Dirichlet series
| L(s) = 1 | + 24·2-s + 192·4-s − 70·5-s − 895·7-s − 1.02e3·8-s − 1.68e3·10-s + 2.42e3·11-s − 3.83e3·13-s − 2.14e4·14-s − 3.68e4·16-s + 2.45e4·17-s + 1.35e3·19-s − 1.34e4·20-s + 5.82e4·22-s + 9.66e4·23-s + 5.20e4·25-s − 9.21e4·26-s − 1.71e5·28-s + 2.28e5·29-s − 2.21e5·31-s − 2.94e5·32-s + 5.88e5·34-s + 6.26e4·35-s − 2.49e5·37-s + 3.24e4·38-s + 7.16e4·40-s − 1.22e6·41-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3/2·4-s − 0.250·5-s − 0.986·7-s − 0.707·8-s − 0.531·10-s + 0.550·11-s − 0.484·13-s − 2.09·14-s − 9/4·16-s + 1.20·17-s + 0.0452·19-s − 0.375·20-s + 1.16·22-s + 1.65·23-s + 0.666·25-s − 1.02·26-s − 1.47·28-s + 1.73·29-s − 1.33·31-s − 1.59·32-s + 2.56·34-s + 0.246·35-s − 0.811·37-s + 0.0959·38-s + 0.177·40-s − 2.76·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 3^{12} \cdot 7^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.71847\times 10^{9}\) |
| Root analytic conductor: | \(6.27379\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{6} \cdot 3^{12} \cdot 7^{6} ,\ ( \ : [7/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(5.502117845\) |
| \(L(\frac12)\) | \(\approx\) | \(5.502117845\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{3} T + p^{6} T^{2} )^{3} \) |
| 3 | \( 1 \) | |
| 7 | \( 1 + 895 T + 179972 p T^{2} + 1933405 p^{3} T^{3} + 179972 p^{8} T^{4} + 895 p^{14} T^{5} + p^{21} T^{6} \) | |
| good | 5 | \( 1 + 14 p T - 1886 p^{2} T^{2} - 2343608 p^{2} T^{3} - 28283066 p^{3} T^{4} + 2364130018 p^{4} T^{5} + 2688590397526 p^{4} T^{6} + 2364130018 p^{11} T^{7} - 28283066 p^{17} T^{8} - 2343608 p^{23} T^{9} - 1886 p^{30} T^{10} + 14 p^{36} T^{11} + p^{42} T^{12} \) |
| 11 | \( 1 - 2428 T - 29709662 T^{2} + 147331160156 T^{3} + 290058748592834 T^{4} - 1714602191236684264 T^{5} + \)\(15\!\cdots\!26\)\( T^{6} - 1714602191236684264 p^{7} T^{7} + 290058748592834 p^{14} T^{8} + 147331160156 p^{21} T^{9} - 29709662 p^{28} T^{10} - 2428 p^{35} T^{11} + p^{42} T^{12} \) | |
| 13 | \( ( 1 + 1919 T + 126281803 T^{2} + 299521970998 T^{3} + 126281803 p^{7} T^{4} + 1919 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 17 | \( 1 - 24508 T - 194164643 T^{2} + 6031351522172 T^{3} - 7027579989153274 T^{4} + \)\(17\!\cdots\!96\)\( T^{5} - \)\(59\!\cdots\!03\)\( T^{6} + \)\(17\!\cdots\!96\)\( p^{7} T^{7} - 7027579989153274 p^{14} T^{8} + 6031351522172 p^{21} T^{9} - 194164643 p^{28} T^{10} - 24508 p^{35} T^{11} + p^{42} T^{12} \) | |
| 19 | \( 1 - 1353 T - 1649735196 T^{2} - 22592829165385 T^{3} + 1265153973374033460 T^{4} + \)\(19\!\cdots\!07\)\( T^{5} - \)\(90\!\cdots\!26\)\( T^{6} + \)\(19\!\cdots\!07\)\( p^{7} T^{7} + 1265153973374033460 p^{14} T^{8} - 22592829165385 p^{21} T^{9} - 1649735196 p^{28} T^{10} - 1353 p^{35} T^{11} + p^{42} T^{12} \) | |
| 23 | \( 1 - 96628 T + 6077834875 T^{2} - 208819379742676 T^{3} - 5601149599840042906 T^{4} + \)\(85\!\cdots\!44\)\( T^{5} - \)\(62\!\cdots\!73\)\( T^{6} + \)\(85\!\cdots\!44\)\( p^{7} T^{7} - 5601149599840042906 p^{14} T^{8} - 208819379742676 p^{21} T^{9} + 6077834875 p^{28} T^{10} - 96628 p^{35} T^{11} + p^{42} T^{12} \) | |
