Dirichlet series
| L(s) = 1 | − 3-s − 21·4-s − 12·5-s + 59·7-s + 12·8-s − 117·9-s + 21·12-s + 61·13-s + 12·15-s + 168·16-s − 107·17-s − 152·19-s + 252·20-s − 59·21-s − 259·23-s − 12·24-s − 624·25-s + 58·27-s − 1.23e3·28-s − 61·29-s − 276·31-s − 288·32-s − 708·35-s + 2.45e3·36-s − 204·37-s − 61·39-s − 144·40-s + ⋯ |
| L(s) = 1 | − 0.192·3-s − 2.62·4-s − 1.07·5-s + 3.18·7-s + 0.530·8-s − 4.33·9-s + 0.505·12-s + 1.30·13-s + 0.206·15-s + 21/8·16-s − 1.52·17-s − 1.83·19-s + 2.81·20-s − 0.613·21-s − 2.34·23-s − 0.102·24-s − 4.99·25-s + 0.413·27-s − 8.36·28-s − 0.390·29-s − 1.59·31-s − 1.59·32-s − 3.41·35-s + 91/8·36-s − 0.906·37-s − 0.250·39-s − 0.569·40-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(11^{16} \cdot 19^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.14614\times 10^{17}\) |
| Root analytic conductor: | \(11.6466\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 11^{16} \cdot 19^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 11 | \( 1 \) |
| 19 | \( ( 1 + p T )^{8} \) | |
| good | 2 | \( 1 + 21 T^{2} - 3 p^{2} T^{3} + 273 T^{4} - 27 p^{3} T^{5} + 2815 T^{6} - 501 p^{2} T^{7} + 12793 p T^{8} - 501 p^{5} T^{9} + 2815 p^{6} T^{10} - 27 p^{12} T^{11} + 273 p^{12} T^{12} - 3 p^{17} T^{13} + 21 p^{18} T^{14} + p^{24} T^{16} \) |
| 3 | \( 1 + T + 118 T^{2} + 59 p T^{3} + 80 p^{4} T^{4} + 5071 p T^{5} + 233795 T^{6} + 722168 T^{7} + 6760328 T^{8} + 722168 p^{3} T^{9} + 233795 p^{6} T^{10} + 5071 p^{10} T^{11} + 80 p^{16} T^{12} + 59 p^{16} T^{13} + 118 p^{18} T^{14} + p^{21} T^{15} + p^{24} T^{16} \) | |
| 5 | \( 1 + 12 T + 768 T^{2} + 1498 p T^{3} + 271824 T^{4} + 2231954 T^{5} + 59633697 T^{6} + 415930724 T^{7} + 8925142012 T^{8} + 415930724 p^{3} T^{9} + 59633697 p^{6} T^{10} + 2231954 p^{9} T^{11} + 271824 p^{12} T^{12} + 1498 p^{16} T^{13} + 768 p^{18} T^{14} + 12 p^{21} T^{15} + p^{24} T^{16} \) | |
| 7 | \( 1 - 59 T + 3203 T^{2} - 113164 T^{3} + 3721687 T^{4} - 97490223 T^{5} + 2410949483 T^{6} - 50510646866 T^{7} + 1007848498724 T^{8} - 50510646866 p^{3} T^{9} + 2410949483 p^{6} T^{10} - 97490223 p^{9} T^{11} + 3721687 p^{12} T^{12} - 113164 p^{15} T^{13} + 3203 p^{18} T^{14} - 59 p^{21} T^{15} + p^{24} T^{16} \) | |
| 13 | \( 1 - 61 T + 11247 T^{2} - 430436 T^{3} + 55305969 T^{4} - 1620335257 T^{5} + 190329605315 T^{6} - 4897726778718 T^{7} + 493418844534712 T^{8} - 4897726778718 p^{3} T^{9} + 190329605315 p^{6} T^{10} - 1620335257 p^{9} T^{11} + 55305969 p^{12} T^{12} - 430436 p^{15} T^{13} + 11247 p^{18} T^{14} - 61 p^{21} T^{15} + p^{24} T^{16} \) | |
| 17 | \( 1 + 107 T + 27750 T^{2} + 2013541 T^{3} + 316686864 T^{4} + 16540945823 T^{5} + 2177048934426 T^{6} + 89368247586737 T^{7} + 11514263156818654 T^{8} + 89368247586737 p^{3} T^{9} + 2177048934426 p^{6} T^{10} + 16540945823 p^{9} T^{11} + 316686864 p^{12} T^{12} + 2013541 p^{15} T^{13} + 27750 p^{18} T^{14} + 107 p^{21} T^{15} + p^{24} T^{16} \) | |
