Dirichlet series
| L(s) = 1 | − 4·4-s + 4·11-s + 4·16-s − 84·17-s + 108·19-s − 12·23-s + 10·25-s − 72·29-s − 132·31-s − 96·37-s − 112·43-s − 16·44-s + 24·47-s + 78·49-s − 32·53-s − 132·59-s + 96·61-s + 16·64-s − 120·67-s + 336·68-s − 8·71-s + 24·73-s − 432·76-s + 12·79-s − 492·89-s + 48·92-s − 40·100-s + ⋯ |
| L(s) = 1 | − 4-s + 4/11·11-s + 1/4·16-s − 4.94·17-s + 5.68·19-s − 0.521·23-s + 2/5·25-s − 2.48·29-s − 4.25·31-s − 2.59·37-s − 2.60·43-s − 0.363·44-s + 0.510·47-s + 1.59·49-s − 0.603·53-s − 2.23·59-s + 1.57·61-s + 1/4·64-s − 1.79·67-s + 4.94·68-s − 0.112·71-s + 0.328·73-s − 5.68·76-s + 0.151·79-s − 5.52·89-s + 0.521·92-s − 2/5·100-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.54058\times 10^{9}\) |
| Root analytic conductor: | \(4.14321\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.6913566347\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6913566347\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | \( 1 \) | |
| 5 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | |
| 7 | \( 1 - 78 T^{2} + 107 p^{2} T^{4} - 78 p^{4} T^{6} + p^{8} T^{8} \) | |
| good | 11 | \( 1 - 4 T - 168 T^{2} - 1928 T^{3} + 16250 T^{4} + 341700 T^{5} + 2353408 T^{6} - 32486668 T^{7} - 296735661 T^{8} - 32486668 p^{2} T^{9} + 2353408 p^{4} T^{10} + 341700 p^{6} T^{11} + 16250 p^{8} T^{12} - 1928 p^{10} T^{13} - 168 p^{12} T^{14} - 4 p^{14} T^{15} + p^{16} T^{16} \) |
| 13 | \( 1 - 860 T^{2} + 357498 T^{4} - 95438800 T^{6} + 18541152803 T^{8} - 95438800 p^{4} T^{10} + 357498 p^{8} T^{12} - 860 p^{12} T^{14} + p^{16} T^{16} \) | |
| 17 | \( 1 + 84 T + 4208 T^{2} + 155904 T^{3} + 4746250 T^{4} + 123749124 T^{5} + 2818194464 T^{6} + 56792172564 T^{7} + 1020121281523 T^{8} + 56792172564 p^{2} T^{9} + 2818194464 p^{4} T^{10} + 123749124 p^{6} T^{11} + 4746250 p^{8} T^{12} + 155904 p^{10} T^{13} + 4208 p^{12} T^{14} + 84 p^{14} T^{15} + p^{16} T^{16} \) | |
| 19 | \( 1 - 108 T + 6142 T^{2} - 243432 T^{3} + 7548345 T^{4} - 198492408 T^{5} + 4660679966 T^{6} - 100240172052 T^{7} + 1986343374068 T^{8} - 100240172052 p^{2} T^{9} + 4660679966 p^{4} T^{10} - 198492408 p^{6} T^{11} + 7548345 p^{8} T^{12} - 243432 p^{10} T^{13} + 6142 p^{12} T^{14} - 108 p^{14} T^{15} + p^{16} T^{16} \) | |
| 23 | \( 1 + 12 T - 1512 T^{2} - 15144 T^{3} + 1340890 T^{4} + 9662580 T^{5} - 850652928 T^{6} - 2479269564 T^{7} + 454065720819 T^{8} - 2479269564 p^{2} T^{9} - 850652928 p^{4} T^{10} + 9662580 p^{6} T^{11} + 1340890 p^{8} T^{12} - 15144 p^{10} T^{13} - 1512 p^{12} T^{14} + 12 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( ( 1 + 36 T + 2904 T^{2} + 82956 T^{3} + 3490070 T^{4} + 82956 p^{2} T^{5} + 2904 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 31 | \( 1 + 132 T + 11278 T^{2} + 722040 T^{3} + 38602473 T^{4} + 1778529960 T^{5} + 72524765390 T^{6} + 2637318955548 T^{7} + 86278260959732 T^{8} + 2637318955548 p^{2} T^{9} + 72524765390 p^{4} T^{10} + 1778529960 p^{6} T^{11} + 38602473 p^{8} T^{12} + 722040 p^{10} T^{13} + 11278 p^{12} T^{14} + 132 p^{14} T^{15} + p^{16} T^{16} \) | |
