Dirichlet series
| L(s) = 1 | + 56·7-s + 3.17e3·25-s + 496·37-s − 3.44e3·43-s + 4.60e3·49-s − 1.33e4·67-s + 2.40e4·79-s − 8.12e3·109-s − 1.24e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.94e5·169-s + 173-s + 1.77e5·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 8/7·7-s + 5.08·25-s + 0.362·37-s − 1.86·43-s + 1.91·49-s − 2.96·67-s + 3.85·79-s − 0.684·109-s − 8.47·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 6.82·169-s + 3.34e−5·173-s + 5.80·175-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{64} \cdot 3^{32} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.93053\times 10^{32}\) |
| Root analytic conductor: | \(10.2076\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{64} \cdot 3^{32} \cdot 7^{16} ,\ ( \ : [2]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(198.8100919\) |
| \(L(\frac12)\) | \(\approx\) | \(198.8100919\) |
| \(L(3)\) | 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 | \( 1 \) | |
| 7 | \( ( 1 - 4 p T - 1128 T^{2} + 16948 p T^{3} - 1534 p^{3} T^{4} + 16948 p^{5} T^{5} - 1128 p^{8} T^{6} - 4 p^{13} T^{7} + p^{16} T^{8} )^{2} \) | |
| good | 5 | \( ( 1 - 1588 T^{2} + 1013256 T^{4} - 471982556 T^{6} + 271367874446 T^{8} - 471982556 p^{8} T^{10} + 1013256 p^{16} T^{12} - 1588 p^{24} T^{14} + p^{32} T^{16} )^{2} \) |
| 11 | \( ( 1 + 62036 T^{2} + 2164640328 T^{4} + 50725616920636 T^{6} + 7155671852468990 p^{2} T^{8} + 50725616920636 p^{8} T^{10} + 2164640328 p^{16} T^{12} + 62036 p^{24} T^{14} + p^{32} T^{16} )^{2} \) | |
| 13 | \( ( 1 - 97480 T^{2} + 5951161884 T^{4} - 250531879591544 T^{6} + 8155307175330796742 T^{8} - 250531879591544 p^{8} T^{10} + 5951161884 p^{16} T^{12} - 97480 p^{24} T^{14} + p^{32} T^{16} )^{2} \) | |
| 17 | \( ( 1 - 410100 T^{2} + 89208759048 T^{4} - 12542080696661148 T^{6} + \)\(12\!\cdots\!14\)\( T^{8} - 12542080696661148 p^{8} T^{10} + 89208759048 p^{16} T^{12} - 410100 p^{24} T^{14} + p^{32} T^{16} )^{2} \) | |
| 19 | \( ( 1 - 472384 T^{2} + 127108105212 T^{4} - 23539117059502784 T^{6} + \)\(34\!\cdots\!14\)\( T^{8} - 23539117059502784 p^{8} T^{10} + 127108105212 p^{16} T^{12} - 472384 p^{24} T^{14} + p^{32} T^{16} )^{2} \) | |
| 23 | \( ( 1 + 656244 T^{2} + 215741987848 T^{4} + 54680033773776924 T^{6} + \)\(14\!\cdots\!94\)\( T^{8} + 54680033773776924 p^{8} T^{10} + 215741987848 p^{16} T^{12} + 656244 p^{24} T^{14} + p^{32} T^{16} )^{2} \) | |
| 29 | \( ( 1 + 3528872 T^{2} + 5370385213020 T^{4} + 4957426340643396760 T^{6} + \)\(36\!\cdots\!78\)\( T^{8} + 4957426340643396760 p^{8} T^{10} + 5370385213020 p^{16} T^{12} + 3528872 p^{24} T^{14} + p^{32} T^{16} )^{2} \) | |
| 31 | \( ( 1 - 6281704 T^{2} + 17924726758236 T^{4} - 30590413634412977240 T^{6} + \)\(34\!\cdots\!58\)\( T^{8} - 30590413634412977240 p^{8} T^{10} + 17924726758236 p^{16} T^{12} - 6281704 p^{24} T^{14} + p^{32} T^{16} )^{2} \) | |
| 37 | \( ( 1 - 124 T + 2711368 T^{2} + 302369996 T^{3} + 7075501277134 T^{4} + 302369996 p^{4} T^{5} + 2711368 p^{8} T^{6} - 124 p^{12} T^{7} + p^{16} T^{8} )^{4} \) | |
| 41 | \( ( 1 - 4603188 T^{2} + 7584869271048 T^{4} - 5407017391684085340 T^{6} + \)\(19\!\cdots\!74\)\( T^{8} - 5407017391684085340 p^{8} T^{10} + 7584869271048 p^{16} T^{12} - 4603188 p^{24} T^{14} + p^{32} T^{16} )^{2} \) | |
| 43 | \( ( 1 + 20 p T + 10679848 T^{2} + 6095636084 T^{3} + 50162800409038 T^{4} + 6095636084 p^{4} T^{5} + 10679848 p^{8} T^{6} + 20 p^{13} T^{7} + p^{16} T^{8} )^{4} \) | |
| 47 | \( ( 1 - 20930248 T^{2} + 223750814787228 T^{4} - \)\(16\!\cdots\!12\)\( T^{6} + \)\(88\!\cdots\!78\)\( T^{8} - \)\(16\!\cdots\!12\)\( p^{8} T^{10} + 223750814787228 p^{16} T^{12} - 20930248 p^{24} T^{14} + p^{32} T^{16} )^{2} \) | |
| 53 | \( ( 1 + 40836200 T^{2} + 817708278666204 T^{4} + \)\(10\!\cdots\!24\)\( T^{6} + \)\(97\!\cdots\!58\)\( T^{8} + \)\(10\!\cdots\!24\)\( p^{8} T^{10} + 817708278666204 p^{16} T^{12} + 40836200 p^{24} T^{14} + p^{32} T^{16} )^{2} \) | |