| 23 | \( 1 + 259 T + 73367 T^{2} + 9196416 T^{3} + 1229379126 T^{4} + 26227930970 T^{5} - 3981639422823 T^{6} - 2360605775901259 T^{7} - 237877490710417134 T^{8} - 2360605775901259 p^{3} T^{9} - 3981639422823 p^{6} T^{10} + 26227930970 p^{9} T^{11} + 1229379126 p^{12} T^{12} + 9196416 p^{15} T^{13} + 73367 p^{18} T^{14} + 259 p^{21} T^{15} + p^{24} T^{16} \) | |
| 29 | \( 1 + 61 T + 97913 T^{2} + 847252 T^{3} + 5029735327 T^{4} - 71393098753 T^{5} + 189960179832165 T^{6} - 2842429290720144 T^{7} + 5454531539808714724 T^{8} - 2842429290720144 p^{3} T^{9} + 189960179832165 p^{6} T^{10} - 71393098753 p^{9} T^{11} + 5029735327 p^{12} T^{12} + 847252 p^{15} T^{13} + 97913 p^{18} T^{14} + 61 p^{21} T^{15} + p^{24} T^{16} \) | |
| 31 | \( 1 + 276 T + 201644 T^{2} + 41588200 T^{3} + 17791449780 T^{4} + 2958997181234 T^{5} + 945704353632209 T^{6} + 130490279666944742 T^{7} + 33811106324899431160 T^{8} + 130490279666944742 p^{3} T^{9} + 945704353632209 p^{6} T^{10} + 2958997181234 p^{9} T^{11} + 17791449780 p^{12} T^{12} + 41588200 p^{15} T^{13} + 201644 p^{18} T^{14} + 276 p^{21} T^{15} + p^{24} T^{16} \) | |
| 37 | \( 1 + 204 T + 252749 T^{2} + 53322746 T^{3} + 32916635154 T^{4} + 6432492148066 T^{5} + 2792913130004395 T^{6} + 484864967605413152 T^{7} + \)\(16\!\cdots\!10\)\( T^{8} + 484864967605413152 p^{3} T^{9} + 2792913130004395 p^{6} T^{10} + 6432492148066 p^{9} T^{11} + 32916635154 p^{12} T^{12} + 53322746 p^{15} T^{13} + 252749 p^{18} T^{14} + 204 p^{21} T^{15} + p^{24} T^{16} \) | |
| 41 | \( 1 - 402 T + 360215 T^{2} - 97930486 T^{3} + 56584436951 T^{4} - 12116728500822 T^{5} + 5826736277022865 T^{6} - 1050290925517741042 T^{7} + \)\(45\!\cdots\!64\)\( T^{8} - 1050290925517741042 p^{3} T^{9} + 5826736277022865 p^{6} T^{10} - 12116728500822 p^{9} T^{11} + 56584436951 p^{12} T^{12} - 97930486 p^{15} T^{13} + 360215 p^{18} T^{14} - 402 p^{21} T^{15} + p^{24} T^{16} \) | |
| 43 | \( 1 - 598 T + 623983 T^{2} - 256697228 T^{3} + 154899488663 T^{4} - 48981741680876 T^{5} + 22070524301673481 T^{6} - 5694663684552238286 T^{7} + \)\(20\!\cdots\!32\)\( T^{8} - 5694663684552238286 p^{3} T^{9} + 22070524301673481 p^{6} T^{10} - 48981741680876 p^{9} T^{11} + 154899488663 p^{12} T^{12} - 256697228 p^{15} T^{13} + 623983 p^{18} T^{14} - 598 p^{21} T^{15} + p^{24} T^{16} \) | |
| 47 | \( 1 + 892 T + 617268 T^{2} + 271207740 T^{3} + 86333610148 T^{4} + 14649970907596 T^{5} - 2044184759540468 T^{6} - 2751920317890835828 T^{7} - \)\(11\!\cdots\!46\)\( T^{8} - 2751920317890835828 p^{3} T^{9} - 2044184759540468 p^{6} T^{10} + 14649970907596 p^{9} T^{11} + 86333610148 p^{12} T^{12} + 271207740 p^{15} T^{13} + 617268 p^{18} T^{14} + 892 p^{21} T^{15} + p^{24} T^{16} \) | |
| 53 | \( 1 + 147 T + 623060 T^{2} + 101437437 T^{3} + 167601769204 T^{4} + 35413692341427 T^{5} + 27300243639180300 T^{6} + 7910189755297031253 T^{7} + \)\(38\!\cdots\!42\)\( T^{8} + 7910189755297031253 p^{3} T^{9} + 27300243639180300 p^{6} T^{10} + 35413692341427 p^{9} T^{11} + 167601769204 p^{12} T^{12} + 101437437 p^{15} T^{13} + 623060 p^{18} T^{14} + 147 p^{21} T^{15} + p^{24} T^{16} \) | |
