Dirichlet series
| L(s) = 1 | + 10·4-s + 4·5-s + 21·9-s − 22·11-s + 46·16-s + 40·20-s + 10·25-s − 2·29-s + 16·31-s + 210·36-s + 26·41-s − 220·44-s + 84·45-s + 66·49-s − 88·55-s − 10·59-s − 30·61-s + 132·64-s − 20·71-s − 12·79-s + 184·80-s + 204·81-s − 462·99-s + 100·100-s − 124·101-s + 4·109-s − 20·116-s + ⋯ |
| L(s) = 1 | + 5·4-s + 1.78·5-s + 7·9-s − 6.63·11-s + 23/2·16-s + 8.94·20-s + 2·25-s − 0.371·29-s + 2.87·31-s + 35·36-s + 4.06·41-s − 33.1·44-s + 12.5·45-s + 66/7·49-s − 11.8·55-s − 1.30·59-s − 3.84·61-s + 33/2·64-s − 2.37·71-s − 1.35·79-s + 20.5·80-s + 68/3·81-s − 46.4·99-s + 10·100-s − 12.3·101-s + 0.383·109-s − 1.85·116-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 19^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 19^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(5^{16} \cdot 19^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.46791\times 10^{18}\) |
| Root analytic conductor: | \(3.79644\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 5^{16} \cdot 19^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(24.53847385\) |
| \(L(\frac12)\) | \(\approx\) | \(24.53847385\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 - 4 T + 6 T^{2} - 12 T^{3} + 44 T^{4} - 92 T^{5} + 218 T^{6} - 788 T^{7} + 2086 T^{8} - 788 p T^{9} + 218 p^{2} T^{10} - 92 p^{3} T^{11} + 44 p^{4} T^{12} - 12 p^{5} T^{13} + 6 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 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
−2.28850634092594267405731050535, −2.22919534993742406097048594251, −2.20491231914552228705720756657, −2.09803389982137216318938034902, −2.08101576329719828411181012256, −2.04544298910915224805657090898, −1.99265392155761593335848282461, −1.94161431310825731272023104444, −1.77560749882316357066124097119, −1.68544183079392578049080933889, −1.45406198568978807795027148954, −1.41087028953531384202095350138, −1.36993092933857482439690655228, −1.34673575090483823734860535035, −1.32126691646878489923596407779, −1.25231887561894662944115664442, −1.22834855028384112158023005930, −1.05717180622457846228864775271, −1.00451971208845258222304363303, −0.941820457394683352348054101935, −0.72839586671991602492562013115, −0.54298255299591879098580783434, −0.51519133432543722610503739808, −0.20776865064613175046376301525, −0.06396976677633202598887019051, 0.06396976677633202598887019051, 0.20776865064613175046376301525, 0.51519133432543722610503739808, 0.54298255299591879098580783434, 0.72839586671991602492562013115, 0.941820457394683352348054101935, 1.00451971208845258222304363303, 1.05717180622457846228864775271, 1.22834855028384112158023005930, 1.25231887561894662944115664442, 1.32126691646878489923596407779, 1.34673575090483823734860535035, 1.36993092933857482439690655228, 1.41087028953531384202095350138, 1.45406198568978807795027148954, 1.68544183079392578049080933889, 1.77560749882316357066124097119, 1.94161431310825731272023104444, 1.99265392155761593335848282461, 2.04544298910915224805657090898, 2.08101576329719828411181012256, 2.09803389982137216318938034902, 2.20491231914552228705720756657, 2.22919534993742406097048594251, 2.28850634092594267405731050535
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.