Dirichlet series
| L(s) = 1 | − 1.96e3·4-s + 1.92e4·7-s − 9.45e4·13-s + 1.74e6·16-s − 2.08e6·19-s − 6.07e6·25-s − 3.77e7·28-s − 8.13e6·31-s + 1.56e7·37-s − 6.17e6·43-s + 2.07e8·49-s + 1.85e8·52-s + 6.32e7·61-s − 9.08e8·64-s − 5.55e8·67-s − 3.66e8·73-s + 4.10e9·76-s − 4.90e8·79-s − 1.81e9·91-s − 1.40e9·97-s + 1.19e10·100-s − 6.90e9·103-s − 1.27e9·109-s + 3.34e10·112-s − 1.49e10·121-s + 1.59e10·124-s + 127-s + ⋯ |
| L(s) = 1 | − 3.83·4-s + 3.02·7-s − 0.918·13-s + 6.63·16-s − 3.67·19-s − 3.11·25-s − 11.5·28-s − 1.58·31-s + 1.36·37-s − 0.275·43-s + 36/7·49-s + 3.52·52-s + 0.585·61-s − 6.76·64-s − 3.36·67-s − 1.51·73-s + 14.1·76-s − 1.41·79-s − 2.77·91-s − 1.60·97-s + 11.9·100-s − 6.04·103-s − 0.864·109-s + 20.0·112-s − 6.35·121-s + 6.06·124-s − 11.1·133-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{24} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.06109\times 10^{15}\) |
| Root analytic conductor: | \(9.86619\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 3^{24} \cdot 7^{8} ,\ ( \ : [9/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 7 | \( ( 1 - p^{4} T )^{8} \) | |
| good | 2 | \( 1 + 1963 T^{2} + 528289 p^{2} T^{4} + 25630819 p^{6} T^{6} + 1850831 p^{19} T^{8} + 25630819 p^{24} T^{10} + 528289 p^{38} T^{12} + 1963 p^{54} T^{14} + p^{72} T^{16} \) |
| 5 | \( 1 + 6079168 T^{2} + 16065515630206 T^{4} + 1277376463146776824 p^{2} T^{6} + \)\(10\!\cdots\!79\)\( p^{4} T^{8} + 1277376463146776824 p^{20} T^{10} + 16065515630206 p^{36} T^{12} + 6079168 p^{54} T^{14} + p^{72} T^{16} \) | |
| 11 | \( 1 + 14986207504 T^{2} + \)\(10\!\cdots\!02\)\( T^{4} + \)\(43\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!31\)\( T^{8} + \)\(43\!\cdots\!40\)\( p^{18} T^{10} + \)\(10\!\cdots\!02\)\( p^{36} T^{12} + 14986207504 p^{54} T^{14} + p^{72} T^{16} \) | |
| 13 | \( ( 1 + 47278 T + 27538690852 T^{2} + 1376493618944482 T^{3} + \)\(39\!\cdots\!86\)\( T^{4} + 1376493618944482 p^{9} T^{5} + 27538690852 p^{18} T^{6} + 47278 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 17 | \( 1 + 199453340584 T^{2} + \)\(19\!\cdots\!36\)\( p T^{4} + \)\(51\!\cdots\!00\)\( T^{6} + \)\(80\!\cdots\!86\)\( T^{8} + \)\(51\!\cdots\!00\)\( p^{18} T^{10} + \)\(19\!\cdots\!36\)\( p^{37} T^{12} + 199453340584 p^{54} T^{14} + p^{72} T^{16} \) | |
| 19 | \( ( 1 + 1044964 T + 1018304076682 T^{2} + 728224758044237464 T^{3} + \)\(51\!\cdots\!95\)\( T^{4} + 728224758044237464 p^{9} T^{5} + 1018304076682 p^{18} T^{6} + 1044964 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 23 | \( 1 + 10105922945296 T^{2} + \)\(46\!\cdots\!02\)\( T^{4} + \)\(13\!\cdots\!28\)\( T^{6} + \)\(27\!\cdots\!75\)\( T^{8} + \)\(13\!\cdots\!28\)\( p^{18} T^{10} + \)\(46\!\cdots\!02\)\( p^{36} T^{12} + 10105922945296 p^{54} T^{14} + p^{72} T^{16} \) | |
| 29 | \( 1 + 57053062194292 T^{2} + \)\(17\!\cdots\!32\)\( T^{4} + \)\(37\!\cdots\!40\)\( T^{6} + \)\(61\!\cdots\!34\)\( T^{8} + \)\(37\!\cdots\!40\)\( p^{18} T^{10} + \)\(17\!\cdots\!32\)\( p^{36} T^{12} + 57053062194292 p^{54} T^{14} + p^{72} T^{16} \) | |