| 59 | \( 1 + 1193 T + 1386767 T^{2} + 1065913712 T^{3} + 821761660978 T^{4} + 505331258311510 T^{5} + 303273189504960681 T^{6} + \)\(15\!\cdots\!55\)\( T^{7} + \)\(74\!\cdots\!18\)\( T^{8} + \)\(15\!\cdots\!55\)\( p^{3} T^{9} + 303273189504960681 p^{6} T^{10} + 505331258311510 p^{9} T^{11} + 821761660978 p^{12} T^{12} + 1065913712 p^{15} T^{13} + 1386767 p^{18} T^{14} + 1193 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( 1 - 1216 T + 1149076 T^{2} - 1016877504 T^{3} + 757140074900 T^{4} - 483444842580288 T^{5} + 292059633231509132 T^{6} - \)\(15\!\cdots\!96\)\( T^{7} + \)\(78\!\cdots\!82\)\( T^{8} - \)\(15\!\cdots\!96\)\( p^{3} T^{9} + 292059633231509132 p^{6} T^{10} - 483444842580288 p^{9} T^{11} + 757140074900 p^{12} T^{12} - 1016877504 p^{15} T^{13} + 1149076 p^{18} T^{14} - 1216 p^{21} T^{15} + p^{24} T^{16} \) | |
| 67 | \( 1 - 589 T + 1332104 T^{2} - 625911821 T^{3} + 914827337056 T^{4} - 391011083265879 T^{5} + 435171510750716207 T^{6} - \)\(16\!\cdots\!36\)\( T^{7} + \)\(15\!\cdots\!40\)\( T^{8} - \)\(16\!\cdots\!36\)\( p^{3} T^{9} + 435171510750716207 p^{6} T^{10} - 391011083265879 p^{9} T^{11} + 914827337056 p^{12} T^{12} - 625911821 p^{15} T^{13} + 1332104 p^{18} T^{14} - 589 p^{21} T^{15} + p^{24} T^{16} \) | |
| 71 | \( 1 + 538 T + 2388660 T^{2} + 1215319754 T^{3} + 2662134604632 T^{4} + 1211367892828006 T^{5} + 1784968501621534789 T^{6} + \)\(69\!\cdots\!26\)\( T^{7} + \)\(78\!\cdots\!40\)\( T^{8} + \)\(69\!\cdots\!26\)\( p^{3} T^{9} + 1784968501621534789 p^{6} T^{10} + 1211367892828006 p^{9} T^{11} + 2662134604632 p^{12} T^{12} + 1215319754 p^{15} T^{13} + 2388660 p^{18} T^{14} + 538 p^{21} T^{15} + p^{24} T^{16} \) | |
| 73 | \( 1 - 367 T + 1215690 T^{2} - 102222021 T^{3} + 924848439344 T^{4} - 118531439105167 T^{5} + 572494675856107446 T^{6} - 40814804499725046125 T^{7} + \)\(23\!\cdots\!18\)\( T^{8} - 40814804499725046125 p^{3} T^{9} + 572494675856107446 p^{6} T^{10} - 118531439105167 p^{9} T^{11} + 924848439344 p^{12} T^{12} - 102222021 p^{15} T^{13} + 1215690 p^{18} T^{14} - 367 p^{21} T^{15} + p^{24} T^{16} \) | |
| 79 | \( 1 - 2368 T + 5022540 T^{2} - 6956108704 T^{3} + 8873195174516 T^{4} - 8984869184308704 T^{5} + 8547885088368856180 T^{6} - \)\(68\!\cdots\!92\)\( T^{7} + \)\(52\!\cdots\!58\)\( T^{8} - \)\(68\!\cdots\!92\)\( p^{3} T^{9} + 8547885088368856180 p^{6} T^{10} - 8984869184308704 p^{9} T^{11} + 8873195174516 p^{12} T^{12} - 6956108704 p^{15} T^{13} + 5022540 p^{18} T^{14} - 2368 p^{21} T^{15} + p^{24} T^{16} \) | |
| 83 | \( 1 - 918 T + 2945843 T^{2} - 1984611324 T^{3} + 3756982610291 T^{4} - 1921102112447724 T^{5} + 2978619439542530297 T^{6} - \)\(12\!\cdots\!34\)\( T^{7} + \)\(18\!\cdots\!92\)\( T^{8} - \)\(12\!\cdots\!34\)\( p^{3} T^{9} + 2978619439542530297 p^{6} T^{10} - 1921102112447724 p^{9} T^{11} + 3756982610291 p^{12} T^{12} - 1984611324 p^{15} T^{13} + 2945843 p^{18} T^{14} - 918 p^{21} T^{15} + p^{24} T^{16} \) | |