| 37 | \( 1 + 96 T + 30 p T^{2} - 46464 T^{3} + 7652401 T^{4} + 360085920 T^{5} - 3532027338 T^{6} + 170035213440 T^{7} + 29111429055396 T^{8} + 170035213440 p^{2} T^{9} - 3532027338 p^{4} T^{10} + 360085920 p^{6} T^{11} + 7652401 p^{8} T^{12} - 46464 p^{10} T^{13} + 30 p^{13} T^{14} + 96 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( 1 - 5456 T^{2} + 486924 p T^{4} - 49821420976 T^{6} + 96897341176550 T^{8} - 49821420976 p^{4} T^{10} + 486924 p^{9} T^{12} - 5456 p^{12} T^{14} + p^{16} T^{16} \) | |
| 43 | \( ( 1 + 56 T + 5082 T^{2} + 180688 T^{3} + 11078435 T^{4} + 180688 p^{2} T^{5} + 5082 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 47 | \( 1 - 24 T + 3580 T^{2} - 81312 T^{3} + 3430986 T^{4} - 200307384 T^{5} + 115413136 p T^{6} - 942687331128 T^{7} + 25778360177363 T^{8} - 942687331128 p^{2} T^{9} + 115413136 p^{5} T^{10} - 200307384 p^{6} T^{11} + 3430986 p^{8} T^{12} - 81312 p^{10} T^{13} + 3580 p^{12} T^{14} - 24 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( 1 + 32 T - 3572 T^{2} - 570944 T^{3} - 366070 T^{4} + 2018388320 T^{5} + 123180800432 T^{6} - 3809526158624 T^{7} - 445561676340941 T^{8} - 3809526158624 p^{2} T^{9} + 123180800432 p^{4} T^{10} + 2018388320 p^{6} T^{11} - 366070 p^{8} T^{12} - 570944 p^{10} T^{13} - 3572 p^{12} T^{14} + 32 p^{14} T^{15} + p^{16} T^{16} \) | |
| 59 | \( 1 + 132 T + 15632 T^{2} + 1296768 T^{3} + 88851370 T^{4} + 4629146772 T^{5} + 218386477856 T^{6} + 7743785317668 T^{7} + 393482793133843 T^{8} + 7743785317668 p^{2} T^{9} + 218386477856 p^{4} T^{10} + 4629146772 p^{6} T^{11} + 88851370 p^{8} T^{12} + 1296768 p^{10} T^{13} + 15632 p^{12} T^{14} + 132 p^{14} T^{15} + p^{16} T^{16} \) | |
| 61 | \( 1 - 96 T + 11380 T^{2} - 797568 T^{3} + 60840138 T^{4} - 3862327392 T^{5} + 197844384848 T^{6} - 13588740778464 T^{7} + 649379857320947 T^{8} - 13588740778464 p^{2} T^{9} + 197844384848 p^{4} T^{10} - 3862327392 p^{6} T^{11} + 60840138 p^{8} T^{12} - 797568 p^{10} T^{13} + 11380 p^{12} T^{14} - 96 p^{14} T^{15} + p^{16} T^{16} \) | |
| 67 | \( 1 + 120 T - 4522 T^{2} - 749040 T^{3} + 46360561 T^{4} + 4026690720 T^{5} - 240439443082 T^{6} - 2848051223400 T^{7} + 1781423144538724 T^{8} - 2848051223400 p^{2} T^{9} - 240439443082 p^{4} T^{10} + 4026690720 p^{6} T^{11} + 46360561 p^{8} T^{12} - 749040 p^{10} T^{13} - 4522 p^{12} T^{14} + 120 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( ( 1 + 4 T + 13176 T^{2} + 338828 T^{3} + 79144886 T^{4} + 338828 p^{2} T^{5} + 13176 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 73 | \( 1 - 24 T + 9630 T^{2} - 226512 T^{3} + 47274361 T^{4} + 2709573216 T^{5} - 18252715698 T^{6} + 34150744827912 T^{7} - 1287593588654412 T^{8} + 34150744827912 p^{2} T^{9} - 18252715698 p^{4} T^{10} + 2709573216 p^{6} T^{11} + 47274361 p^{8} T^{12} - 226512 p^{10} T^{13} + 9630 p^{12} T^{14} - 24 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( 1 - 12 T - 2418 T^{2} + 386472 T^{3} - 58917335 T^{4} - 324907032 T^{5} + 66125106126 T^{6} - 10581169939908 T^{7} + 1991860414252788 T^{8} - 10581169939908 p^{2} T^{9} + 66125106126 p^{4} T^{10} - 324907032 p^{6} T^{11} - 58917335 p^{8} T^{12} + 386472 p^{10} T^{13} - 2418 p^{12} T^{14} - 12 p^{14} T^{15} + p^{16} T^{16} \) | |
