Dirichlet series
| L(s) = 1 | − 10·4-s − 21·9-s − 22·11-s + 46·16-s − 2·29-s − 16·31-s + 210·36-s − 26·41-s + 220·44-s − 66·49-s − 10·59-s − 30·61-s − 132·64-s + 20·71-s − 12·79-s + 204·81-s + 462·99-s − 124·101-s + 4·109-s + 20·116-s + 159·121-s + 160·124-s + 127-s + 131-s + 137-s + 139-s − 966·144-s + ⋯ |
| L(s) = 1 | − 5·4-s − 7·9-s − 6.63·11-s + 23/2·16-s − 0.371·29-s − 2.87·31-s + 35·36-s − 4.06·41-s + 33.1·44-s − 9.42·49-s − 1.30·59-s − 3.84·61-s − 16.5·64-s + 2.37·71-s − 1.35·79-s + 68/3·81-s + 46.4·99-s − 12.3·101-s + 0.383·109-s + 1.85·116-s + 14.4·121-s + 14.3·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 80.5·144-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{32} \cdot 19^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{32} \cdot 19^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(5^{32} \cdot 19^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.29162\times 10^{29}\) |
| Root analytic conductor: | \(8.48910\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(16\) |
| Selberg data: | \((32,\ 5^{32} \cdot 19^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) | |
| good | 2 | \( 1 + 5 p T^{2} + 27 p T^{4} + 53 p^{2} T^{6} + 339 p T^{8} + 931 p T^{10} + 4563 T^{12} + 5 p^{11} T^{14} + 2659 p^{3} T^{16} + 5 p^{13} T^{18} + 4563 p^{4} T^{20} + 931 p^{7} T^{22} + 339 p^{9} T^{24} + 53 p^{12} T^{26} + 27 p^{13} T^{28} + 5 p^{15} T^{30} + p^{16} T^{32} \) |
| 3 | \( 1 + 7 p T^{2} + 79 p T^{4} + 1871 T^{6} + 11450 T^{8} + 19135 p T^{10} + 243650 T^{12} + 893438 T^{14} + 2862415 T^{16} + 893438 p^{2} T^{18} + 243650 p^{4} T^{20} + 19135 p^{7} T^{22} + 11450 p^{8} T^{24} + 1871 p^{10} T^{26} + 79 p^{13} T^{28} + 7 p^{15} T^{30} + p^{16} T^{32} \) | |
| 7 | \( 1 + 66 T^{2} + 2081 T^{4} + 42012 T^{6} + 616866 T^{8} + 7124626 T^{10} + 68478540 T^{12} + 570368870 T^{14} + 4219284829 T^{16} + 570368870 p^{2} T^{18} + 68478540 p^{4} T^{20} + 7124626 p^{6} T^{22} + 616866 p^{8} T^{24} + 42012 p^{10} T^{26} + 2081 p^{12} T^{28} + 66 p^{14} T^{30} + p^{16} T^{32} \) | |
| 11 | \( ( 1 + p T + 102 T^{2} + 658 T^{3} + 3846 T^{4} + 18509 T^{5} + 82041 T^{6} + 313492 T^{7} + 1112696 T^{8} + 313492 p T^{9} + 82041 p^{2} T^{10} + 18509 p^{3} T^{11} + 3846 p^{4} T^{12} + 658 p^{5} T^{13} + 102 p^{6} T^{14} + p^{8} T^{15} + p^{8} T^{16} )^{2} \) | |
| 13 | \( 1 + 84 T^{2} + 3917 T^{4} + 128052 T^{6} + 3279065 T^{8} + 69339672 T^{10} + 1253169478 T^{12} + 19707109224 T^{14} + 272761050454 T^{16} + 19707109224 p^{2} T^{18} + 1253169478 p^{4} T^{20} + 69339672 p^{6} T^{22} + 3279065 p^{8} T^{24} + 128052 p^{10} T^{26} + 3917 p^{12} T^{28} + 84 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 + 137 T^{2} + 9369 T^{4} + 433207 T^{6} + 15386898 T^{8} + 447213445 T^{10} + 10971277495 T^{12} + 230927231611 T^{14} + 4210574211770 T^{16} + 230927231611 p^{2} T^{18} + 10971277495 p^{4} T^{20} + 447213445 p^{6} T^{22} + 15386898 p^{8} T^{24} + 433207 p^{10} T^{26} + 9369 p^{12} T^{28} + 137 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 + 199 T^{2} + 18309 T^{4} + 1051021 T^{6} + 43287438 T^{8} + 1406829971 T^{10} + 38874444562 T^{12} + 969173355394 T^{14} + 22766596327475 T^{16} + 969173355394 p^{2} T^{18} + 38874444562 p^{4} T^{20} + 1406829971 p^{6} T^{22} + 43287438 p^{8} T^{24} + 1051021 p^{10} T^{26} + 18309 p^{12} T^{28} + 199 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 + T + 4 p T^{2} - 178 T^{3} + 5934 T^{4} - 23275 T^{5} + 216593 T^{6} - 1121766 T^{7} + 6797608 T^{8} - 