| 31 | \( ( 1 + 4068298 T - 8390538453392 T^{2} - 45744645086994852200 T^{3} + \)\(41\!\cdots\!81\)\( T^{4} - 45744645086994852200 p^{9} T^{5} - 8390538453392 p^{18} T^{6} + 4068298 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 37 | \( ( 1 - 7804610 T + 216273038846140 T^{2} - \)\(26\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!37\)\( T^{4} - \)\(26\!\cdots\!60\)\( p^{9} T^{5} + 216273038846140 p^{18} T^{6} - 7804610 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 41 | \( 1 + 941202615658240 T^{2} + \)\(60\!\cdots\!94\)\( T^{4} + \)\(28\!\cdots\!52\)\( T^{6} + \)\(10\!\cdots\!19\)\( T^{8} + \)\(28\!\cdots\!52\)\( p^{18} T^{10} + \)\(60\!\cdots\!94\)\( p^{36} T^{12} + 941202615658240 p^{54} T^{14} + p^{72} T^{16} \) | |
| 43 | \( ( 1 + 3086830 T + 1057201196352112 T^{2} + \)\(43\!\cdots\!58\)\( T^{3} + \)\(60\!\cdots\!22\)\( T^{4} + \)\(43\!\cdots\!58\)\( p^{9} T^{5} + 1057201196352112 p^{18} T^{6} + 3086830 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 47 | \( 1 - 102284284056308 T^{2} + \)\(10\!\cdots\!40\)\( T^{4} + \)\(24\!\cdots\!80\)\( T^{6} - \)\(70\!\cdots\!94\)\( T^{8} + \)\(24\!\cdots\!80\)\( p^{18} T^{10} + \)\(10\!\cdots\!40\)\( p^{36} T^{12} - 102284284056308 p^{54} T^{14} + p^{72} T^{16} \) | |
| 53 | \( 1 + 8585271284037688 T^{2} + \)\(26\!\cdots\!92\)\( T^{4} - \)\(71\!\cdots\!84\)\( T^{6} - \)\(21\!\cdots\!66\)\( T^{8} - \)\(71\!\cdots\!84\)\( p^{18} T^{10} + \)\(26\!\cdots\!92\)\( p^{36} T^{12} + 8585271284037688 p^{54} T^{14} + p^{72} T^{16} \) | |
| 59 | \( 1 + 26776750535661052 T^{2} + \)\(47\!\cdots\!72\)\( T^{4} + \)\(55\!\cdots\!40\)\( T^{6} + \)\(15\!\cdots\!34\)\( p^{2} T^{8} + \)\(55\!\cdots\!40\)\( p^{18} T^{10} + \)\(47\!\cdots\!72\)\( p^{36} T^{12} + 26776750535661052 p^{54} T^{14} + p^{72} T^{16} \) | |
| 61 | \( ( 1 - 31647752 T + 21902035437348148 T^{2} - \)\(57\!\cdots\!60\)\( T^{3} + \)\(23\!\cdots\!46\)\( T^{4} - \)\(57\!\cdots\!60\)\( p^{9} T^{5} + 21902035437348148 p^{18} T^{6} - 31647752 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 67 | \( ( 1 + 277559002 T + 96985273599608572 T^{2} + \)\(19\!\cdots\!90\)\( T^{3} + \)\(38\!\cdots\!66\)\( T^{4} + \)\(19\!\cdots\!90\)\( p^{9} T^{5} + 96985273599608572 p^{18} T^{6} + 277559002 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 71 | \( 1 + 246899786199858136 T^{2} + \)\(30\!\cdots\!10\)\( T^{4} + \)\(24\!\cdots\!44\)\( T^{6} + \)\(13\!\cdots\!19\)\( T^{8} + \)\(24\!\cdots\!44\)\( p^{18} T^{10} + \)\(30\!\cdots\!10\)\( p^{36} T^{12} + 246899786199858136 p^{54} T^{14} + p^{72} T^{16} \) | |
