Dirichlet series
| L(s) = 1 | + 12·5-s − 4·7-s + 32·11-s + 4·13-s + 52·17-s + 40·23-s + 30·25-s − 96·31-s − 48·35-s + 60·37-s − 152·41-s − 88·43-s + 16·47-s + 8·49-s − 108·53-s + 384·55-s − 264·61-s + 48·65-s − 216·67-s + 240·71-s − 208·73-s − 128·77-s − 18·81-s + 336·83-s + 624·85-s − 16·91-s − 208·97-s + ⋯ |
| L(s) = 1 | + 12/5·5-s − 4/7·7-s + 2.90·11-s + 4/13·13-s + 3.05·17-s + 1.73·23-s + 6/5·25-s − 3.09·31-s − 1.37·35-s + 1.62·37-s − 3.70·41-s − 2.04·43-s + 0.340·47-s + 8/49·49-s − 2.03·53-s + 6.98·55-s − 4.32·61-s + 0.738·65-s − 3.22·67-s + 3.38·71-s − 2.84·73-s − 1.66·77-s − 2/9·81-s + 4.04·83-s + 7.34·85-s − 0.175·91-s − 2.14·97-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{48} \cdot 3^{8} \cdot 5^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.19204\times 10^{11}\) |
| Root analytic conductor: | \(5.11449\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{48} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(12.37349744\) |
| \(L(\frac12)\) | \(\approx\) | \(12.37349744\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p^{2} T^{4} )^{2} \) | |
| 5 | \( 1 - 12 T + 114 T^{2} - 708 T^{3} + 826 p T^{4} - 708 p^{2} T^{5} + 114 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) | |
| good | 7 | \( 1 + 4 T + 8 T^{2} + 316 T^{3} - 2816 T^{4} - 2108 T^{5} + 64024 T^{6} + 120828 T^{7} + 11331966 T^{8} + 120828 p^{2} T^{9} + 64024 p^{4} T^{10} - 2108 p^{6} T^{11} - 2816 p^{8} T^{12} + 316 p^{10} T^{13} + 8 p^{12} T^{14} + 4 p^{14} T^{15} + p^{16} T^{16} \) |
| 11 | \( ( 1 - 16 T + 126 T^{2} + 1888 T^{3} - 31078 T^{4} + 1888 p^{2} T^{5} + 126 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 13 | \( 1 - 4 T + 8 T^{2} - 124 p T^{3} + 1072 p T^{4} + 328076 T^{5} - 124520 T^{6} + 28503876 T^{7} - 763526370 T^{8} + 28503876 p^{2} T^{9} - 124520 p^{4} T^{10} + 328076 p^{6} T^{11} + 1072 p^{9} T^{12} - 124 p^{11} T^{13} + 8 p^{12} T^{14} - 4 p^{14} T^{15} + p^{16} T^{16} \) | |
| 17 | \( 1 - 52 T + 1352 T^{2} - 28204 T^{3} + 490768 T^{4} - 6510964 T^{5} + 72784600 T^{6} - 481409004 T^{7} + 154737438 T^{8} - 481409004 p^{2} T^{9} + 72784600 p^{4} T^{10} - 6510964 p^{6} T^{11} + 490768 p^{8} T^{12} - 28204 p^{10} T^{13} + 1352 p^{12} T^{14} - 52 p^{14} T^{15} + p^{16} T^{16} \) | |
| 19 | \( 1 - 2000 T^{2} + 1889820 T^{4} - 1128357424 T^{6} + 476406585350 T^{8} - 1128357424 p^{4} T^{10} + 1889820 p^{8} T^{12} - 2000 p^{12} T^{14} + p^{16} T^{16} \) | |
| 23 | \( 1 - 40 T + 800 T^{2} - 18248 T^{3} - 19516 T^{4} + 7340008 T^{5} - 111492768 T^{6} + 1524968904 T^{7} - 5167201274 T^{8} + 1524968904 p^{2} T^{9} - 111492768 p^{4} T^{10} + 7340008 p^{6} T^{11} - 19516 p^{8} T^{12} - 18248 p^{10} T^{13} + 800 p^{12} T^{14} - 40 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( 1 - 2228 T^{2} + 1857576 T^{4} - 965861788 T^{6} + 647582808974 T^{8} - 965861788 p^{4} T^{10} + 1857576 p^{8} T^{12} - 2228 p^{12} T^{14} + p^{16} T^{16} \) | |
