Dirichlet series
| L(s) = 1 | + 2·5-s + 18·11-s − 2·13-s + 4·17-s + 26·19-s + 2·25-s + 202·29-s − 368·31-s − 10·37-s + 838·43-s − 944·47-s + 1.76e3·49-s + 378·53-s + 36·55-s + 1.70e3·59-s + 910·61-s − 4·65-s − 1.94e3·67-s + 4.41e3·79-s − 2.56e3·83-s + 8·85-s + 52·95-s − 4·97-s + 1.56e3·101-s − 4.03e3·107-s − 330·109-s + 668·113-s + ⋯ |
| L(s) = 1 | + 0.178·5-s + 0.493·11-s − 0.0426·13-s + 0.0570·17-s + 0.313·19-s + 0.0159·25-s + 1.29·29-s − 2.13·31-s − 0.0444·37-s + 2.97·43-s − 2.92·47-s + 5.13·49-s + 0.979·53-s + 0.0882·55-s + 3.76·59-s + 1.91·61-s − 0.00763·65-s − 3.54·67-s + 6.28·79-s − 3.38·83-s + 0.0102·85-s + 0.0561·95-s − 0.00418·97-s + 1.53·101-s − 3.64·107-s − 0.289·109-s + 0.556·113-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{60} \cdot 3^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.05534\times 10^{15}\) |
| Root analytic conductor: | \(5.82967\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{60} \cdot 3^{20} ,\ ( \ : [3/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.1455368027\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1455368027\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + 2 T^{2} + 966 T^{3} - 13723 T^{4} - 3608 p T^{5} + 530104 T^{6} + 11981288 T^{7} - 28535006 T^{8} - 2301854348 T^{9} + 23672040908 T^{10} - 2301854348 p^{3} T^{11} - 28535006 p^{6} T^{12} + 11981288 p^{9} T^{13} + 530104 p^{12} T^{14} - 3608 p^{16} T^{15} - 13723 p^{18} T^{16} + 966 p^{21} T^{17} + 2 p^{24} T^{18} - 2 p^{27} T^{19} + p^{30} T^{20} \) |
| 7 | \( 1 - 1762 T^{2} + 219995 p T^{4} - 133155368 p T^{6} + 440869947922 T^{8} - 168121217547916 T^{10} + 440869947922 p^{6} T^{12} - 133155368 p^{13} T^{14} + 219995 p^{19} T^{16} - 1762 p^{24} T^{18} + p^{30} T^{20} \) | |
| 11 | \( 1 - 18 T + 162 T^{2} - 122934 T^{3} + 4077397 T^{4} + 79597000 T^{5} + 5463099864 T^{6} - 313798751208 T^{7} - 56924680478 p^{2} T^{8} + 9237744161500 p T^{9} + 18755914132083020 T^{10} + 9237744161500 p^{4} T^{11} - 56924680478 p^{8} T^{12} - 313798751208 p^{9} T^{13} + 5463099864 p^{12} T^{14} + 79597000 p^{15} T^{15} + 4077397 p^{18} T^{16} - 122934 p^{21} T^{17} + 162 p^{24} T^{18} - 18 p^{27} T^{19} + p^{30} T^{20} \) | |
| 13 | \( 1 + 2 T + 2 T^{2} - 45206 T^{3} - 184727 p T^{4} + 29395960 T^{5} + 1085386040 T^{6} - 29556007880 p T^{7} - 9400034983966 T^{8} + 1531455561616908 T^{9} + 23489689415409228 T^{10} + 1531455561616908 p^{3} T^{11} - 9400034983966 p^{6} T^{12} - 29556007880 p^{10} T^{13} + 1085386040 p^{12} T^{14} + 29395960 p^{15} T^{15} - 184727 p^{19} T^{16} - 45206 p^{21} T^{17} + 2 p^{24} T^{18} + 2 p^{27} T^{19} + p^{30} T^{20} \) | |
