Dirichlet series
| L(s) = 1 | − 8·2-s − 12·3-s + 24·4-s − 20·5-s + 96·6-s − 100·7-s + 54·9-s + 160·10-s − 288·12-s + 59·13-s + 800·14-s + 240·15-s − 240·16-s − 81·17-s − 432·18-s + 19-s − 480·20-s + 1.20e3·21-s − 185·23-s + 150·25-s − 472·26-s − 2.40e3·28-s + 403·29-s − 1.92e3·30-s − 534·31-s + 768·32-s + 648·34-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 3·4-s − 1.78·5-s + 6.53·6-s − 5.39·7-s + 2·9-s + 5.05·10-s − 6.92·12-s + 1.25·13-s + 15.2·14-s + 4.13·15-s − 3.75·16-s − 1.15·17-s − 5.65·18-s + 0.0120·19-s − 5.36·20-s + 12.4·21-s − 1.67·23-s + 6/5·25-s − 3.56·26-s − 16.1·28-s + 2.58·29-s − 11.6·30-s − 3.09·31-s + 4.24·32-s + 3.26·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.63654\times 10^{12}\) |
| Root analytic conductor: | \(5.79923\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.003531667412\) |
| \(L(\frac12)\) | \(\approx\) | \(0.003531667412\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p T + p^{2} T^{2} )^{4} \) |
| 3 | \( ( 1 + p T + p^{2} T^{2} )^{4} \) | |
| 5 | \( ( 1 + p T + p^{2} T^{2} )^{4} \) | |
| 19 | \( 1 - T - 4781 T^{2} + 23176 p T^{3} + 121808 p^{2} T^{4} + 23176 p^{4} T^{5} - 4781 p^{6} T^{6} - p^{9} T^{7} + p^{12} T^{8} \) | |
| good | 7 | \( ( 1 + 50 T + 1567 T^{2} + 35836 T^{3} + 696680 T^{4} + 35836 p^{3} T^{5} + 1567 p^{6} T^{6} + 50 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 11 | \( ( 1 + 317 T^{2} + 34578 T^{3} - 1201092 T^{4} + 34578 p^{3} T^{5} + 317 p^{6} T^{6} + p^{12} T^{8} )^{2} \) | |
| 13 | \( 1 - 59 T - 1591 T^{2} + 11858 T^{3} + 4745255 T^{4} + 261956879 T^{5} - 6864635958 T^{6} - 439250484141 T^{7} + 12493993781320 T^{8} - 439250484141 p^{3} T^{9} - 6864635958 p^{6} T^{10} + 261956879 p^{9} T^{11} + 4745255 p^{12} T^{12} + 11858 p^{15} T^{13} - 1591 p^{18} T^{14} - 59 p^{21} T^{15} + p^{24} T^{16} \) | |
| 17 | \( 1 + 81 T + 2933 T^{2} - 736698 T^{3} - 81116233 T^{4} - 4110560325 T^{5} + 44982717222 T^{6} + 25403020039527 T^{7} + 2111821543313756 T^{8} + 25403020039527 p^{3} T^{9} + 44982717222 p^{6} T^{10} - 4110560325 p^{9} T^{11} - 81116233 p^{12} T^{12} - 736698 p^{15} T^{13} + 2933 p^{18} T^{14} + 81 p^{21} T^{15} + p^{24} T^{16} \) | |
| 23 | \( 1 + 185 T + 2293 T^{2} - 93654 T^{3} + 60735497 T^{4} - 17091898177 T^{5} - 1191216841974 T^{6} + 87876479040313 T^{7} + 3955688056913194 T^{8} + 87876479040313 p^{3} T^{9} - 1191216841974 p^{6} T^{10} - 17091898177 p^{9} T^{11} + 60735497 p^{12} T^{12} - 93654 p^{15} T^{13} + 2293 p^{18} T^{14} + 185 p^{21} T^{15} + p^{24} T^{16} \) | |
| 29 | \( 1 - 403 T + 23359 T^{2} + 4311948 T^{3} + 1592951075 T^{4} - 432921922543 T^{5} + 3222544321518 T^{6} - 2939773242997979 T^{7} + 1805598692897527996 T^{8} - 2939773242997979 p^{3} T^{9} + 3222544321518 p^{6} T^{10} - 432921922543 p^{9} T^{11} + 1592951075 p^{12} T^{12} + 4311948 p^{15} T^{13} + 23359 p^{18} T^{14} - 403 p^{21} T^{15} + p^{24} T^{16} \) | |
| 31 | \( ( 1 + 267 T + 137471 T^{2} + 24127128 T^{3} + 6414226848 T^{4} + 24127128 p^{3} T^{5} + 137471 p^{6} T^{6} + 267 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 37 | \( ( 1 + 530 T + 91129 T^{2} - 9969614 T^{3} - 5975356744 T^{4} - 9969614 p^{3} T^{5} + 91129 p^{6} T^{6} + 530 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 41 | \( 1 + 263 T + 65213 T^{2} - 13991702 T^{3} - 9704270381 T^{4} - 3525172417767 T^{5} - 72795892731710 T^{6} + 145143416365184441 T^{7} + 78715172883066370220 T^{8} + 145143416365184441 p^{3} T^{9} - 72795892731710 p^{6} T^{10} - 3525172417767 p^{9} T^{11} - 9704270381 p^{12} T^{12} - 13991702 p^{15} T^{13} + 65213 p^{18} T^{14} + 263 p^{21} T^{15} + p^{24} T^{16} \) | |