| 31 | \( ( 1 + 48 T + 1740 T^{2} - 1968 T^{3} - 318442 T^{4} - 1968 p^{2} T^{5} + 1740 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 37 | \( 1 - 60 T + 1800 T^{2} + 32652 T^{3} - 6760976 T^{4} + 249442548 T^{5} - 2263719528 T^{6} - 292468480836 T^{7} + 19178941117086 T^{8} - 292468480836 p^{2} T^{9} - 2263719528 p^{4} T^{10} + 249442548 p^{6} T^{11} - 6760976 p^{8} T^{12} + 32652 p^{10} T^{13} + 1800 p^{12} T^{14} - 60 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( ( 1 + 76 T + 5776 T^{2} + 267748 T^{3} + 13703518 T^{4} + 267748 p^{2} T^{5} + 5776 p^{4} T^{6} + 76 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 43 | \( 1 + 88 T + 3872 T^{2} + 100696 T^{3} + 1714468 T^{4} + 169137736 T^{5} + 13315542880 T^{6} + 642392563656 T^{7} + 29742475153158 T^{8} + 642392563656 p^{2} T^{9} + 13315542880 p^{4} T^{10} + 169137736 p^{6} T^{11} + 1714468 p^{8} T^{12} + 100696 p^{10} T^{13} + 3872 p^{12} T^{14} + 88 p^{14} T^{15} + p^{16} T^{16} \) | |
| 47 | \( 1 - 16 T + 128 T^{2} + 128816 T^{3} + 2412964 T^{4} - 504683248 T^{5} + 16062853504 T^{6} - 177987325872 T^{7} - 50194759450554 T^{8} - 177987325872 p^{2} T^{9} + 16062853504 p^{4} T^{10} - 504683248 p^{6} T^{11} + 2412964 p^{8} T^{12} + 128816 p^{10} T^{13} + 128 p^{12} T^{14} - 16 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( 1 + 108 T + 5832 T^{2} + 412884 T^{3} + 23090224 T^{4} + 11383740 p T^{5} + 15734940120 T^{6} + 223612868052 T^{7} - 24915462238434 T^{8} + 223612868052 p^{2} T^{9} + 15734940120 p^{4} T^{10} + 11383740 p^{7} T^{11} + 23090224 p^{8} T^{12} + 412884 p^{10} T^{13} + 5832 p^{12} T^{14} + 108 p^{14} T^{15} + p^{16} T^{16} \) | |
| 59 | \( 1 - 25500 T^{2} + 290547640 T^{4} - 1939529752884 T^{6} + 8319211197765870 T^{8} - 1939529752884 p^{4} T^{10} + 290547640 p^{8} T^{12} - 25500 p^{12} T^{14} + p^{16} T^{16} \) | |
| 61 | \( ( 1 + 132 T + 13872 T^{2} + 941868 T^{3} + 61614014 T^{4} + 941868 p^{2} T^{5} + 13872 p^{4} T^{6} + 132 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 67 | \( 1 + 216 T + 23328 T^{2} + 2220888 T^{3} + 213731236 T^{4} + 17658282696 T^{5} + 1294438543200 T^{6} + 97283636525256 T^{7} + 7010705853400710 T^{8} + 97283636525256 p^{2} T^{9} + 1294438543200 p^{4} T^{10} + 17658282696 p^{6} T^{11} + 213731236 p^{8} T^{12} + 2220888 p^{10} T^{13} + 23328 p^{12} T^{14} + 216 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( ( 1 - 120 T + 20340 T^{2} - 1543656 T^{3} + 149872550 T^{4} - 1543656 p^{2} T^{5} + 20340 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 73 | \( 1 + 208 T + 21632 T^{2} + 2121136 T^{3} + 133201276 T^{4} + 248473360 T^{5} - 580118578304 T^{6} - 88473819716496 T^{7} - 8873634681591930 T^{8} - 88473819716496 p^{2} T^{9} - 580118578304 p^{4} T^{10} + 248473360 p^{6} T^{11} + 133201276 p^{8} T^{12} + 2121136 p^{10} T^{13} + 21632 p^{12} T^{14} + 208 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( 1 - 32504 T^{2} + 462697980 T^{4} - 4035012008776 T^{6} + 27090604999745414 T^{8} - 4035012008776 p^{4} T^{10} + 462697980 p^{8} T^{12} - 32504 p^{12} T^{14} + p^{16} T^{16} \) | |