| 17 | \( ( 1 - 2 T + 12653 T^{2} - 102520 T^{3} + 98460610 T^{4} - 354493580 T^{5} + 98460610 p^{3} T^{6} - 102520 p^{6} T^{7} + 12653 p^{9} T^{8} - 2 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 19 | \( 1 - 26 T + 338 T^{2} + 339906 T^{3} - 64153371 T^{4} - 2461461784 T^{5} + 143449890200 T^{6} - 36783398837960 T^{7} + 1011857007777554 T^{8} + 259766590630759364 T^{9} - 7366645907488092948 T^{10} + 259766590630759364 p^{3} T^{11} + 1011857007777554 p^{6} T^{12} - 36783398837960 p^{9} T^{13} + 143449890200 p^{12} T^{14} - 2461461784 p^{15} T^{15} - 64153371 p^{18} T^{16} + 339906 p^{21} T^{17} + 338 p^{24} T^{18} - 26 p^{27} T^{19} + p^{30} T^{20} \) | |
| 23 | \( 1 - 76386 T^{2} + 2913757597 T^{4} - 73253961622040 T^{6} + 1342371312768300946 T^{8} - \)\(18\!\cdots\!84\)\( T^{10} + 1342371312768300946 p^{6} T^{12} - 73253961622040 p^{12} T^{14} + 2913757597 p^{18} T^{16} - 76386 p^{24} T^{18} + p^{30} T^{20} \) | |
| 29 | \( 1 - 202 T + 20402 T^{2} + 1177934 T^{3} + 398569397 T^{4} - 239164019416 T^{5} + 40873283338616 T^{6} - 2529271278095288 T^{7} + 194871598558001506 T^{8} - \)\(12\!\cdots\!32\)\( T^{9} + \)\(30\!\cdots\!36\)\( T^{10} - \)\(12\!\cdots\!32\)\( p^{3} T^{11} + 194871598558001506 p^{6} T^{12} - 2529271278095288 p^{9} T^{13} + 40873283338616 p^{12} T^{14} - 239164019416 p^{15} T^{15} + 398569397 p^{18} T^{16} + 1177934 p^{21} T^{17} + 20402 p^{24} T^{18} - 202 p^{27} T^{19} + p^{30} T^{20} \) | |
| 31 | \( ( 1 + 184 T + 134043 T^{2} + 19809056 T^{3} + 7638677322 T^{4} + 852982867024 T^{5} + 7638677322 p^{3} T^{6} + 19809056 p^{6} T^{7} + 134043 p^{9} T^{8} + 184 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 37 | \( 1 + 10 T + 50 T^{2} + 1972962 T^{3} + 1630465317 T^{4} - 153991562664 T^{5} + 324850634232 T^{6} - 5252842710654600 T^{7} + 3474549392106364962 T^{8} + 27985624577691139772 T^{9} + \)\(10\!\cdots\!24\)\( T^{10} + 27985624577691139772 p^{3} T^{11} + 3474549392106364962 p^{6} T^{12} - 5252842710654600 p^{9} T^{13} + 324850634232 p^{12} T^{14} - 153991562664 p^{15} T^{15} + 1630465317 p^{18} T^{16} + 1972962 p^{21} T^{17} + 50 p^{24} T^{18} + 10 p^{27} T^{19} + p^{30} T^{20} \) | |
| 41 | \( 1 - 441018 T^{2} + 97166156061 T^{4} - 13934678680622904 T^{6} + \)\(14\!\cdots\!14\)\( T^{8} - \)\(11\!\cdots\!88\)\( T^{10} + \)\(14\!\cdots\!14\)\( p^{6} T^{12} - 13934678680622904 p^{12} T^{14} + 97166156061 p^{18} T^{16} - 441018 p^{24} T^{18} + p^{30} T^{20} \) | |
| 43 | \( 1 - 838 T + 351122 T^{2} - 132133650 T^{3} + 56398378005 T^{4} - 20936462157416 T^{5} + 6471694737204248 T^{6} - 1952595983380873720 T^{7} + \)\(59\!\cdots\!10\)\( T^{8} - \)\(17\!\cdots\!68\)\( T^{9} + \)\(49\!\cdots\!52\)\( T^{10} - \)\(17\!\cdots\!68\)\( p^{3} T^{11} + \)\(59\!\cdots\!10\)\( p^{6} T^{12} - 1952595983380873720 p^{9} T^{13} + 6471694737204248 p^{12} T^{14} - 20936462157416 p^{15} T^{15} + 56398378005 p^{18} T^{16} - 132133650 p^{21} T^{17} + 351122 p^{24} T^{18} - 838 p^{27} T^{19} + p^{30} T^{20} \) | |
