Dirichlet series
| L(s) = 1 | − 3·3-s + 5·5-s + 7-s + 21·9-s + 155·11-s + 7·13-s − 15·15-s + 161·17-s + 272·19-s − 3·21-s − 628·23-s + 129·25-s + 59·27-s + 33·29-s − 323·31-s − 465·33-s + 5·35-s + 49·37-s − 21·39-s + 361·41-s − 1.44e3·43-s + 105·45-s + 1.06e3·47-s − 11·49-s − 483·51-s − 281·53-s + 775·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.0539·7-s + 7/9·9-s + 4.24·11-s + 0.149·13-s − 0.258·15-s + 2.29·17-s + 3.28·19-s − 0.0311·21-s − 5.69·23-s + 1.03·25-s + 0.420·27-s + 0.211·29-s − 1.87·31-s − 2.45·33-s + 0.0241·35-s + 0.217·37-s − 0.0862·39-s + 1.37·41-s − 5.11·43-s + 0.347·45-s + 3.31·47-s − 0.0320·49-s − 1.32·51-s − 0.728·53-s + 1.90·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.35216\times 10^{8}\) |
| Root analytic conductor: | \(3.22247\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 11^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.6782414147\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6782414147\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - 155 T + 13111 T^{2} - 68645 p T^{3} + 264456 p^{2} T^{4} - 68645 p^{4} T^{5} + 13111 p^{6} T^{6} - 155 p^{9} T^{7} + p^{12} T^{8} \) | |
| good | 3 | \( 1 + p T - 4 p T^{2} - 158 T^{3} - 1228 T^{4} - 6473 T^{5} + 959 T^{6} + 225344 T^{7} + 1236232 T^{8} + 225344 p^{3} T^{9} + 959 p^{6} T^{10} - 6473 p^{9} T^{11} - 1228 p^{12} T^{12} - 158 p^{15} T^{13} - 4 p^{19} T^{14} + p^{22} T^{15} + p^{24} T^{16} \) |
| 5 | \( 1 - p T - 104 T^{2} - 326 p T^{3} + 10006 T^{4} + 56777 p T^{5} + 2835571 T^{6} - 1265534 p^{2} T^{7} - 387953844 T^{8} - 1265534 p^{5} T^{9} + 2835571 p^{6} T^{10} + 56777 p^{10} T^{11} + 10006 p^{12} T^{12} - 326 p^{16} T^{13} - 104 p^{18} T^{14} - p^{22} T^{15} + p^{24} T^{16} \) | |
| 7 | \( 1 - T + 12 T^{2} + 1104 T^{3} - 87888 T^{4} + 922401 T^{5} + 28062801 T^{6} - 211635072 T^{7} + 13215739812 T^{8} - 211635072 p^{3} T^{9} + 28062801 p^{6} T^{10} + 922401 p^{9} T^{11} - 87888 p^{12} T^{12} + 1104 p^{15} T^{13} + 12 p^{18} T^{14} - p^{21} T^{15} + p^{24} T^{16} \) | |
| 13 | \( 1 - 7 T + 1930 T^{2} + 79384 T^{3} + 10224596 T^{4} + 326819897 T^{5} + 769971197 p T^{6} + 1235524128730 T^{7} + 47709276128944 T^{8} + 1235524128730 p^{3} T^{9} + 769971197 p^{7} T^{10} + 326819897 p^{9} T^{11} + 10224596 p^{12} T^{12} + 79384 p^{15} T^{13} + 1930 p^{18} T^{14} - 7 p^{21} T^{15} + p^{24} T^{16} \) | |
| 17 | \( 1 - 161 T + 5020 T^{2} + 211188 T^{3} + 21130156 T^{4} - 1323457999 T^{5} - 265862283281 T^{6} + 778694329080 p T^{7} + 277440196391544 T^{8} + 778694329080 p^{4} T^{9} - 265862283281 p^{6} T^{10} - 1323457999 p^{9} T^{11} + 21130156 p^{12} T^{12} + 211188 p^{15} T^{13} + 5020 p^{18} T^{14} - 161 p^{21} T^{15} + p^{24} T^{16} \) | |
| 19 | \( 1 - 272 T + 33201 T^{2} - 3039584 T^{3} + 292957015 T^{4} - 21884364992 T^{5} + 909384182595 T^{6} - 32108296725040 T^{7} + 2715838419478696 T^{8} - 32108296725040 p^{3} T^{9} + 909384182595 p^{6} T^{10} - 21884364992 p^{9} T^{11} + 292957015 p^{12} T^{12} - 3039584 p^{15} T^{13} + 33201 p^{18} T^{14} - 272 p^{21} T^{15} + p^{24} T^{16} \) | |
| 23 | \( ( 1 + 314 T + 61816 T^{2} + 8042722 T^{3} + 950275950 T^{4} + 8042722 p^{3} T^{5} + 61816 p^{6} T^{6} + 314 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 29 | \( 1 - 33 T + 14176 T^{2} + 2091144 T^{3} + 419909220 T^{4} + 81926257017 T^{5} + 7198320043835 T^{6} + 2719607629962300 T^{7} + 63828590693142176 T^{8} + 2719607629962300 p^{3} T^{9} + 7198320043835 p^{6} T^{10} + 81926257017 p^{9} T^{11} + 419909220 p^{12} T^{12} + 2091144 p^{15} T^{13} + 14176 p^{18} T^{14} - 33 p^{21} T^{15} + p^{24} T^{16} \) | |
