Dirichlet series
| L(s) = 1 | + 2·5-s − 11·9-s − 128·13-s − 6·17-s + 61·25-s + 128·29-s − 66·37-s + 384·41-s − 22·45-s − 6·49-s − 2·53-s − 434·61-s − 256·65-s − 10·73-s + 238·81-s − 12·85-s − 522·89-s − 768·97-s − 254·101-s + 178·109-s + 448·113-s + 1.40e3·117-s − 283·121-s + 442·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 2/5·5-s − 1.22·9-s − 9.84·13-s − 0.352·17-s + 2.43·25-s + 4.41·29-s − 1.78·37-s + 9.36·41-s − 0.488·45-s − 0.122·49-s − 0.0377·53-s − 7.11·61-s − 3.93·65-s − 0.136·73-s + 2.93·81-s − 0.141·85-s − 5.86·89-s − 7.91·97-s − 2.51·101-s + 1.63·109-s + 3.96·113-s + 12.0·117-s − 2.33·121-s + 3.53·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{60} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.67298\times 10^{9}\) |
| Root analytic conductor: | \(2.47053\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{60} \cdot 7^{12} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.03269804514\) |
| \(L(\frac12)\) | \(\approx\) | \(0.03269804514\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 6 T^{2} + 4335 T^{4} + 71444 T^{6} + 4335 p^{4} T^{8} + 6 p^{8} T^{10} + p^{12} T^{12} \) | |
| good | 3 | \( 1 + 11 T^{2} - 13 p^{2} T^{4} - 884 T^{6} + 6035 p T^{8} + 64793 T^{10} - 1253162 T^{12} + 64793 p^{4} T^{14} + 6035 p^{9} T^{16} - 884 p^{12} T^{18} - 13 p^{18} T^{20} + 11 p^{20} T^{22} + p^{24} T^{24} \) |
| 5 | \( ( 1 - T - 29 T^{2} - 132 T^{3} + 201 T^{4} + 481 p T^{5} + 10726 T^{6} + 481 p^{3} T^{7} + 201 p^{4} T^{8} - 132 p^{6} T^{9} - 29 p^{8} T^{10} - p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 11 | \( 1 + 283 T^{2} + 46555 T^{4} + 3528284 T^{6} - 210042215 T^{8} - 117264051335 T^{10} - 18469284955402 T^{12} - 117264051335 p^{4} T^{14} - 210042215 p^{8} T^{16} + 3528284 p^{12} T^{18} + 46555 p^{16} T^{20} + 283 p^{20} T^{22} + p^{24} T^{24} \) | |
| 13 | \( ( 1 + 32 T + 803 T^{2} + 11632 T^{3} + 803 p^{2} T^{4} + 32 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
| 17 | \( ( 1 + 3 T - 181 T^{2} + 11996 T^{3} - 1671 T^{4} - 1164367 T^{5} + 69728918 T^{6} - 1164367 p^{2} T^{7} - 1671 p^{4} T^{8} + 11996 p^{6} T^{9} - 181 p^{8} T^{10} + 3 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 19 | \( 1 + 843 T^{2} + 169755 T^{4} + 10759612 T^{6} + 34418758617 T^{8} + 15878060416809 T^{10} + 3783625584664182 T^{12} + 15878060416809 p^{4} T^{14} + 34418758617 p^{8} T^{16} + 10759612 p^{12} T^{18} + 169755 p^{16} T^{20} + 843 p^{20} T^{22} + p^{24} T^{24} \) | |
| 23 | \( 1 + 2659 T^{2} + 3876571 T^{4} + 4127666252 T^{6} + 3503571279769 T^{8} + 2424695206910689 T^{10} + 1397846635041452342 T^{12} + 2424695206910689 p^{4} T^{14} + 3503571279769 p^{8} T^{16} + 4127666252 p^{12} T^{18} + 3876571 p^{16} T^{20} + 2659 p^{20} T^{22} + p^{24} T^{24} \) | |
| 29 | \( ( 1 - 32 T - 109 T^{2} + 37136 T^{3} - 109 p^{2} T^{4} - 32 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
| 31 | \( 1 + 2963 T^{2} + 4291147 T^{4} + 3752683836 T^{6} + 1856892008697 T^{8} - 613860419386879 T^{10} - 1734915478854125546 T^{12} - 613860419386879 p^{4} T^{14} + 1856892008697 p^{8} T^{16} + 3752683836 p^{12} T^{18} + 4291147 p^{16} T^{20} + 2963 p^{20} T^{22} + p^{24} T^{24} \) | |
| 37 | \( ( 1 + 33 T - 1333 T^{2} - 122132 T^{3} - 783591 T^{4} + 87702163 T^{5} + 5266681334 T^{6} + 87702163 p^{2} T^{7} - 783591 p^{4} T^{8} - 122132 p^{6} T^{9} - 1333 p^{8} T^{10} + 33 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 41 | \( ( 1 - 96 T + 179 p T^{2} - 328752 T^{3} + 179 p^{3} T^{4} - 96 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
| 43 | \( ( 1 - 5206 T^{2} + 17138783 T^{4} - 37018566196 T^{6} + 17138783 p^{4} T^{8} - 5206 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 47 | \( 1 + 6995 T^{2} + 22181707 T^{4} + 50014251324 T^{6} + 105476166484281 T^{8} + 174006129657696449 T^{10} + \)\(27\!\cdots\!62\)\( T^{12} + 174006129657696449 p^{4} T^{14} + 105476166484281 p^{8} T^{16} + 50014251324 p^{12} T^{18} + 22181707 p^{16} T^{20} + 6995 p^{20} T^{22} + p^{24} T^{24} \) | |