| 83 | \( 1 - 2384 T^{2} + 54393900 T^{4} - 170213772976 T^{6} + 3204332319622694 T^{8} - 170213772976 p^{4} T^{10} + 54393900 p^{8} T^{12} - 2384 p^{12} T^{14} + p^{16} T^{16} \) | |
| 89 | \( 1 + 492 T + 137248 T^{2} + 27827520 T^{3} + 4512312618 T^{4} + 615183153660 T^{5} + 72837035721440 T^{6} + 7634393359602348 T^{7} + 716618506839255347 T^{8} + 7634393359602348 p^{2} T^{9} + 72837035721440 p^{4} T^{10} + 615183153660 p^{6} T^{11} + 4512312618 p^{8} T^{12} + 27827520 p^{10} T^{13} + 137248 p^{12} T^{14} + 492 p^{14} T^{15} + p^{16} T^{16} \) | |
| 97 | \( 1 - 35880 T^{2} + 724155484 T^{4} - 9811041724440 T^{6} + 103976225413471686 T^{8} - 9811041724440 p^{4} T^{10} + 724155484 p^{8} T^{12} - 35880 p^{12} T^{14} + p^{16} T^{16} \) | |
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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
−4.41214172026144661286215952721, −4.02884020317118029858642148745, −4.00771310845367409344065922247, −3.98101722927368960907893966862, −3.91973624536493682119435379438, −3.64975498269709834723372149565, −3.61418916343563796700818055233, −3.59879803432869415016879837411, −3.20513070416504467756176661030, −3.05992101995921515205985393654, −2.91151327300464044594860437931, −2.77768009862058358132874165466, −2.71587914308965195007552624963, −2.48003899883849490757146405178, −2.37519188733602670015242962638, −1.80086940559862466425125420329, −1.75064008421820999431197560904, −1.65427316412905603397328642757, −1.61050367448338281964114430888, −1.55911381183855691757920152119, −1.34380699926776485565407898054, −0.791910833649271225249856522932, −0.41355896970281087137682904852, −0.24996681915512855531569785918, −0.20336021582573748816525337982, 0.20336021582573748816525337982, 0.24996681915512855531569785918, 0.41355896970281087137682904852, 0.791910833649271225249856522932, 1.34380699926776485565407898054, 1.55911381183855691757920152119, 1.61050367448338281964114430888, 1.65427316412905603397328642757, 1.75064008421820999431197560904, 1.80086940559862466425125420329, 2.37519188733602670015242962638, 2.48003899883849490757146405178, 2.71587914308965195007552624963, 2.77768009862058358132874165466, 2.91151327300464044594860437931, 3.05992101995921515205985393654, 3.20513070416504467756176661030, 3.59879803432869415016879837411, 3.61418916343563796700818055233, 3.64975498269709834723372149565, 3.91973624536493682119435379438, 3.98101722927368960907893966862, 4.00771310845367409344065922247, 4.02884020317118029858642148745, 4.41214172026144661286215952721
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.