| 31 | \( 1 + 323 T - 4098 T^{2} - 5799192 T^{3} + 305184598 T^{4} + 93424183073 T^{5} + 1165032156515 p T^{6} + 3925905573337100 T^{7} - 564631402757797204 T^{8} + 3925905573337100 p^{3} T^{9} + 1165032156515 p^{7} T^{10} + 93424183073 p^{9} T^{11} + 305184598 p^{12} T^{12} - 5799192 p^{15} T^{13} - 4098 p^{18} T^{14} + 323 p^{21} T^{15} + p^{24} T^{16} \) | |
| 37 | \( 1 - 49 T - 7868 T^{2} + 4260746 T^{3} - 566286138 T^{4} + 1042572020189 T^{5} + 91349107260671 T^{6} - 14416097444880138 T^{7} + 5705343181934634812 T^{8} - 14416097444880138 p^{3} T^{9} + 91349107260671 p^{6} T^{10} + 1042572020189 p^{9} T^{11} - 566286138 p^{12} T^{12} + 4260746 p^{15} T^{13} - 7868 p^{18} T^{14} - 49 p^{21} T^{15} + p^{24} T^{16} \) | |
| 41 | \( 1 - 361 T - 19140 T^{2} + 12862580 T^{3} + 3853225260 T^{4} - 605378813703 T^{5} - 283363657590617 T^{6} + 99783171614693080 T^{7} - 22290482644766813160 T^{8} + 99783171614693080 p^{3} T^{9} - 283363657590617 p^{6} T^{10} - 605378813703 p^{9} T^{11} + 3853225260 p^{12} T^{12} + 12862580 p^{15} T^{13} - 19140 p^{18} T^{14} - 361 p^{21} T^{15} + p^{24} T^{16} \) | |
| 43 | \( ( 1 + 721 T + 420117 T^{2} + 154459221 T^{3} + 51447883420 T^{4} + 154459221 p^{3} T^{5} + 420117 p^{6} T^{6} + 721 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 47 | \( 1 - 1069 T + 301070 T^{2} + 43368042 T^{3} - 24642745634 T^{4} - 6223420821981 T^{5} + 4635059307315839 T^{6} - 746453682627128500 T^{7} + 29698413240018564184 T^{8} - 746453682627128500 p^{3} T^{9} + 4635059307315839 p^{6} T^{10} - 6223420821981 p^{9} T^{11} - 24642745634 p^{12} T^{12} + 43368042 p^{15} T^{13} + 301070 p^{18} T^{14} - 1069 p^{21} T^{15} + p^{24} T^{16} \) | |
| 53 | \( 1 + 281 T - 343802 T^{2} - 60864958 T^{3} + 33213391926 T^{4} + 4932564267181 T^{5} + 8874412003121445 T^{6} + 107096647673733916 T^{7} - \)\(25\!\cdots\!56\)\( T^{8} + 107096647673733916 p^{3} T^{9} + 8874412003121445 p^{6} T^{10} + 4932564267181 p^{9} T^{11} + 33213391926 p^{12} T^{12} - 60864958 p^{15} T^{13} - 343802 p^{18} T^{14} + 281 p^{21} T^{15} + p^{24} T^{16} \) | |
| 59 | \( 1 - 128 T - 87119 T^{2} + 50674984 T^{3} + 39669673335 T^{4} - 496570131448 T^{5} - 4049042036547885 T^{6} + 22790010341236160 T^{7} + \)\(27\!\cdots\!36\)\( T^{8} + 22790010341236160 p^{3} T^{9} - 4049042036547885 p^{6} T^{10} - 496570131448 p^{9} T^{11} + 39669673335 p^{12} T^{12} + 50674984 p^{15} T^{13} - 87119 p^{18} T^{14} - 128 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( 1 + 617 T - 60798 T^{2} - 112995618 T^{3} + 22253617738 T^{4} + 1960695737 p^{2} T^{5} - 13214872371636415 T^{6} + 3429112403997460700 T^{7} + \)\(68\!\cdots\!36\)\( T^{8} + 3429112403997460700 p^{3} T^{9} - 13214872371636415 p^{6} T^{10} + 1960695737 p^{11} T^{11} + 22253617738 p^{12} T^{12} - 112995618 p^{15} T^{13} - 60798 p^{18} T^{14} + 617 p^{21} T^{15} + p^{24} T^{16} \) | |
| 67 | \( ( 1 + 289 T + 686735 T^{2} + 243308709 T^{3} + 267199450216 T^{4} + 243308709 p^{3} T^{5} + 686735 p^{6} T^{6} + 289 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 71 | \( 1 + 115 T - 138428 T^{2} - 42960000 T^{3} + 77925723838 T^{4} + 71653925274035 T^{5} + 12191505393571699 T^{6} - 3285417296443548450 T^{7} + \)\(11\!\cdots\!80\)\( T^{8} - 3285417296443548450 p^{3} T^{9} + 12191505393571699 p^{6} T^{10} + 71653925274035 p^{9} T^{11} + 77925723838 p^{12} T^{12} - 42960000 p^{15} T^{13} - 138428 p^{18} T^{14} + 115 p^{21} T^{15} + p^{24} T^{16} \) | |