| 43 | \( 1 - 859 T + 186949 T^{2} - 30529502 T^{3} + 39331556517 T^{4} - 12343625227317 T^{5} + 53415359807186 T^{6} - 421645757548600695 T^{7} + \)\(36\!\cdots\!46\)\( T^{8} - 421645757548600695 p^{3} T^{9} + 53415359807186 p^{6} T^{10} - 12343625227317 p^{9} T^{11} + 39331556517 p^{12} T^{12} - 30529502 p^{15} T^{13} + 186949 p^{18} T^{14} - 859 p^{21} T^{15} + p^{24} T^{16} \) | |
| 47 | \( 1 + 471 T - 169139 T^{2} - 49000530 T^{3} + 40640983273 T^{4} + 3689515723041 T^{5} - 6091573314812510 T^{6} - 270361343108017113 T^{7} + \)\(60\!\cdots\!62\)\( T^{8} - 270361343108017113 p^{3} T^{9} - 6091573314812510 p^{6} T^{10} + 3689515723041 p^{9} T^{11} + 40640983273 p^{12} T^{12} - 49000530 p^{15} T^{13} - 169139 p^{18} T^{14} + 471 p^{21} T^{15} + p^{24} T^{16} \) | |
| 53 | \( 1 - 328 T - 410911 T^{2} + 90321448 T^{3} + 113506643545 T^{4} - 13004571545280 T^{5} - 24819353985305258 T^{6} + 584790303345838784 T^{7} + \)\(44\!\cdots\!70\)\( T^{8} + 584790303345838784 p^{3} T^{9} - 24819353985305258 p^{6} T^{10} - 13004571545280 p^{9} T^{11} + 113506643545 p^{12} T^{12} + 90321448 p^{15} T^{13} - 410911 p^{18} T^{14} - 328 p^{21} T^{15} + p^{24} T^{16} \) | |
| 59 | \( 1 - 97 T - 597049 T^{2} + 60155848 T^{3} + 189437389369 T^{4} - 14815950624105 T^{5} - 51149037269347154 T^{6} + 1230870974692422797 T^{7} + \)\(11\!\cdots\!82\)\( T^{8} + 1230870974692422797 p^{3} T^{9} - 51149037269347154 p^{6} T^{10} - 14815950624105 p^{9} T^{11} + 189437389369 p^{12} T^{12} + 60155848 p^{15} T^{13} - 597049 p^{18} T^{14} - 97 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( 1 + 406 T - 462399 T^{2} - 478782 p T^{3} + 167576635101 T^{4} - 15356787450960 T^{5} - 33819560261221390 T^{6} + 1330293516876469064 T^{7} + \)\(46\!\cdots\!34\)\( T^{8} + 1330293516876469064 p^{3} T^{9} - 33819560261221390 p^{6} T^{10} - 15356787450960 p^{9} T^{11} + 167576635101 p^{12} T^{12} - 478782 p^{16} T^{13} - 462399 p^{18} T^{14} + 406 p^{21} T^{15} + p^{24} T^{16} \) | |
| 67 | \( 1 - 1561 T + 891295 T^{2} - 147756512 T^{3} - 83499929919 T^{4} + 108271167181143 T^{5} - 58271669112005122 T^{6} - 5539819718095077195 T^{7} + \)\(18\!\cdots\!66\)\( T^{8} - 5539819718095077195 p^{3} T^{9} - 58271669112005122 p^{6} T^{10} + 108271167181143 p^{9} T^{11} - 83499929919 p^{12} T^{12} - 147756512 p^{15} T^{13} + 891295 p^{18} T^{14} - 1561 p^{21} T^{15} + p^{24} T^{16} \) | |
| 71 | \( 1 - 1021 T - 142933 T^{2} + 236376916 T^{3} + 105903230065 T^{4} + 28650734259267 T^{5} - 102068625138090326 T^{6} + 6339760947406775585 T^{7} + \)\(26\!\cdots\!06\)\( T^{8} + 6339760947406775585 p^{3} T^{9} - 102068625138090326 p^{6} T^{10} + 28650734259267 p^{9} T^{11} + 105903230065 p^{12} T^{12} + 236376916 p^{15} T^{13} - 142933 p^{18} T^{14} - 1021 p^{21} T^{15} + p^{24} T^{16} \) | |
| 73 | \( 1 - 183 T - 1456915 T^{2} + 140926502 T^{3} + 1325175868915 T^{4} - 74600933099417 T^{5} - 800025567415854846 T^{6} + 10272033877457957255 T^{7} + \)\(36\!\cdots\!56\)\( T^{8} + 10272033877457957255 p^{3} T^{9} - 800025567415854846 p^{6} T^{10} - 74600933099417 p^{9} T^{11} + 1325175868915 p^{12} T^{12} + 140926502 p^{15} T^{13} - 1456915 p^{18} T^{14} - 183 p^{21} T^{15} + p^{24} T^{16} \) | |