| 29 | \( ( 1 - 114112 T + 9419826450 T^{2} - 2374628747215540 T^{3} + 9419826450 p^{7} T^{4} - 114112 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 31 | \( 1 + 221395 T - 44218377843 T^{2} - 4019607964239930 T^{3} + \)\(33\!\cdots\!51\)\( T^{4} + \)\(18\!\cdots\!35\)\( T^{5} - \)\(82\!\cdots\!02\)\( T^{6} + \)\(18\!\cdots\!35\)\( p^{7} T^{7} + \)\(33\!\cdots\!51\)\( p^{14} T^{8} - 4019607964239930 p^{21} T^{9} - 44218377843 p^{28} T^{10} + 221395 p^{35} T^{11} + p^{42} T^{12} \) | |
| 37 | \( 1 + 249987 T - 214909914018 T^{2} - 25679418067857883 T^{3} + \)\(38\!\cdots\!36\)\( T^{4} + \)\(22\!\cdots\!51\)\( T^{5} - \)\(38\!\cdots\!68\)\( T^{6} + \)\(22\!\cdots\!51\)\( p^{7} T^{7} + \)\(38\!\cdots\!36\)\( p^{14} T^{8} - 25679418067857883 p^{21} T^{9} - 214909914018 p^{28} T^{10} + 249987 p^{35} T^{11} + p^{42} T^{12} \) | |
| 41 | \( ( 1 + 610926 T + 337401342555 T^{2} + 125227407587641980 T^{3} + 337401342555 p^{7} T^{4} + 610926 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 43 | \( ( 1 - 600243 T + 860312262129 T^{2} - 316769755052672318 T^{3} + 860312262129 p^{7} T^{4} - 600243 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 47 | \( 1 + 123114 T - 1387760812485 T^{2} - 70783646176082958 T^{3} + \)\(12\!\cdots\!14\)\( T^{4} + \)\(27\!\cdots\!58\)\( T^{5} - \)\(72\!\cdots\!57\)\( T^{6} + \)\(27\!\cdots\!58\)\( p^{7} T^{7} + \)\(12\!\cdots\!14\)\( p^{14} T^{8} - 70783646176082958 p^{21} T^{9} - 1387760812485 p^{28} T^{10} + 123114 p^{35} T^{11} + p^{42} T^{12} \) | |
| 53 | \( 1 - 3004752 T + 3617300839710 T^{2} - 2951125291233682824 T^{3} + \)\(29\!\cdots\!94\)\( T^{4} - \)\(19\!\cdots\!84\)\( T^{5} + \)\(66\!\cdots\!22\)\( T^{6} - \)\(19\!\cdots\!84\)\( p^{7} T^{7} + \)\(29\!\cdots\!94\)\( p^{14} T^{8} - 2951125291233682824 p^{21} T^{9} + 3617300839710 p^{28} T^{10} - 3004752 p^{35} T^{11} + p^{42} T^{12} \) | |
| 59 | \( 1 + 2852938 T + 3702521326114 T^{2} + 280739840897568580 T^{3} - \)\(90\!\cdots\!10\)\( T^{4} - \)\(17\!\cdots\!02\)\( T^{5} - \)\(30\!\cdots\!66\)\( T^{6} - \)\(17\!\cdots\!02\)\( p^{7} T^{7} - \)\(90\!\cdots\!10\)\( p^{14} T^{8} + 280739840897568580 p^{21} T^{9} + 3702521326114 p^{28} T^{10} + 2852938 p^{35} T^{11} + p^{42} T^{12} \) | |
| 61 | \( 1 + 665386 T - 64464724419 p T^{2} + 5106467503962313590 T^{3} + \)\(61\!\cdots\!50\)\( T^{4} - \)\(16\!\cdots\!34\)\( T^{5} - \)\(12\!\cdots\!39\)\( T^{6} - \)\(16\!\cdots\!34\)\( p^{7} T^{7} + \)\(61\!\cdots\!50\)\( p^{14} T^{8} + 5106467503962313590 p^{21} T^{9} - 64464724419 p^{29} T^{10} + 665386 p^{35} T^{11} + p^{42} T^{12} \) | |
| 67 | \( 1 + 10545857 T + 57433067065892 T^{2} + \)\(23\!\cdots\!77\)\( T^{3} + \)\(81\!\cdots\!36\)\( T^{4} + \)\(24\!\cdots\!01\)\( T^{5} + \)\(62\!\cdots\!82\)\( T^{6} + \)\(24\!\cdots\!01\)\( p^{7} T^{7} + \)\(81\!\cdots\!36\)\( p^{14} T^{8} + \)\(23\!\cdots\!77\)\( p^{21} T^{9} + 57433067065892 p^{28} T^{10} + 10545857 p^{35} T^{11} + p^{42} T^{12} \) | |