| 73 | \( 1 + 1487 T + 142728 T^{2} - 14316264 p T^{3} - 569519703768 T^{4} + 366060407928453 T^{5} + 393291607153115139 T^{6} - 68019158899540158384 T^{7} - \)\(19\!\cdots\!28\)\( T^{8} - 68019158899540158384 p^{3} T^{9} + 393291607153115139 p^{6} T^{10} + 366060407928453 p^{9} T^{11} - 569519703768 p^{12} T^{12} - 14316264 p^{16} T^{13} + 142728 p^{18} T^{14} + 1487 p^{21} T^{15} + p^{24} T^{16} \) | |
| 79 | \( 1 + 71 T - 500222 T^{2} + 73013624 T^{3} + 350790961998 T^{4} - 97550067497959 T^{5} - 161731343227399015 T^{6} - 8534351363848547400 T^{7} + \)\(59\!\cdots\!76\)\( T^{8} - 8534351363848547400 p^{3} T^{9} - 161731343227399015 p^{6} T^{10} - 97550067497959 p^{9} T^{11} + 350790961998 p^{12} T^{12} + 73013624 p^{15} T^{13} - 500222 p^{18} T^{14} + 71 p^{21} T^{15} + p^{24} T^{16} \) | |
| 83 | \( 1 + 1942 T + 875395 T^{2} - 857035884 T^{3} - 790010610449 T^{4} + 618864428088728 T^{5} + 845503237658086801 T^{6} - \)\(17\!\cdots\!30\)\( T^{7} - \)\(59\!\cdots\!96\)\( T^{8} - \)\(17\!\cdots\!30\)\( p^{3} T^{9} + 845503237658086801 p^{6} T^{10} + 618864428088728 p^{9} T^{11} - 790010610449 p^{12} T^{12} - 857035884 p^{15} T^{13} + 875395 p^{18} T^{14} + 1942 p^{21} T^{15} + p^{24} T^{16} \) | |
| 89 | \( ( 1 + 1101 T + 2406895 T^{2} + 1914547747 T^{3} + 2334666485008 T^{4} + 1914547747 p^{3} T^{5} + 2406895 p^{6} T^{6} + 1101 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 97 | \( 1 + 5128 T + 10343177 T^{2} + 9500759764 T^{3} + 1479496697751 T^{4} - 4959875190278012 T^{5} - 1475957592224572125 T^{6} + \)\(96\!\cdots\!12\)\( T^{7} + \)\(15\!\cdots\!64\)\( T^{8} + \)\(96\!\cdots\!12\)\( p^{3} T^{9} - 1475957592224572125 p^{6} T^{10} - 4959875190278012 p^{9} T^{11} + 1479496697751 p^{12} T^{12} + 9500759764 p^{15} T^{13} + 10343177 p^{18} T^{14} + 5128 p^{21} T^{15} + p^{24} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.25190155933358114982794977689, −5.19258247565194773246574475904, −5.18029375958443141764921577318, −4.64492963635356895862914429028, −4.52883301627833416269927028334, −4.34091561656575601612885957250, −4.29534962455597259147411530934, −4.11397546174268328428678810638, −4.07605791012163892716482239740, −3.54442013394143753479488666645, −3.52823018355529400083771201407, −3.49902630078312899385687228865, −3.36750562196320944481264282089, −3.18541639119534585332478531343, −2.89837934027946008569923001493, −2.51160661282591362530467038833, −2.12167124782638122110487723805, −2.09998270452172694222598994813, −1.53926379329187845730740330808, −1.44605986318679376050451841775, −1.41205310385267054220955367462, −1.37431139745581202428555096447, −0.967777356489846957668027175201, −0.65506171105411039948250730180, −0.06841925235016306106376670684, 0.06841925235016306106376670684, 0.65506171105411039948250730180, 0.967777356489846957668027175201, 1.37431139745581202428555096447, 1.41205310385267054220955367462, 1.44605986318679376050451841775, 1.53926379329187845730740330808, 2.09998270452172694222598994813, 2.12167124782638122110487723805, 2.51160661282591362530467038833, 2.89837934027946008569923001493, 3.18541639119534585332478531343, 3.36750562196320944481264282089, 3.49902630078312899385687228865, 3.52823018355529400083771201407, 3.54442013394143753479488666645, 4.07605791012163892716482239740, 4.11397546174268328428678810638, 4.29534962455597259147411530934, 4.34091561656575601612885957250, 4.52883301627833416269927028334, 4.64492963635356895862914429028, 5.18029375958443141764921577318, 5.19258247565194773246574475904, 5.25190155933358114982794977689
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.