| 59 | \( ( 1 - 35059784 T^{2} + 758018677634332 T^{4} - \)\(11\!\cdots\!88\)\( T^{6} + \)\(15\!\cdots\!02\)\( T^{8} - \)\(11\!\cdots\!88\)\( p^{8} T^{10} + 758018677634332 p^{16} T^{12} - 35059784 p^{24} T^{14} + p^{32} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 53973704 T^{2} + 1670920786679836 T^{4} - \)\(35\!\cdots\!60\)\( T^{6} + \)\(55\!\cdots\!78\)\( T^{8} - \)\(35\!\cdots\!60\)\( p^{8} T^{10} + 1670920786679836 p^{16} T^{12} - 53973704 p^{24} T^{14} + p^{32} T^{16} )^{2} \) | |
| 67 | \( ( 1 + 3332 T + 50282088 T^{2} + 249085594156 T^{3} + 1194060524389838 T^{4} + 249085594156 p^{4} T^{5} + 50282088 p^{8} T^{6} + 3332 p^{12} T^{7} + p^{16} T^{8} )^{4} \) | |
| 71 | \( ( 1 + 84773236 T^{2} + 3511029625160968 T^{4} + \)\(12\!\cdots\!76\)\( T^{6} + \)\(38\!\cdots\!70\)\( T^{8} + \)\(12\!\cdots\!76\)\( p^{8} T^{10} + 3511029625160968 p^{16} T^{12} + 84773236 p^{24} T^{14} + p^{32} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 107436552 T^{2} + 5692934809317660 T^{4} - \)\(20\!\cdots\!44\)\( T^{6} + \)\(58\!\cdots\!50\)\( T^{8} - \)\(20\!\cdots\!44\)\( p^{8} T^{10} + 5692934809317660 p^{16} T^{12} - 107436552 p^{24} T^{14} + p^{32} T^{16} )^{2} \) | |
| 79 | \( ( 1 - 6020 T + 77766744 T^{2} - 566708142028 T^{3} + 3036022092850862 T^{4} - 566708142028 p^{4} T^{5} + 77766744 p^{8} T^{6} - 6020 p^{12} T^{7} + p^{16} T^{8} )^{4} \) | |
| 83 | \( ( 1 - 175547208 T^{2} + 15904004249000988 T^{4} - \)\(10\!\cdots\!32\)\( T^{6} + \)\(55\!\cdots\!38\)\( T^{8} - \)\(10\!\cdots\!32\)\( p^{8} T^{10} + 15904004249000988 p^{16} T^{12} - 175547208 p^{24} T^{14} + p^{32} T^{16} )^{2} \) | |
| 89 | \( ( 1 - 254132724 T^{2} + 32548036235294472 T^{4} - \)\(28\!\cdots\!16\)\( T^{6} + \)\(19\!\cdots\!10\)\( T^{8} - \)\(28\!\cdots\!16\)\( p^{8} T^{10} + 32548036235294472 p^{16} T^{12} - 254132724 p^{24} T^{14} + p^{32} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 252564360 T^{2} + 44607808570949916 T^{4} - \)\(55\!\cdots\!76\)\( T^{6} + \)\(55\!\cdots\!74\)\( T^{8} - \)\(55\!\cdots\!76\)\( p^{8} T^{10} + 44607808570949916 p^{16} T^{12} - 252564360 p^{24} T^{14} + p^{32} T^{16} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.03959122985578669880916282136, −1.90025786698260560906046910265, −1.70124710140622495088697095724, −1.69871698111588554296537948275, −1.60489827875503829299715130497, −1.49634173673765844024556794537, −1.41840221293234485728887757981, −1.33333528098268923113776294385, −1.32256257531115592775827088141, −1.32187069751810357531102204636, −1.31426812634220026370578215226, −1.25091203160871834791380728170, −1.18789495861860534593499561671, −0.964907261331500379837201030903, −0.818393436930533167503000924928, −0.798942637073595233640030368433, −0.62286477561304210558861069855, −0.56149192087108371965163364597, −0.54741754215312950412713931826, −0.54652357532409325363706558581, −0.35379638666481330088516674842, −0.35133194591651949026789259880, −0.31809994238282488055144484998, −0.30816560249173059606900376479, −0.10778274300826144158771442992, 0.10778274300826144158771442992, 0.30816560249173059606900376479, 0.31809994238282488055144484998, 0.35133194591651949026789259880, 0.35379638666481330088516674842, 0.54652357532409325363706558581, 0.54741754215312950412713931826, 0.56149192087108371965163364597, 0.62286477561304210558861069855, 0.798942637073595233640030368433, 0.818393436930533167503000924928, 0.964907261331500379837201030903, 1.18789495861860534593499561671, 1.25091203160871834791380728170, 1.31426812634220026370578215226, 1.32187069751810357531102204636, 1.32256257531115592775827088141, 1.33333528098268923113776294385, 1.41840221293234485728887757981, 1.49634173673765844024556794537, 1.60489827875503829299715130497, 1.69871698111588554296537948275, 1.70124710140622495088697095724, 1.90025786698260560906046910265, 2.03959122985578669880916282136
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.