Dirichlet series
| L(s) = 1 | − 8·2-s − 8·3-s + 36·4-s − 5-s + 64·6-s + 6·7-s − 120·8-s + 36·9-s + 8·10-s − 288·12-s + 6·13-s − 48·14-s + 8·15-s + 330·16-s − 8·17-s − 288·18-s − 7·19-s − 36·20-s − 48·21-s − 5·23-s + 960·24-s − 15·25-s − 48·26-s − 120·27-s + 216·28-s − 15·29-s − 64·30-s + ⋯ |
| L(s) = 1 | − 5.65·2-s − 4.61·3-s + 18·4-s − 0.447·5-s + 26.1·6-s + 2.26·7-s − 42.4·8-s + 12·9-s + 2.52·10-s − 83.1·12-s + 1.66·13-s − 12.8·14-s + 2.06·15-s + 82.5·16-s − 1.94·17-s − 67.8·18-s − 1.60·19-s − 8.04·20-s − 10.4·21-s − 1.04·23-s + 195.·24-s − 3·25-s − 9.41·26-s − 23.0·27-s + 40.8·28-s − 2.78·29-s − 11.6·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 17^{8} \cdot 59^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 17^{8} \cdot 59^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 17^{8} \cdot 59^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.84337\times 10^{13}\) |
| Root analytic conductor: | \(6.93209\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 17^{8} \cdot 59^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1131579513\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1131579513\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{8} \) |
| 3 | \( ( 1 + T )^{8} \) | |
| 17 | \( ( 1 + T )^{8} \) | |
| 59 | \( ( 1 - T )^{8} \) | |
| good | 5 | \( 1 + T + 16 T^{2} + 13 T^{3} + 31 p T^{4} + 169 T^{5} + 224 p T^{6} + 241 p T^{7} + 6024 T^{8} + 241 p^{2} T^{9} + 224 p^{3} T^{10} + 169 p^{3} T^{11} + 31 p^{5} T^{12} + 13 p^{5} T^{13} + 16 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - 6 T + 45 T^{2} - 192 T^{3} + 870 T^{4} - 2981 T^{5} + 10569 T^{6} - 30258 T^{7} + 88824 T^{8} - 30258 p T^{9} + 10569 p^{2} T^{10} - 2981 p^{3} T^{11} + 870 p^{4} T^{12} - 192 p^{5} T^{13} + 45 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 40 T^{2} - 37 T^{3} + 887 T^{4} - 1413 T^{5} + 112 p^{2} T^{6} - 26460 T^{7} + 165104 T^{8} - 26460 p T^{9} + 112 p^{4} T^{10} - 1413 p^{3} T^{11} + 887 p^{4} T^{12} - 37 p^{5} T^{13} + 40 p^{6} T^{14} + p^{8} T^{16} \) | |
| 13 | \( 1 - 6 T + 89 T^{2} - 368 T^{3} + 3167 T^{4} - 9602 T^{5} + 64903 T^{6} - 156108 T^{7} + 950952 T^{8} - 156108 p T^{9} + 64903 p^{2} T^{10} - 9602 p^{3} T^{11} + 3167 p^{4} T^{12} - 368 p^{5} T^{13} + 89 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 7 T + 81 T^{2} + 298 T^{3} + 127 p T^{4} + 6649 T^{5} + 65062 T^{6} + 201479 T^{7} + 1577194 T^{8} + 201479 p T^{9} + 65062 p^{2} T^{10} + 6649 p^{3} T^{11} + 127 p^{5} T^{12} + 298 p^{5} T^{13} + 81 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 5 T + 103 T^{2} + 429 T^{3} + 5130 T^{4} + 18025 T^{5} + 171911 T^{6} + 528498 T^{7} + 4431424 T^{8} + 528498 p T^{9} + 171911 p^{2} T^{10} + 18025 p^{3} T^{11} + 5130 p^{4} T^{12} + 429 p^{5} T^{13} + 103 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 15 T + 234 T^{2} + 2369 T^{3} + 22509 T^{4} + 5995 p T^{5} + 1246686 T^{6} + 7705949 