| 83 | \( 1 - 336 T + 56448 T^{2} - 7845648 T^{3} + 1037744740 T^{4} - 116459238960 T^{5} + 11328785476992 T^{6} - 1064762407254768 T^{7} + 93992795964479238 T^{8} - 1064762407254768 p^{2} T^{9} + 11328785476992 p^{4} T^{10} - 116459238960 p^{6} T^{11} + 1037744740 p^{8} T^{12} - 7845648 p^{10} T^{13} + 56448 p^{12} T^{14} - 336 p^{14} T^{15} + p^{16} T^{16} \) | |
| 89 | \( 1 - 25856 T^{2} + 291349884 T^{4} - 1811616834304 T^{6} + 9967156810684550 T^{8} - 1811616834304 p^{4} T^{10} + 291349884 p^{8} T^{12} - 25856 p^{12} T^{14} + p^{16} T^{16} \) | |
| 97 | \( 1 + 208 T + 21632 T^{2} + 956336 T^{3} - 203065348 T^{4} - 29702194672 T^{5} - 1328057611392 T^{6} + 139759626438000 T^{7} + 32182715530229254 T^{8} + 139759626438000 p^{2} T^{9} - 1328057611392 p^{4} T^{10} - 29702194672 p^{6} T^{11} - 203065348 p^{8} T^{12} + 956336 p^{10} T^{13} + 21632 p^{12} T^{14} + 208 p^{14} T^{15} + p^{16} T^{16} \) | |
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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
−4.05371527180654207206207355446, −3.76838134965731607558421645743, −3.73660696040616318198794468104, −3.66245239908368265561829071525, −3.45745919215482034589927655656, −3.32007384724287212576809728490, −3.30317241227808760400548906744, −3.25414433350991750098136237606, −3.03659984838017652073938272945, −3.01152903926381871594545143044, −2.79852182898296527583185591415, −2.44225971568453772716518243880, −2.24974049631523435540852144840, −2.15335987779485778129413460005, −2.05854793175559144959742431310, −1.86958297945773156469470601551, −1.55507737203613120679764629554, −1.46672640223742759371929411765, −1.44678456012447289212102404805, −1.30813122649260456247396024048, −1.22542057587887873392484207828, −1.14446651728387048762505312516, −0.57483693624640599727664606790, −0.31854129159442069006189263696, −0.23577505461654047839190735307, 0.23577505461654047839190735307, 0.31854129159442069006189263696, 0.57483693624640599727664606790, 1.14446651728387048762505312516, 1.22542057587887873392484207828, 1.30813122649260456247396024048, 1.44678456012447289212102404805, 1.46672640223742759371929411765, 1.55507737203613120679764629554, 1.86958297945773156469470601551, 2.05854793175559144959742431310, 2.15335987779485778129413460005, 2.24974049631523435540852144840, 2.44225971568453772716518243880, 2.79852182898296527583185591415, 3.01152903926381871594545143044, 3.03659984838017652073938272945, 3.25414433350991750098136237606, 3.30317241227808760400548906744, 3.32007384724287212576809728490, 3.45745919215482034589927655656, 3.66245239908368265561829071525, 3.73660696040616318198794468104, 3.76838134965731607558421645743, 4.05371527180654207206207355446
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.