Dirichlet series
| L(s) = 1 | + 3.24e3·2-s − 1.84e6·4-s + 1.95e8·5-s − 1.05e9·7-s − 2.68e10·8-s + 6.33e11·10-s + 6.37e11·11-s + 6.27e12·13-s − 3.41e12·14-s − 8.55e13·16-s + 2.13e14·17-s − 1.70e15·19-s − 3.60e14·20-s + 2.06e15·22-s − 6.29e15·23-s + 2.38e16·25-s + 2.03e16·26-s + 1.94e15·28-s − 3.45e16·29-s − 3.25e16·31-s − 9.21e16·32-s + 6.94e17·34-s − 2.05e17·35-s + 8.91e17·37-s − 5.53e18·38-s − 5.23e18·40-s + 5.33e18·41-s + ⋯ |
| L(s) = 1 | + 1.12·2-s − 0.220·4-s + 1.78·5-s − 0.201·7-s − 1.10·8-s + 2.00·10-s + 0.673·11-s + 0.971·13-s − 0.225·14-s − 1.21·16-s + 1.51·17-s − 3.35·19-s − 0.393·20-s + 0.755·22-s − 1.37·23-s + 2·25-s + 1.08·26-s + 0.0443·28-s − 0.525·29-s − 0.229·31-s − 0.452·32-s + 1.69·34-s − 0.360·35-s + 0.824·37-s − 3.76·38-s − 1.97·40-s + 1.51·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+23/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(4100625\) = \(3^{8} \cdot 5^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.17710\times 10^{8}\) |
| Root analytic conductor: | \(12.2817\) |
| Motivic weight: | \(23\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 4100625,\ (\ :23/2, 23/2, 23/2, 23/2),\ 1)\) |
Particular Values
| \(L(12)\) | \(\approx\) | \(15.90586011\) |
| \(L(\frac12)\) | \(\approx\) | \(15.90586011\) |
| \(L(\frac{25}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | |
|---|---|---|---|
| bad | 3 | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 - p^{11} T )^{4} \) | |
| good | 2 | $C_2 \wr S_4$ | \( 1 - 1623 p T + 774005 p^{4} T^{2} - 18923059 p^{10} T^{3} + 642696813 p^{17} T^{4} - 18923059 p^{33} T^{5} + 774005 p^{50} T^{6} - 1623 p^{70} T^{7} + p^{92} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 150481216 p T + 51728549764415214268 T^{2} - \)\(13\!\cdots\!08\)\( p^{2} T^{3} + \)\(85\!\cdots\!50\)\( p^{5} T^{4} - \)\(13\!\cdots\!08\)\( p^{25} T^{5} + 51728549764415214268 p^{46} T^{6} + 150481216 p^{70} T^{7} + p^{92} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 57952739904 p T + \)\(14\!\cdots\!72\)\( p^{2} T^{2} - \)\(10\!\cdots\!80\)\( p^{4} T^{3} + \)\(14\!\cdots\!86\)\( p^{4} T^{4} - \)\(10\!\cdots\!80\)\( p^{27} T^{5} + \)\(14\!\cdots\!72\)\( p^{48} T^{6} - 57952739904 p^{70} T^{7} + p^{92} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 483071444408 p T + \)\(90\!\cdots\!36\)\( T^{2} - \)\(46\!\cdots\!52\)\( T^{3} + \)\(40\!\cdots\!50\)\( p T^{4} - \)\(46\!\cdots\!52\)\( p^{23} T^{5} + \)\(90\!\cdots\!36\)\( p^{46} T^{6} - 483071444408 p^{70} T^{7} + p^{92} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 12579198521592 p T + \)\(24\!\cdots\!92\)\( p^{2} T^{2} - \)\(26\!\cdots\!60\)\( p^{3} T^{3} + \)\(23\!\cdots\!66\)\( p^{4} T^{4} - \)\(26\!\cdots\!60\)\( p^{26} T^{5} + \)\(24\!\cdots\!92\)\( p^{48} T^{6} - 12579198521592 p^{70} T^{7} + p^{92} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 1704940308311488 T + \)\(79\!\cdots\!28\)\( p T^{2} + \)\(25\!\cdots\!56\)\( p^{2} T^{3} + \)\(70\!\cdots\!46\)\( p^{3} T^{4} + \)\(25\!\cdots\!56\)\( p^{25} T^{5} + \)\(79\!\cdots\!28\)\( p^{47} T^{6} + 1704940308311488 p^{69} T^{7} + p^{92} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 6294627833142096 T + \)\(16\!\cdots\!24\)\( p T^{2} + \)\(37\!\cdots\!36\)\( T^{3} + \)\(25\!\cdots\!90\)\( T^{4} + \)\(37\!\cdots\!36\)\( p^{23} T^{5} + \)\(16\!\cdots\!24\)\( p^{47} T^{6} + 6294627833142096 p^{69} T^{7} + p^{92} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 34510260443456904 T + \)\(11\!