T^{7} + 44401236 T^{8} + 7705949 p T^{9} + 1246686 p^{2} T^{10} + 5995 p^{4} T^{11} + 22509 p^{4} T^{12} + 2369 p^{5} T^{13} + 234 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 21 T + 356 T^{2} - 4191 T^{3} + 42775 T^{4} - 358287 T^{5} + 2698408 T^{6} - 17567237 T^{7} + 104455208 T^{8} - 17567237 p T^{9} + 2698408 p^{2} T^{10} - 358287 p^{3} T^{11} + 42775 p^{4} T^{12} - 4191 p^{5} T^{13} + 356 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 7 T + 226 T^{2} - 1661 T^{3} + 24233 T^{4} - 171303 T^{5} + 1611578 T^{6} - 10128697 T^{7} + 72053348 T^{8} - 10128697 p T^{9} + 1611578 p^{2} T^{10} - 171303 p^{3} T^{11} + 24233 p^{4} T^{12} - 1661 p^{5} T^{13} + 226 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + T + 173 T^{2} - 95 T^{3} + 12846 T^{4} - 36877 T^{5} + 583411 T^{6} - 3265996 T^{7} + 22849452 T^{8} - 3265996 p T^{9} + 583411 p^{2} T^{10} - 36877 p^{3} T^{11} + 12846 p^{4} T^{12} - 95 p^{5} T^{13} + 173 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 14 T + 378 T^{2} - 3989 T^{3} + 59851 T^{4} - 496697 T^{5} + 5279166 T^{6} - 34984358 T^{7} + 285945288 T^{8} - 34984358 p T^{9} + 5279166 p^{2} T^{10} - 496697 p^{3} T^{11} + 59851 p^{4} T^{12} - 3989 p^{5} T^{13} + 378 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 8 T + 174 T^{2} + 1355 T^{3} + 397 p T^{4} + 133669 T^{5} + 1408814 T^{6} + 8757326 T^{7} + 75891024 T^{8} + 8757326 p T^{9} + 1408814 p^{2} T^{10} + 133669 p^{3} T^{11} + 397 p^{5} T^{12} + 1355 p^{5} T^{13} + 174 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 8 T + 257 T^{2} - 2258 T^{3} + 34400 T^{4} - 280917 T^{5} + 3115751 T^{6} - 21346436 T^{7} + 197739264 T^{8} - 21346436 p T^{9} + 3115751 p^{2} T^{10} - 280917 p^{3} T^{11} + 34400 p^{4} T^{12} - 2258 p^{5} T^{13} + 257 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 246 T^{2} + 81 T^{3} + 33189 T^{4} + 4535 T^{5} + 3088722 T^{6} - 6804 T^{7} + 215176596 T^{8} - 6804 p T^{9} + 3088722 p^{2} T^{10} + 4535 p^{3} T^{11} + 33189 p^{4} T^{12} + 81 p^{5} T^{13} + 246 p^{6} T^{14} + p^{8} T^{16} \) | |
| 67 | \( 1 - 15 T + 353 T^{2} - 3872 T^{3} + 57887 T^{4} - 521506 T^{5} + 6166835 T^{6} - 47777769 T^{7} + 478658792 T^{8} - 47777769 p T^{9} + 6166835 p^{2} T^{10} - 521506 p^{3} T^{11} + 57887 p^{4} T^{12} - 3872 p^{5} T^{13} + 353 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 22 T + 514 T^{2} + 7845 T^{3} + 114881 T^{4} + 1373943 T^{5} + 15251750 T^{6} + 147989716 T^{7} + 1321412100 T^{8} + 147989716 p T^{9} + 15251750 p^{2} T^{10} + 1373943 p^{3} T^{11} + 114881 p^{4} T^{12} + 7845 p^{5} T^{13} + 514 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 13 T + 299 T^{2} - 2613 T^{3} + 38508 T^{4} - 247425 T^{5} + 3210865 T^{6} - 16961992 T^{7} + 236132828 T^{8} - 16961992 p T^{9} + 3210865 p^{2} T^{10} - 247425 p^{3} T^{11} + 38508 p^{4} T^{12} - 2613 p^{5} T^{13} + 299 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 