| 47 | \( ( 1 + 472 T + 462219 T^{2} + 171516064 T^{3} + 90105579914 T^{4} + 25593405310224 T^{5} + 90105579914 p^{3} T^{6} + 171516064 p^{6} T^{7} + 462219 p^{9} T^{8} + 472 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 53 | \( 1 - 378 T + 71442 T^{2} + 995350 p T^{3} + 3016286341 T^{4} + 526686651752 T^{5} + 976891435665272 T^{6} + 1674349213754452168 T^{7} - 1581505854923305054 T^{8} + \)\(14\!\cdots\!20\)\( T^{9} + \)\(29\!\cdots\!08\)\( T^{10} + \)\(14\!\cdots\!20\)\( p^{3} T^{11} - 1581505854923305054 p^{6} T^{12} + 1674349213754452168 p^{9} T^{13} + 976891435665272 p^{12} T^{14} + 526686651752 p^{15} T^{15} + 3016286341 p^{18} T^{16} + 995350 p^{22} T^{17} + 71442 p^{24} T^{18} - 378 p^{27} T^{19} + p^{30} T^{20} \) | |
| 59 | \( 1 - 1706 T + 1455218 T^{2} - 989315358 T^{3} + 555128806581 T^{4} - 232658117164632 T^{5} + 78453755015006616 T^{6} - 20092577710244830152 T^{7} + \)\(57\!\cdots\!98\)\( T^{8} + \)\(37\!\cdots\!12\)\( p T^{9} - \)\(35\!\cdots\!80\)\( p^{2} T^{10} + \)\(37\!\cdots\!12\)\( p^{4} T^{11} + \)\(57\!\cdots\!98\)\( p^{6} T^{12} - 20092577710244830152 p^{9} T^{13} + 78453755015006616 p^{12} T^{14} - 232658117164632 p^{15} T^{15} + 555128806581 p^{18} T^{16} - 989315358 p^{21} T^{17} + 1455218 p^{24} T^{18} - 1706 p^{27} T^{19} + p^{30} T^{20} \) | |
| 61 | \( 1 - 910 T + 414050 T^{2} + 45940410 T^{3} + 20471098485 T^{4} - 72823214590920 T^{5} + 58848327585507000 T^{6} - 6001380717052735080 T^{7} + \)\(59\!\cdots\!50\)\( T^{8} - \)\(32\!\cdots\!00\)\( T^{9} + \)\(38\!\cdots\!00\)\( T^{10} - \)\(32\!\cdots\!00\)\( p^{3} T^{11} + \)\(59\!\cdots\!50\)\( p^{6} T^{12} - 6001380717052735080 p^{9} T^{13} + 58848327585507000 p^{12} T^{14} - 72823214590920 p^{15} T^{15} + 20471098485 p^{18} T^{16} + 45940410 p^{21} T^{17} + 414050 p^{24} T^{18} - 910 p^{27} T^{19} + p^{30} T^{20} \) | |
| 67 | \( 1 + 1942 T + 1885682 T^{2} + 1530965298 T^{3} + 1338066168261 T^{4} + 1068751594464168 T^{5} + 724275679990789656 T^{6} + \)\(46\!\cdots\!12\)\( T^{7} + \)\(29\!\cdots\!06\)\( T^{8} + \)\(17\!\cdots\!64\)\( T^{9} + \)\(97\!\cdots\!68\)\( T^{10} + \)\(17\!\cdots\!64\)\( p^{3} T^{11} + \)\(29\!\cdots\!06\)\( p^{6} T^{12} + \)\(46\!\cdots\!12\)\( p^{9} T^{13} + 724275679990789656 p^{12} T^{14} + 1068751594464168 p^{15} T^{15} + 1338066168261 p^{18} T^{16} + 1530965298 p^{21} T^{17} + 1885682 p^{24} T^{18} + 1942 p^{27} T^{19} + p^{30} T^{20} \) | |
| 71 | \( 1 - 2500418 T^{2} + 3072516920573 T^{4} - 2420243241413642648 T^{6} + \)\(13\!\cdots\!66\)\( T^{8} - \)\(55\!\cdots\!32\)\( T^{10} + \)\(13\!\cdots\!66\)\( p^{6} T^{12} - 2420243241413642648 p^{12} T^{14} + 3072516920573 p^{18} T^{16} - 2500418 p^{24} T^{18} + p^{30} T^{20} \) | |
| 73 | \( 1 - 3134282 T^{2} + 4627160201821 T^{4} - 4242124717558516472 T^{6} + \)\(26\!\cdots\!74\)\( T^{8} - \)\(12\!\cdots\!92\)\( T^{10} + \)\(26\!\cdots\!74\)\( p^{6} T^{12} - 4242124717558516472 p^{12} T^{14} + 4627160201821 p^{18} T^{16} - 3134282 p^{24} T^{18} + p^{30} T^{20} \) | |