| 71 | \( ( 1 - 1019762 T + 13751446670721 T^{2} - 12281893848498536948 T^{3} + 13751446670721 p^{7} T^{4} - 1019762 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 73 | \( 1 - 6858031 T + 11406715637558 T^{2} - 5678556792586599249 T^{3} + \)\(57\!\cdots\!36\)\( T^{4} + \)\(31\!\cdots\!97\)\( T^{5} - \)\(28\!\cdots\!92\)\( T^{6} + \)\(31\!\cdots\!97\)\( p^{7} T^{7} + \)\(57\!\cdots\!36\)\( p^{14} T^{8} - 5678556792586599249 p^{21} T^{9} + 11406715637558 p^{28} T^{10} - 6858031 p^{35} T^{11} + p^{42} T^{12} \) | |
| 79 | \( 1 + 1723021 T - 26859061916043 T^{2} - \)\(11\!\cdots\!58\)\( T^{3} + \)\(17\!\cdots\!19\)\( T^{4} + \)\(11\!\cdots\!73\)\( T^{5} + \)\(30\!\cdots\!54\)\( T^{6} + \)\(11\!\cdots\!73\)\( p^{7} T^{7} + \)\(17\!\cdots\!19\)\( p^{14} T^{8} - \)\(11\!\cdots\!58\)\( p^{21} T^{9} - 26859061916043 p^{28} T^{10} + 1723021 p^{35} T^{11} + p^{42} T^{12} \) | |
| 83 | \( ( 1 + 2635804 T + 81306945154278 T^{2} + \)\(14\!\cdots\!98\)\( T^{3} + 81306945154278 p^{7} T^{4} + 2635804 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 89 | \( 1 + 976224 T - 97541008465623 T^{2} + 15887021538672480288 T^{3} + \)\(53\!\cdots\!74\)\( T^{4} - \)\(26\!\cdots\!68\)\( T^{5} - \)\(25\!\cdots\!31\)\( T^{6} - \)\(26\!\cdots\!68\)\( p^{7} T^{7} + \)\(53\!\cdots\!74\)\( p^{14} T^{8} + 15887021538672480288 p^{21} T^{9} - 97541008465623 p^{28} T^{10} + 976224 p^{35} T^{11} + p^{42} T^{12} \) | |
| 97 | \( ( 1 + 8485792 T + 246057278958152 T^{2} + \)\(13\!\cdots\!86\)\( T^{3} + 246057278958152 p^{7} T^{4} + 8485792 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.01522101833558039141085301166, −5.60306827992531495341365073953, −5.57053609673688446739368492546, −5.36462635656583681473692888504, −5.11221657927388943588886070689, −4.98284180438150369278010563071, −4.64009657249656809142716415401, −4.55345979073239916511831768923, −4.30125991756727539008904039243, −4.05716058128555360578176956708, −3.96661579648636113944708434833, −3.45297418541101039229709712292, −3.36521622733832976682106156602, −3.17506328721112795528574044764, −3.10237580559854178431468958789, −2.83263574545765537338400028030, −2.65831165207217807969768679933, −2.07906895669592672888689560775, −2.07786414435011738388608286751, −1.50464806805545855800996905330, −1.29146208240274252206110223230, −0.954143974110808473792124954506, −0.878730424475675404386668394420, −0.25548644625514612256394555726, −0.24035662825923026860846015737, 0.24035662825923026860846015737, 0.25548644625514612256394555726, 0.878730424475675404386668394420, 0.954143974110808473792124954506, 1.29146208240274252206110223230, 1.50464806805545855800996905330, 2.07786414435011738388608286751, 2.07906895669592672888689560775, 2.65831165207217807969768679933, 2.83263574545765537338400028030, 3.10237580559854178431468958789, 3.17506328721112795528574044764, 3.36521622733832976682106156602, 3.45297418541101039229709712292, 3.96661579648636113944708434833, 4.05716058128555360578176956708, 4.30125991756727539008904039243, 4.55345979073239916511831768923, 4.64009657249656809142716415401, 4.98284180438150369278010563071, 5.11221657927388943588886070689, 5.36462635656583681473692888504, 5.57053609673688446739368492546, 5.60306827992531495341365073953, 6.01522101833558039141085301166
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.