| 73 | \( ( 1 + 183380230 T + 164887693399811152 T^{2} + \)\(19\!\cdots\!82\)\( T^{3} + \)\(12\!\cdots\!62\)\( T^{4} + \)\(19\!\cdots\!82\)\( p^{9} T^{5} + 164887693399811152 p^{18} T^{6} + 183380230 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 79 | \( ( 1 + 245057518 T + 235688929600592224 T^{2} + \)\(73\!\cdots\!86\)\( T^{3} + \)\(35\!\cdots\!50\)\( T^{4} + \)\(73\!\cdots\!86\)\( p^{9} T^{5} + 235688929600592224 p^{18} T^{6} + 245057518 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 83 | \( 1 + 804952885144418020 T^{2} + \)\(34\!\cdots\!24\)\( T^{4} + \)\(98\!\cdots\!20\)\( T^{6} + \)\(21\!\cdots\!06\)\( T^{8} + \)\(98\!\cdots\!20\)\( p^{18} T^{10} + \)\(34\!\cdots\!24\)\( p^{36} T^{12} + 804952885144418020 p^{54} T^{14} + p^{72} T^{16} \) | |
| 89 | \( 1 + 255644176219027984 T^{2} + \)\(40\!\cdots\!70\)\( T^{4} + \)\(81\!\cdots\!56\)\( T^{6} + \)\(71\!\cdots\!99\)\( T^{8} + \)\(81\!\cdots\!56\)\( p^{18} T^{10} + \)\(40\!\cdots\!70\)\( p^{36} T^{12} + 255644176219027984 p^{54} T^{14} + p^{72} T^{16} \) | |
| 97 | \( ( 1 + 701439760 T + 2574378986272688260 T^{2} + \)\(14\!\cdots\!40\)\( T^{3} + \)\(27\!\cdots\!38\)\( T^{4} + \)\(14\!\cdots\!40\)\( p^{9} T^{5} + 2574378986272688260 p^{18} T^{6} + 701439760 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 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
−4.60774486343064521718931646134, −4.31934795376837704878769669805, −4.19298060313095401904935560777, −4.14614254131024986950781175261, −4.03783647588706667066473655930, −4.02656847942879107160037448235, −3.95053865539791803953911849458, −3.92467801768293042187791524984, −3.81708129951178704002838570379, −3.19816377994803821946570044095, −3.12954559678333309312324448829, −2.94186596067038033112404682131, −2.59977294594945191821117414551, −2.53634794426327313167841049430, −2.34612114259041238693844142879, −2.27496964412398464451646821264, −2.09409456148431340196845632696, −2.01986042285413241483164050262, −1.72639379304306639183457163168, −1.44116845263649140256043544244, −1.27688350587196947192057449333, −1.24194539254201110023658112698, −1.15701255813966835249303450237, −1.09052455717484414585966528154, −0.927042990203747731981174980392, 0, 0, 0, 0, 0, 0, 0, 0, 0.927042990203747731981174980392, 1.09052455717484414585966528154, 1.15701255813966835249303450237, 1.24194539254201110023658112698, 1.27688350587196947192057449333, 1.44116845263649140256043544244, 1.72639379304306639183457163168, 2.01986042285413241483164050262, 2.09409456148431340196845632696, 2.27496964412398464451646821264, 2.34612114259041238693844142879, 2.53634794426327313167841049430, 2.59977294594945191821117414551, 2.94186596067038033112404682131, 3.12954559678333309312324448829, 3.19816377994803821946570044095, 3.81708129951178704002838570379, 3.92467801768293042187791524984, 3.95053865539791803953911849458, 4.02656847942879107160037448235, 4.03783647588706667066473655930, 4.14614254131024986950781175261, 4.19298060313095401904935560777, 4.31934795376837704878769669805, 4.60774486343064521718931646134
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.