( p^{3} T^{9} + 787320946600626756 p^{6} T^{10} - 1072289764835752 p^{9} T^{11} + 1407763939144 p^{12} T^{12} - 1693750680 p^{15} T^{13} + 1963372 p^{18} T^{14} - 1640 p^{21} T^{15} + p^{24} T^{16} \) | |
| 79 | \( 1 - 3788 T + 9467560 T^{2} - 16726417788 T^{3} + 23873989818352 T^{4} - 27943095275951068 T^{5} + 28027016427546257112 T^{6} - \)\(24\!\cdots\!64\)\( T^{7} + \)\(18\!\cdots\!58\)\( T^{8} - \)\(24\!\cdots\!64\)\( p^{3} T^{9} + 28027016427546257112 p^{6} T^{10} - 27943095275951068 p^{9} T^{11} + 23873989818352 p^{12} T^{12} - 16726417788 p^{15} T^{13} + 9467560 p^{18} T^{14} - 3788 p^{21} T^{15} + p^{24} T^{16} \) | |
| 83 | \( 1 - 2032 T + 5259016 T^{2} - 7305233360 T^{3} + 10942155159788 T^{4} - 11547028601020944 T^{5} + 12572818056838413624 T^{6} - \)\(10\!\cdots\!72\)\( T^{7} + \)\(89\!\cdots\!58\)\( T^{8} - \)\(10\!\cdots\!72\)\( p^{3} T^{9} + 12572818056838413624 p^{6} T^{10} - 11547028601020944 p^{9} T^{11} + 10942155159788 p^{12} T^{12} - 7305233360 p^{15} T^{13} + 5259016 p^{18} T^{14} - 2032 p^{21} T^{15} + p^{24} T^{16} \) | |
| 89 | \( 1 - 1304 T + 1932764 T^{2} - 2811742520 T^{3} + 3063293640792 T^{4} - 3496174617710536 T^{5} + 3362485970490195188 T^{6} - \)\(30\!\cdots\!12\)\( T^{7} + \)\(29\!\cdots\!22\)\( T^{8} - \)\(30\!\cdots\!12\)\( p^{3} T^{9} + 3362485970490195188 p^{6} T^{10} - 3496174617710536 p^{9} T^{11} + 3063293640792 p^{12} T^{12} - 2811742520 p^{15} T^{13} + 1932764 p^{18} T^{14} - 1304 p^{21} T^{15} + p^{24} T^{16} \) | |
| 97 | \( 1 - 2376 T + 4905092 T^{2} - 8362258920 T^{3} + 12708012691752 T^{4} - 16422796465233816 T^{5} + 20013930093094231116 T^{6} - \)\(21\!\cdots\!60\)\( T^{7} + \)\(21\!\cdots\!94\)\( T^{8} - \)\(21\!\cdots\!60\)\( p^{3} T^{9} + 20013930093094231116 p^{6} T^{10} - 16422796465233816 p^{9} T^{11} + 12708012691752 p^{12} T^{12} - 8362258920 p^{15} T^{13} + 4905092 p^{18} T^{14} - 2376 p^{21} T^{15} + p^{24} T^{16} \) | |
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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
−4.26423553218514399437061229047, −3.97090412246981400276264309861, −3.61239791370839853058101288951, −3.59639516438629840931695917224, −3.50087326733974504425758124654, −3.47360231145699902685670889479, −3.45421102107020757393911246170, −2.72010003575625260299577993923, −2.43869417492715141720574806299, −2.40814547540840589436942782994, −2.39639179743175505477789919273, −2.34383303608831939890124466103, −1.94596664101583330402589763544, −1.93849190430178970019788871075, −1.85968594692416734975420460860, −1.72866509290757693351716419777, −1.54089702532931521927779434504, −1.07536149442455199685869557666, −0.804625687307297145432836592434, −0.70793595915932271805022676123, −0.68411049885920301926806797793, −0.62341823930582948686358208052, −0.59944922518115691687701849437, −0.54966502523905339189153694296, −0.23281608035654492526553497295, 0.23281608035654492526553497295, 0.54966502523905339189153694296, 0.59944922518115691687701849437, 0.62341823930582948686358208052, 0.68411049885920301926806797793, 0.70793595915932271805022676123, 0.804625687307297145432836592434, 1.07536149442455199685869557666, 1.54089702532931521927779434504, 1.72866509290757693351716419777, 1.85968594692416734975420460860, 1.93849190430178970019788871075, 1.94596664101583330402589763544, 2.34383303608831939890124466103, 2.39639179743175505477789919273, 2.40814547540840589436942782994, 2.43869417492715141720574806299, 2.72010003575625260299577993923, 3.45421102107020757393911246170, 3.47360231145699902685670889479, 3.50087326733974504425758124654, 3.59639516438629840931695917224, 3.61239791370839853058101288951, 3.97090412246981400276264309861, 4.26423553218514399437061229047
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.