Dirichlet series
| L(s) = 1 | + 8·2-s + 3·3-s + 36·4-s + 8·5-s + 24·6-s + 7·7-s + 120·8-s − 3·9-s + 64·10-s + 10·11-s + 108·12-s − 8·13-s + 56·14-s + 24·15-s + 330·16-s + 11·17-s − 24·18-s + 2·19-s + 288·20-s + 21·21-s + 80·22-s + 12·23-s + 360·24-s + 36·25-s − 64·26-s − 22·27-s + 252·28-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 1.73·3-s + 18·4-s + 3.57·5-s + 9.79·6-s + 2.64·7-s + 42.4·8-s − 9-s + 20.2·10-s + 3.01·11-s + 31.1·12-s − 2.21·13-s + 14.9·14-s + 6.19·15-s + 82.5·16-s + 2.66·17-s − 5.65·18-s + 0.458·19-s + 64.3·20-s + 4.58·21-s + 17.0·22-s + 2.50·23-s + 73.4·24-s + 36/5·25-s − 12.5·26-s − 4.23·27-s + 47.6·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 5^{8} \cdot 13^{8} \cdot 31^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.14989\times 10^{12}\) |
| Root analytic conductor: | \(5.67271\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 5^{8} \cdot 13^{8} \cdot 31^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(38159.18873\) |
| \(L(\frac12)\) | \(\approx\) | \(38159.18873\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{8} \) |
| 5 | \( ( 1 - T )^{8} \) | |
| 13 | \( ( 1 + T )^{8} \) | |
| 31 | \( ( 1 - T )^{8} \) | |
| good | 3 | \( 1 - p T + 4 p T^{2} - 23 T^{3} + 20 p T^{4} - 85 T^{5} + 205 T^{6} - 89 p T^{7} + 640 T^{8} - 89 p^{2} T^{9} + 205 p^{2} T^{10} - 85 p^{3} T^{11} + 20 p^{5} T^{12} - 23 p^{5} T^{13} + 4 p^{7} T^{14} - p^{8} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - p T + 55 T^{2} - 5 p^{2} T^{3} + 1180 T^{4} - 4076 T^{5} + 14985 T^{6} - 42356 T^{7} + 126587 T^{8} - 42356 p T^{9} + 14985 p^{2} T^{10} - 4076 p^{3} T^{11} + 1180 p^{4} T^{12} - 5 p^{7} T^{13} + 55 p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 10 T + 95 T^{2} - 564 T^{3} + 3220 T^{4} - 14376 T^{5} + 5708 p T^{6} - 229940 T^{7} + 826220 T^{8} - 229940 p T^{9} + 5708 p^{3} T^{10} - 14376 p^{3} T^{11} + 3220 p^{4} T^{12} - 564 p^{5} T^{13} + 95 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 11 T + 142 T^{2} - 966 T^{3} + 7252 T^{4} - 36512 T^{5} + 207239 T^{6} - 855149 T^{7} + 4088897 T^{8} - 855149 p T^{9} + 207239 p^{2} T^{10} - 36512 p^{3} T^{11} + 7252 p^{4} T^{12} - 966 p^{5} T^{13} + 142 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 2 T + 101 T^{2} - 138 T^{3} + 4448 T^{4} - 3201 T^{5} + 119593 T^{6} - 1544 p T^{7} + 2457947 T^{8} - 1544 p^{2} T^{9} + 119593 p^{2} T^{10} - 3201 p^{3} T^{11} + 4448 p^{4} T^{12} - 138 p^{5} T^{13} + 101 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 12 T + 8 p T^{2} - 1547 T^{3} + 13591 T^{4} - 89049 T^{5} + 572659 T^{6} - 133750 p T^{7} + 15878439 T^{8} - 133750 p^{2} T^{9} + 572659 p^{2} T^{10} - 89049 p^{3} T^{11} + 13591 p^{4} T^{12} - 1547 p^{5} T^{13} + 8 p^{7} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 9 T + 196 T^{2} - 1226 T^{3} + 15640 T^{4} - 73316 T^{5} + 730147 T^{6} - 2760641 T^{7} + 24176065 T^{8} - 2760641 p T^{9} + 730147 p^{2} T^{10} - 73316 p^{3} T^{11} + 15640 p^{4} T^{12} - 1226 p^{5} T^{13} + 196 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 19 T + 328 T^{2} - 3534 T^{3} + 35889 T^{4} - 277312 T^{5} + 2119891 T^{6} - 13382404 T^{7} + 88040891 T^{8} - 13382404 p T^{9} + 2119891 p^{2} T^{10} - 277312 p^{3} T^{11} + 35889 p^{4} T^{12} - 3534 p^{5} T^{13} + 328 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 6 T + 217 T^{2} - 965 T^{3} + 22336 T^{4} - 80410 T^{5} + 1504132 T^{6} - 4599821 T^{7} + 72415420 T^{8} - 4599821 p T^{9} + 1504132 p^{2} T^{10} - 80410 p^{3} T^{11} + 22336 p^{4} T^{12} - 965 p^{5} T^{13} + 217 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 21 T + 383 T^{2} - 4597 T^{3} + 48445 T^{4} - 411643 T^{5} + 3219476 T^{6} - 22327957 T^{7} + 151790370 T^{8} - 22327957 p T^{9} + 3219476 p^{2} T^{10} - 411643 p^{3} T^{11} + 48445 p^{4} T^{12} - 4597 p^{5} T^{13} + 383 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - T + 187 T^{2} - 187 T^{3} + 16778 T^{4} - 10978 T^{5} + 989509 T^{6} - 293802 T^{7} + 48237421 T^{8} - 293802 p T^{9} + 989509 p^{2} T^{10} - 10978 p^{3} T^{11} + 16778 p^{4} T^{12} - 187 p^{5} T^{13} + 187 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 18 T + 439 T^{2} - 5739 T^{3} + 82066 T^{4} - 829046 T^{5} + 8684734 T^{6} - 69925473 T^{7} + 574436848 T^{8} - 69925473 p T^{9} + 8684734 p^{2} T^{10} - 829046 p^{3} T^{11} + 82066 p^{4} T^{12} - 