1121766 p T^{9} + 216593 p^{2} T^{10} - 23275 p^{3} T^{11} + 5934 p^{4} T^{12} - 178 p^{5} T^{13} + 4 p^{7} T^{14} + p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 8 T + 92 T^{2} + 586 T^{3} + 6264 T^{4} + 34676 T^{5} + 268931 T^{6} + 1355730 T^{7} + 9846500 T^{8} + 1355730 p T^{9} + 268931 p^{2} T^{10} + 34676 p^{3} T^{11} + 6264 p^{4} T^{12} + 586 p^{5} T^{13} + 92 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 + 330 T^{2} + 56078 T^{4} + 6449298 T^{6} + 558693730 T^{8} + 38529548684 T^{10} + 59026283145 p T^{12} + 2802228080552 p T^{14} + 4164209714713156 T^{16} + 2802228080552 p^{3} T^{18} + 59026283145 p^{5} T^{20} + 38529548684 p^{6} T^{22} + 558693730 p^{8} T^{24} + 6449298 p^{10} T^{26} + 56078 p^{12} T^{28} + 330 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 + 13 T + 216 T^{2} + 1846 T^{3} + 19030 T^{4} + 123315 T^{5} + 1012028 T^{6} + 5615571 T^{7} + 43184091 T^{8} + 5615571 p T^{9} + 1012028 p^{2} T^{10} + 123315 p^{3} T^{11} + 19030 p^{4} T^{12} + 1846 p^{5} T^{13} + 216 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( 1 + 317 T^{2} + 57282 T^{4} + 7258360 T^{6} + 709382250 T^{8} + 55923208015 T^{10} + 3653329372707 T^{12} + 200804783070476 T^{14} + 9362969451989696 T^{16} + 200804783070476 p^{2} T^{18} + 3653329372707 p^{4} T^{20} + 55923208015 p^{6} T^{22} + 709382250 p^{8} T^{24} + 7258360 p^{10} T^{26} + 57282 p^{12} T^{28} + 317 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( 1 + 281 T^{2} + 41338 T^{4} + 4232968 T^{6} + 7213892 p T^{8} + 22767200557 T^{10} + 1343116171204 T^{12} + 71670967359385 T^{14} + 3509332551662605 T^{16} + 71670967359385 p^{2} T^{18} + 1343116171204 p^{4} T^{20} + 22767200557 p^{6} T^{22} + 7213892 p^{9} T^{24} + 4232968 p^{10} T^{26} + 41338 p^{12} T^{28} + 281 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 + 481 T^{2} + 109454 T^{4} + 15882300 T^{6} + 1680870930 T^{8} + 141286268115 T^{10} + 10019387404115 T^{12} + 623789454740344 T^{14} + 34812964295858984 T^{16} + 623789454740344 p^{2} T^{18} + 10019387404115 p^{4} T^{20} + 141286268115 p^{6} T^{22} + 1680870930 p^{8} T^{24} + 15882300 p^{10} T^{26} + 109454 p^{12} T^{28} + 481 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 + 5 T + 208 T^{2} + 686 T^{3} + 22036 T^{4} + 66773 T^{5} + 1698647 T^{6} + 4679056 T^{7} + 104953836 T^{8} + 4679056 p T^{9} + 1698647 p^{2} T^{10} + 66773 p^{3} T^{11} + 22036 p^{4} T^{12} + 686 p^{5} T^{13} + 208 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 + 15 T + 397 T^{2} + 4699 T^{3} + 71495 T^{4} + 700428 T^{5} + 7806960 T^{6} + 64209582 T^{7} + 572907406 T^{8} + 64209582 p T^{9} + 7806960 p^{2} T^{10} + 700428 p^{3} T^{11} + 71495 p^{4} T^{12} + 4699 p^{5} T^{13} + 397 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 + 634 T^{2} + 205109 T^{4} + 44542972 T^{6} + 107951390 p T^{8} + 928174660730 T^{10} + 97174717460772 T^{12} + 8455007581562702 T^{14} + 617407611474126261 T^{16} + 8455007581562702 p^{2} T^{18} + 97174717460772 p^{4} T^{20} + 928174660730 p^{6} T^{22} + 107951390 p^{9} T^{24} + 44542972 p^{10} T^{26} + 205109 p^{12} T^{28} + 634 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 - 10 T + 351 T^{2} - 2664 T^{3} + 55293 T^{4} - 327926 T^{5} + 5463262 T^{6} - 27012630 T^{7} + 419355862 T^{8} - 27012630 p T^{9} + 5463262 p^{2} T^{10} - 327926 p^{3} T^{11} + 55293 p^{4} T^{12} - 2664 p^{5} T^{13} + 351 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 + 684 T^{2} + 238605 T^{4} + 55671944 T^{6} + 9689214777 