| 89 | \( 1 - 2212 T + 6290669 T^{2} - 8822950262 T^{3} + 14151518901786 T^{4} - 14451015742760230 T^{5} + 17051619162071443027 T^{6} - \)\(14\!\cdots\!20\)\( T^{7} + \)\(13\!\cdots\!90\)\( T^{8} - \)\(14\!\cdots\!20\)\( p^{3} T^{9} + 17051619162071443027 p^{6} T^{10} - 14451015742760230 p^{9} T^{11} + 14151518901786 p^{12} T^{12} - 8822950262 p^{15} T^{13} + 6290669 p^{18} T^{14} - 2212 p^{21} T^{15} + p^{24} T^{16} \) | |
| 97 | \( 1 + 3136 T + 10596753 T^{2} + 21258961494 T^{3} + 41304997956158 T^{4} + 60298573868666950 T^{5} + 83479929405854686911 T^{6} + \)\(93\!\cdots\!40\)\( T^{7} + \)\(98\!\cdots\!50\)\( T^{8} + \)\(93\!\cdots\!40\)\( p^{3} T^{9} + 83479929405854686911 p^{6} T^{10} + 60298573868666950 p^{9} T^{11} + 41304997956158 p^{12} T^{12} + 21258961494 p^{15} T^{13} + 10596753 p^{18} T^{14} + 3136 p^{21} T^{15} + p^{24} T^{16} \) | |
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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
−3.86438132509843975065232884196, −3.85780381578231016457310103409, −3.59409001950147522908266797440, −3.53345437229906797248503695974, −3.46620134200892673892988610244, −3.44323758366423620287623766496, −3.30281666718687477825499600952, −3.09390287729420003593792945171, −3.00319267334293318071524092196, −2.75926053390109530553946577415, −2.41404084987473483583859148673, −2.31579113204373707989699679954, −2.30237386961933977862702351293, −2.21033698229624676225038456199, −2.15383914176899099383007040419, −1.93631706296074515787161062392, −1.91880964411420405349538284430, −1.87006356336074994044572087861, −1.75701558822926785668216969958, −1.38150258948432060032229060652, −1.33535704843811503177134241096, −1.11265356361935787601540785542, −0.842604865151429187682878297335, −0.77843434913521087940726417302, −0.76021155900683177512334811130, 0, 0, 0, 0, 0, 0, 0, 0, 0.76021155900683177512334811130, 0.77843434913521087940726417302, 0.842604865151429187682878297335, 1.11265356361935787601540785542, 1.33535704843811503177134241096, 1.38150258948432060032229060652, 1.75701558822926785668216969958, 1.87006356336074994044572087861, 1.91880964411420405349538284430, 1.93631706296074515787161062392, 2.15383914176899099383007040419, 2.21033698229624676225038456199, 2.30237386961933977862702351293, 2.31579113204373707989699679954, 2.41404084987473483583859148673, 2.75926053390109530553946577415, 3.00319267334293318071524092196, 3.09390287729420003593792945171, 3.30281666718687477825499600952, 3.44323758366423620287623766496, 3.46620134200892673892988610244, 3.53345437229906797248503695974, 3.59409001950147522908266797440, 3.85780381578231016457310103409, 3.86438132509843975065232884196
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.