26 T + 640 T^{2} - 9656 T^{3} + 142699 T^{4} - 1596408 T^{5} + 18223156 T^{6} - 169570774 T^{7} + 1652607816 T^{8} - 169570774 p T^{9} + 18223156 p^{2} T^{10} - 1596408 p^{3} T^{11} + 142699 p^{4} T^{12} - 9656 p^{5} T^{13} + 640 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 30 T + 740 T^{2} - 12480 T^{3} + 194182 T^{4} - 2481911 T^{5} + 29776469 T^{6} - 306563522 T^{7} + 2989193580 T^{8} - 306563522 p T^{9} + 29776469 p^{2} T^{10} - 2481911 p^{3} T^{11} + 194182 p^{4} T^{12} - 12480 p^{5} T^{13} + 740 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 6 T + 412 T^{2} + 2490 T^{3} + 80192 T^{4} + 540605 T^{5} + 10393669 T^{6} + 74826734 T^{7} + 1035585538 T^{8} + 74826734 p T^{9} + 10393669 p^{2} T^{10} + 540605 p^{3} T^{11} + 80192 p^{4} T^{12} + 2490 p^{5} T^{13} + 412 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 23 T + 706 T^{2} - 10817 T^{3} + 195228 T^{4} - 2300379 T^{5} + 31448105 T^{6} - 308141370 T^{7} + 3535616038 T^{8} - 308141370 p T^{9} + 31448105 p^{2} T^{10} - 2300379 p^{3} T^{11} + 195228 p^{4} T^{12} - 10817 p^{5} T^{13} + 706 p^{6} T^{14} - 23 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.45460592769075059758918390245, −2.83516164587858749466743708660, −2.73434567353762631263410320455, −2.71112287003026102152004186711, −2.65684491659361176589115526737, −2.47920776415777301988879461710, −2.47627761529744505218754265473, −2.40238664839854412918965908282, −2.17263108204109544275346973771, −1.85006164052778256076019243859, −1.83621545593640256717639528314, −1.73638030187544234009863948986, −1.63586034635095896749478367775, −1.63003159252559521698055418141, −1.60390380624061051200791625532, −1.53090666887310962380605096890, −1.47211075368750667006714875011, −0.939954484996958543302151157345, −0.865074412963826817840833062495, −0.74934660791775561309221091588, −0.64279924297663149402705860424, −0.54849450905845530650093455334, −0.46922048604302318215758841935, −0.33975553865699725850739058636, −0.21141054323432542135705877913, 0.21141054323432542135705877913, 0.33975553865699725850739058636, 0.46922048604302318215758841935, 0.54849450905845530650093455334, 0.64279924297663149402705860424, 0.74934660791775561309221091588, 0.865074412963826817840833062495, 0.939954484996958543302151157345, 1.47211075368750667006714875011, 1.53090666887310962380605096890, 1.60390380624061051200791625532, 1.63003159252559521698055418141, 1.63586034635095896749478367775, 1.73638030187544234009863948986, 1.83621545593640256717639528314, 1.85006164052778256076019243859, 2.17263108204109544275346973771, 2.40238664839854412918965908282, 2.47627761529744505218754265473, 2.47920776415777301988879461710, 2.65684491659361176589115526737, 2.71112287003026102152004186711, 2.73434567353762631263410320455, 2.83516164587858749466743708660, 3.45460592769075059758918390245
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.