T^{8} + 1332390305892 T^{10} + 149822696019470 T^{12} + 14066053781332756 T^{14} + 1115296531673056654 T^{16} + 14066053781332756 p^{2} T^{18} + 149822696019470 p^{4} T^{20} + 1332390305892 p^{6} T^{22} + 9689214777 p^{8} T^{24} + 55671944 p^{10} T^{26} + 238605 p^{12} T^{28} + 684 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 + 6 T + 199 T^{2} + 392 T^{3} + 23778 T^{4} + 37484 T^{5} + 2584965 T^{6} + 5446706 T^{7} + 239055058 T^{8} + 5446706 p T^{9} + 2584965 p^{2} T^{10} + 37484 p^{3} T^{11} + 23778 p^{4} T^{12} + 392 p^{5} T^{13} + 199 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 + 924 T^{2} + 410251 T^{4} + 117109132 T^{6} + 24269891594 T^{8} + 3905116220884 T^{10} + 508110457511029 T^{12} + 54771814258547460 T^{14} + 4953530917648975370 T^{16} + 54771814258547460 p^{2} T^{18} + 508110457511029 p^{4} T^{20} + 3905116220884 p^{6} T^{22} + 24269891594 p^{8} T^{24} + 117109132 p^{10} T^{26} + 410251 p^{12} T^{28} + 924 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 451 T^{2} - 344 T^{3} + 94842 T^{4} - 117908 T^{5} + 12746144 T^{6} - 18552024 T^{7} + 1275998013 T^{8} - 18552024 p T^{9} + 12746144 p^{2} T^{10} - 117908 p^{3} T^{11} + 94842 p^{4} T^{12} - 344 p^{5} T^{13} + 451 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 + 614 T^{2} + 178770 T^{4} + 34051218 T^{6} + 5054132534 T^{8} + 650710057888 T^{10} + 75143609093297 T^{12} + 7859270288485044 T^{14} + 774097356913814580 T^{16} + 7859270288485044 p^{2} T^{18} + 75143609093297 p^{4} T^{20} + 650710057888 p^{6} T^{22} + 5054132534 p^{8} T^{24} + 34051218 p^{10} T^{26} + 178770 p^{12} T^{28} + 614 p^{14} T^{30} + p^{16} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.33831166464360547662437562624, −2.14813474532044611602592624204, −2.13095226853377100031144017251, −2.11864093324874737666671223403, −2.08922053640095308518378444641, −2.01917970637333893410417112026, −1.96439191448579266652932001273, −1.87594896197693265967583791920, −1.82672528952274915031887333538, −1.77361538753419354148013067369, −1.68788901280239520578005570583, −1.42927430079409441769030177941, −1.41650539008234798568828215753, −1.38506024186434631509837679118, −1.38497080976757700598432207782, −1.29516974146485662848553423654, −1.22607251775874176624513232471, −1.19257384326480504186000835444, −1.13942639791128945953576926656, −1.09446211953257282323205254037, −0.965067941120881258736942863448, −0.916639263236409492891138029761, −0.893047017823122366879717555199, −0.69862851590904290930413787088, −0.62645511976521079566177151506, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.62645511976521079566177151506, 0.69862851590904290930413787088, 0.893047017823122366879717555199, 0.916639263236409492891138029761, 0.965067941120881258736942863448, 1.09446211953257282323205254037, 1.13942639791128945953576926656, 1.19257384326480504186000835444, 1.22607251775874176624513232471, 1.29516974146485662848553423654, 1.38497080976757700598432207782, 1.38506024186434631509837679118, 1.41650539008234798568828215753, 1.42927430079409441769030177941, 1.68788901280239520578005570583, 1.77361538753419354148013067369, 1.82672528952274915031887333538, 1.87594896197693265967583791920, 1.96439191448579266652932001273, 2.01917970637333893410417112026, 2.08922053640095308518378444641, 2.11864093324874737666671223403, 2.13095226853377100031144017251, 2.14813474532044611602592624204, 2.33831166464360547662437562624
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.