| 79 | \( 1 - 340 T - 1251849 T^{2} + 430880488 T^{3} + 822291521069 T^{4} - 224495638315152 T^{5} - 433972045076791934 T^{6} + 49166146047601190780 T^{7} + \)\(21\!\cdots\!58\)\( T^{8} + 49166146047601190780 p^{3} T^{9} - 433972045076791934 p^{6} T^{10} - 224495638315152 p^{9} T^{11} + 822291521069 p^{12} T^{12} + 430880488 p^{15} T^{13} - 1251849 p^{18} T^{14} - 340 p^{21} T^{15} + p^{24} T^{16} \) | |
| 83 | \( ( 1 - 539 T + 1099350 T^{2} - 1063061979 T^{3} + 703744664818 T^{4} - 1063061979 p^{3} T^{5} + 1099350 p^{6} T^{6} - 539 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 89 | \( 1 - 334 T - 2149315 T^{2} + 531173518 T^{3} + 2656627789021 T^{4} - 411101739251472 T^{5} - 2555140899951044414 T^{6} + \)\(11\!\cdots\!92\)\( T^{7} + \)\(20\!\cdots\!98\)\( T^{8} + \)\(11\!\cdots\!92\)\( p^{3} T^{9} - 2555140899951044414 p^{6} T^{10} - 411101739251472 p^{9} T^{11} + 2656627789021 p^{12} T^{12} + 531173518 p^{15} T^{13} - 2149315 p^{18} T^{14} - 334 p^{21} T^{15} + p^{24} T^{16} \) | |
| 97 | \( 1 - 1656 T - 566044 T^{2} + 15980912 p T^{3} + 704483353354 T^{4} - 192081059187848 T^{5} - 2075400268885992432 T^{6} - \)\(13\!\cdots\!20\)\( T^{7} + \)\(26\!\cdots\!47\)\( T^{8} - \)\(13\!\cdots\!20\)\( p^{3} T^{9} - 2075400268885992432 p^{6} T^{10} - 192081059187848 p^{9} T^{11} + 704483353354 p^{12} T^{12} + 15980912 p^{16} T^{13} - 566044 p^{18} T^{14} - 1656 p^{21} T^{15} + p^{24} T^{16} \) | |
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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
−4.09036988173189944443722941882, −4.04495827511822232680710986674, −3.74162259148194850279461303320, −3.73476055142095537701442979308, −3.67451978994481904889459846357, −3.66029300724567310134239840757, −3.58275832830737525601202421092, −3.19415780357023185883062002692, −3.08279239465821718362081451861, −2.91785647002478863659449257865, −2.73982450290689302884467494928, −2.67045290059403931863192412007, −2.35120936589515564833324443644, −2.15099703285418587142740959011, −2.00642282156011195799494885261, −1.67180641561275458581093784062, −1.60878227999397870289471524868, −1.18745948844585688052198590213, −1.10060444995179369430461233395, −0.947756745113377242311051686581, −0.59358688679681237083216695537, −0.38030635774710374023389642934, −0.25376794385321949137383923359, −0.17454772135754081309284386550, −0.15728810637547423489553830068, 0.15728810637547423489553830068, 0.17454772135754081309284386550, 0.25376794385321949137383923359, 0.38030635774710374023389642934, 0.59358688679681237083216695537, 0.947756745113377242311051686581, 1.10060444995179369430461233395, 1.18745948844585688052198590213, 1.60878227999397870289471524868, 1.67180641561275458581093784062, 2.00642282156011195799494885261, 2.15099703285418587142740959011, 2.35120936589515564833324443644, 2.67045290059403931863192412007, 2.73982450290689302884467494928, 2.91785647002478863659449257865, 3.08279239465821718362081451861, 3.19415780357023185883062002692, 3.58275832830737525601202421092, 3.66029300724567310134239840757, 3.67451978994481904889459846357, 3.73476055142095537701442979308, 3.74162259148194850279461303320, 4.04495827511822232680710986674, 4.09036988173189944443722941882
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.