5739 p^{5} T^{13} + 439 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 4 T + 418 T^{2} + 1433 T^{3} + 79107 T^{4} + 230889 T^{5} + 8857621 T^{6} + 21667484 T^{7} + 641920257 T^{8} + 21667484 p T^{9} + 8857621 p^{2} T^{10} + 230889 p^{3} T^{11} + 79107 p^{4} T^{12} + 1433 p^{5} T^{13} + 418 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 10 T + 225 T^{2} - 1458 T^{3} + 20965 T^{4} - 81381 T^{5} + 1246679 T^{6} - 2967465 T^{7} + 69904815 T^{8} - 2967465 p T^{9} + 1246679 p^{2} T^{10} - 81381 p^{3} T^{11} + 20965 p^{4} T^{12} - 1458 p^{5} T^{13} + 225 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 8 T + 304 T^{2} - 2676 T^{3} + 43966 T^{4} - 422457 T^{5} + 4314533 T^{6} - 41675589 T^{7} + 4865452 p T^{8} - 41675589 p T^{9} + 4314533 p^{2} T^{10} - 422457 p^{3} T^{11} + 43966 p^{4} T^{12} - 2676 p^{5} T^{13} + 304 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 18 T + 527 T^{2} - 7570 T^{3} + 122445 T^{4} - 1440428 T^{5} + 16651934 T^{6} - 161127696 T^{7} + 1456474046 T^{8} - 161127696 p T^{9} + 16651934 p^{2} T^{10} - 1440428 p^{3} T^{11} + 122445 p^{4} T^{12} - 7570 p^{5} T^{13} + 527 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 9 T + 303 T^{2} - 2147 T^{3} + 39938 T^{4} - 294562 T^{5} + 3956964 T^{6} - 32581124 T^{7} + 331313988 T^{8} - 32581124 p T^{9} + 3956964 p^{2} T^{10} - 294562 p^{3} T^{11} + 39938 p^{4} T^{12} - 2147 p^{5} T^{13} + 303 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 14 T + 319 T^{2} - 2462 T^{3} + 39204 T^{4} - 240732 T^{5} + 4363626 T^{6} - 29468184 T^{7} + 436816552 T^{8} - 29468184 p T^{9} + 4363626 p^{2} T^{10} - 240732 p^{3} T^{11} + 39204 p^{4} T^{12} - 2462 p^{5} T^{13} + 319 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 3 T + 459 T^{2} - 1253 T^{3} + 104568 T^{4} - 255908 T^{5} + 15058067 T^{6} - 32213220 T^{7} + 1491796993 T^{8} - 32213220 p T^{9} + 15058067 p^{2} T^{10} - 255908 p^{3} T^{11} + 104568 p^{4} T^{12} - 1253 p^{5} T^{13} + 459 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 15 T + 470 T^{2} + 5306 T^{3} + 104658 T^{4} + 1013512 T^{5} + 15440651 T^{6} + 127982973 T^{7} + 1608768743 T^{8} + 127982973 p T^{9} + 15440651 p^{2} T^{10} + 1013512 p^{3} T^{11} + 104658 p^{4} T^{12} + 5306 p^{5} T^{13} + 470 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 12 T + 682 T^{2} - 6848 T^{3} + 207942 T^{4} - 1758100 T^{5} + 37549240 T^{6} - 266055261 T^{7} + 4439344935 T^{8} - 266055261 p T^{9} + 37549240 p^{2} T^{10} - 1758100 p^{3} T^{11} + 207942 p^{4} T^{12} - 6848 p^{5} T^{13} + 682 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.50819373218455952106131919871, −3.17582602567474468408391298201, −3.07328232453547136503686914844, −3.05607377135438782341217245183, −3.02878086812519098660646202260, −2.92828672532311069981419289351, −2.91848540369528754040120803799, −2.86828545006809012591128447903, −2.70778172136651699414502215096, −2.47779221065759808672096932713, −2.29620828640807967646105151458, −2.28514204344973910164240606252, −2.14462105918445385475997133386, −2.13366273257959018739690427252, −2.12646045493740817109931144271, −2.08405388200332494819477319077, −1.67067941269524647013771938929, −1.54508063794505684767962710398, −1.28102628392591708266309555309, −1.27681013149333131580429538508, −1.23980391904717940228434224677, −0.999134019258481287522032672134, −0.975708623473008345219339238701, −0.818365851654603833771118762692, −0.68032654907572420449214613122, 0.68032654907572420449214613122, 0.818365851654603833771118762692, 0.975708623473008345219339238701, 0.999134019258481287522032672134, 1.23980391904717940228434224677, 1.27681013149333131580429538508, 1.28102628392591708266309555309, 1.54508063794505684767962710398, 1.67067941269524647013771938929, 2.08405388200332494819477319077, 2.12646045493740817109931144271, 2.13366273257959018739690427252, 2.14462105918445385475997133386, 2.28514204344973910164240606252, 2.29620828640807967646105151458, 2.47779221065759808672096932713, 2.70778172136651699414502215096, 2.86828545006809012591128447903, 2.91848540369528754040120803799, 2.92828672532311069981419289351, 3.02878086812519098660646202260, 3.05607377135438782341217245183, 3.07328232453547136503686914844, 3.17582602567474468408391298201, 3.50819373218455952106131919871
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.