| 53 | \( ( 1 + T - 2837 T^{2} - 53748 T^{3} + 54009 T^{4} + 72242131 T^{5} + 22496954230 T^{6} + 72242131 p^{2} T^{7} + 54009 p^{4} T^{8} - 53748 p^{6} T^{9} - 2837 p^{8} T^{10} + p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 59 | \( 1 + 18811 T^{2} + 200120491 T^{4} + 1494527327372 T^{6} + 8628303999799801 T^{8} + 40087667805525876841 T^{10} + \)\(15\!\cdots\!62\)\( T^{12} + 40087667805525876841 p^{4} T^{14} + 8628303999799801 p^{8} T^{16} + 1494527327372 p^{12} T^{18} + 200120491 p^{16} T^{20} + 18811 p^{20} T^{22} + p^{24} T^{24} \) | |
| 61 | \( ( 1 + 217 T + 20699 T^{2} + 1814604 T^{3} + 173780089 T^{4} + 12006150107 T^{5} + 685616953942 T^{6} + 12006150107 p^{2} T^{7} + 173780089 p^{4} T^{8} + 1814604 p^{6} T^{9} + 20699 p^{8} T^{10} + 217 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 67 | \( 1 + 9899 T^{2} + 24620203 T^{4} - 79278218964 T^{6} - 82489071404871 T^{8} + 4300819315073397209 T^{10} + \)\(30\!\cdots\!86\)\( T^{12} + 4300819315073397209 p^{4} T^{14} - 82489071404871 p^{8} T^{16} - 79278218964 p^{12} T^{18} + 24620203 p^{16} T^{20} + 9899 p^{20} T^{22} + p^{24} T^{24} \) | |
| 71 | \( ( 1 - 12262 T^{2} + 100517039 T^{4} - 635265930196 T^{6} + 100517039 p^{4} T^{8} - 12262 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 73 | \( ( 1 + 5 T - 12813 T^{2} - 167716 T^{3} + 95820777 T^{4} + 946458799 T^{5} - 562891511514 T^{6} + 946458799 p^{2} T^{7} + 95820777 p^{4} T^{8} - 167716 p^{6} T^{9} - 12813 p^{8} T^{10} + 5 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 79 | \( 1 + 28083 T^{2} + 421723515 T^{4} + 4556484807052 T^{6} + 39635396888713497 T^{8} + \)\(29\!\cdots\!89\)\( T^{10} + \)\(19\!\cdots\!22\)\( T^{12} + \)\(29\!\cdots\!89\)\( p^{4} T^{14} + 39635396888713497 p^{8} T^{16} + 4556484807052 p^{12} T^{18} + 421723515 p^{16} T^{20} + 28083 p^{20} T^{22} + p^{24} T^{24} \) | |
| 83 | \( ( 1 - 29238 T^{2} + 396631039 T^{4} - 3342660983284 T^{6} + 396631039 p^{4} T^{8} - 29238 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 89 | \( ( 1 + 261 T + 22931 T^{2} + 2459452 T^{3} + 465950601 T^{4} + 40453273231 T^{5} + 2330338704422 T^{6} + 40453273231 p^{2} T^{7} + 465950601 p^{4} T^{8} + 2459452 p^{6} T^{9} + 22931 p^{8} T^{10} + 261 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 97 | \( ( 1 + 192 T + 28555 T^{2} + 2621616 T^{3} + 28555 p^{2} T^{4} + 192 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.11390019977588875200821331188, −4.03167934144141303161376519369, −4.00495615710315413574525906923, −3.59923024623062535306256826707, −3.27849913265425099870509849679, −3.05811686380007414422785036111, −2.99859289818434013836885913273, −2.93222160767252822877972983254, −2.92591021326245570299227580543, −2.79201539927769463267627367311, −2.63455237363815592576812552998, −2.62261286639164846997530643369, −2.51474787235862252122013793510, −2.49464669565529612056725093576, −2.34612763309079209074312743483, −2.31144865538848013884383016585, −1.93405331746657200162983139011, −1.82293202511150938399695233913, −1.45441890342099510512690112604, −1.18815922331347407384212677794, −1.06397037607025677737678880048, −0.993141563742037017109697750920, −0.49436543176195877492833863613, −0.11728393919227077960525064222, −0.083600454682232193233879584370, 0.083600454682232193233879584370, 0.11728393919227077960525064222, 0.49436543176195877492833863613, 0.993141563742037017109697750920, 1.06397037607025677737678880048, 1.18815922331347407384212677794, 1.45441890342099510512690112604, 1.82293202511150938399695233913, 1.93405331746657200162983139011, 2.31144865538848013884383016585, 2.34612763309079209074312743483, 2.49464669565529612056725093576, 2.51474787235862252122013793510, 2.62261286639164846997530643369, 2.63455237363815592576812552998, 2.79201539927769463267627367311, 2.92591021326245570299227580543, 2.93222160767252822877972983254, 2.99859289818434013836885913273, 3.05811686380007414422785036111, 3.27849913265425099870509849679, 3.59923024623062535306256826707, 4.00495615710315413574525906923, 4.03167934144141303161376519369, 4.11390019977588875200821331188
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.