| 79 | \( ( 1 - 2208 T + 3816107 T^{2} - 54694528 p T^{3} + 4245684014154 T^{4} - 3176789940661184 T^{5} + 4245684014154 p^{3} T^{6} - 54694528 p^{7} T^{7} + 3816107 p^{9} T^{8} - 2208 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 83 | \( 1 + 2562 T + 3281922 T^{2} + 2891460918 T^{3} + 1934799974629 T^{4} + 1191348439341176 T^{5} + 882645219418437336 T^{6} + \)\(79\!\cdots\!28\)\( T^{7} + \)\(89\!\cdots\!74\)\( T^{8} + \)\(97\!\cdots\!88\)\( T^{9} + \)\(82\!\cdots\!76\)\( T^{10} + \)\(97\!\cdots\!88\)\( p^{3} T^{11} + \)\(89\!\cdots\!74\)\( p^{6} T^{12} + \)\(79\!\cdots\!28\)\( p^{9} T^{13} + 882645219418437336 p^{12} T^{14} + 1191348439341176 p^{15} T^{15} + 1934799974629 p^{18} T^{16} + 2891460918 p^{21} T^{17} + 3281922 p^{24} T^{18} + 2562 p^{27} T^{19} + p^{30} T^{20} \) | |
| 89 | \( 1 - 3643178 T^{2} + 6505439011133 T^{4} - 7720859292932177528 T^{6} + \)\(69\!\cdots\!98\)\( T^{8} - \)\(52\!\cdots\!28\)\( T^{10} + \)\(69\!\cdots\!98\)\( p^{6} T^{12} - 7720859292932177528 p^{12} T^{14} + 6505439011133 p^{18} T^{16} - 3643178 p^{24} T^{18} + p^{30} T^{20} \) | |
| 97 | \( ( 1 + 2 T + 2565789 T^{2} - 723000584 T^{3} + 3231145430658 T^{4} - 1257303020547316 T^{5} + 3231145430658 p^{3} T^{6} - 723000584 p^{6} T^{7} + 2565789 p^{9} T^{8} + 2 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.56151082193274882812772300729, −3.44738815671800516790049584893, −3.31709989167626574448445144906, −3.13467187713724162990569767233, −3.09015463394317440452915097546, −3.08522178207019048314004738374, −2.71544365626811988175121895301, −2.44846767143039716238292518371, −2.43336924552341363781255351242, −2.43190673308079844446669745931, −2.31321271905972455609493655089, −2.13345846336532752118324263438, −2.07519297875702675573573797471, −1.98300546976644516659445085855, −1.94075529887901435059212776240, −1.41451703920941481485613579523, −1.19876379811931008539505127431, −1.19680559789134606055581613003, −1.10296005975909847114705600785, −0.980004016658408306913373385107, −0.912778749431110011884182015611, −0.796072023031706854653421869869, −0.45383839934688294791849973641, −0.13581468507461638303729159335, −0.03625185614507454039738692329, 0.03625185614507454039738692329, 0.13581468507461638303729159335, 0.45383839934688294791849973641, 0.796072023031706854653421869869, 0.912778749431110011884182015611, 0.980004016658408306913373385107, 1.10296005975909847114705600785, 1.19680559789134606055581613003, 1.19876379811931008539505127431, 1.41451703920941481485613579523, 1.94075529887901435059212776240, 1.98300546976644516659445085855, 2.07519297875702675573573797471, 2.13345846336532752118324263438, 2.31321271905972455609493655089, 2.43190673308079844446669745931, 2.43336924552341363781255351242, 2.44846767143039716238292518371, 2.71544365626811988175121895301, 3.08522178207019048314004738374, 3.09015463394317440452915097546, 3.13467187713724162990569767233, 3.31709989167626574448445144906, 3.44738815671800516790049584893, 3.56151082193274882812772300729
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.