\cdots\!80\)\( T^{2} + \)\(27\!\cdots\!08\)\( T^{3} + \)\(59\!\cdots\!98\)\( T^{4} + \)\(27\!\cdots\!08\)\( p^{23} T^{5} + \)\(11\!\cdots\!80\)\( p^{46} T^{6} + 34510260443456904 p^{69} T^{7} + p^{92} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 32516973391589776 T + \)\(46\!\cdots\!08\)\( T^{2} + \)\(17\!\cdots\!48\)\( T^{3} + \)\(12\!\cdots\!94\)\( T^{4} + \)\(17\!\cdots\!48\)\( p^{23} T^{5} + \)\(46\!\cdots\!08\)\( p^{46} T^{6} + 32516973391589776 p^{69} T^{7} + p^{92} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 891824674798536728 T - \)\(36\!\cdots\!52\)\( T^{2} - \)\(36\!\cdots\!12\)\( T^{3} + \)\(23\!\cdots\!70\)\( T^{4} - \)\(36\!\cdots\!12\)\( p^{23} T^{5} - \)\(36\!\cdots\!52\)\( p^{46} T^{6} - 891824674798536728 p^{69} T^{7} + p^{92} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 5333989455991248696 T + \)\(11\!\cdots\!88\)\( p T^{2} - \)\(15\!\cdots\!08\)\( T^{3} + \)\(85\!\cdots\!54\)\( T^{4} - \)\(15\!\cdots\!08\)\( p^{23} T^{5} + \)\(11\!\cdots\!88\)\( p^{47} T^{6} - 5333989455991248696 p^{69} T^{7} + p^{92} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 48519132390046480 p T + \)\(11\!\cdots\!20\)\( T^{2} + \)\(55\!\cdots\!60\)\( p T^{3} + \)\(60\!\cdots\!98\)\( T^{4} + \)\(55\!\cdots\!60\)\( p^{24} T^{5} + \)\(11\!\cdots\!20\)\( p^{46} T^{6} + 48519132390046480 p^{70} T^{7} + p^{92} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 3076110902091815760 T + \)\(43\!\cdots\!80\)\( T^{2} + \)\(25\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!58\)\( T^{4} + \)\(25\!\cdots\!20\)\( p^{23} T^{5} + \)\(43\!\cdots\!80\)\( p^{46} T^{6} - 3076110902091815760 p^{69} T^{7} + p^{92} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 99435256235497891224 T + \)\(20\!\cdots\!52\)\( T^{2} - \)\(13\!\cdots\!44\)\( T^{3} + \)\(14\!\cdots\!50\)\( T^{4} - \)\(13\!\cdots\!44\)\( p^{23} T^{5} + \)\(20\!\cdots\!52\)\( p^{46} T^{6} - 99435256235497891224 p^{69} T^{7} + p^{92} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - \)\(12\!\cdots\!52\)\( T + \)\(13\!\cdots\!92\)\( T^{2} - \)\(94\!\cdots\!36\)\( p T^{3} + \)\(81\!\cdots\!54\)\( T^{4} - \)\(94\!\cdots\!36\)\( p^{24} T^{5} + \)\(13\!\cdots\!92\)\( p^{46} T^{6} - \)\(12\!\cdots\!52\)\( p^{69} T^{7} + p^{92} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + \)\(96\!\cdots\!60\)\( T + \)\(77\!\cdots\!76\)\( T^{2} + \)\(37\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!66\)\( T^{4} + \)\(37\!\cdots\!40\)\( p^{23} T^{5} + \)\(77\!\cdots\!76\)\( p^{46} T^{6} + \)\(96\!\cdots\!60\)\( p^{69} T^{7} + p^{92} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + \)\(71\!\cdots\!24\)\( T + \)\(38\!\cdots\!68\)\( T^{2} + \)\(20\!\cdots\!00\)\( T^{3} + \)\(57\!\cdots\!26\)\( T^{4} + \)\(20\!\cdots\!00\)\( p^{23} T^{5} + \)\(38\!\cdots\!68\)\( p^{46} T^{6} + \)\(71\!\cdots\!24\)\( p^{69} T^{7} + p^{92} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - \)\(52\!\cdots\!32\)\( T + \)\(23\!\cdots\!28\)\( T^{2} - \)\(62\!\cdots\!04\)\( T^{3} + \)\(14\!\cdots\!70\)\( T^{4} - \)\(62\!\cdots\!04\)\( p^{23} T^{5} + \)\(23\!\cdots\!28\)\( p^{46} T^{6} - \)\(52\!\cdots\!32\)\( p^{69} T^{7} + p^{92} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + \)\(89\!\cdots\!84\)\( T + \)\(56\!\cdots\!52\)\( T^{2} + \)\(22\!\cdots\!24\)\( T^{3} + \)\(72\!\cdots\!90\)\( T^{4} + \)\(22\!\cdots\!24\)\( p^{23} T^{5} + \)\(56\!\cdots\!52\)\( p^{46} T^{6} + \)\(89\!\cdots\!84\)\( p^{69} T^{7} + p^{92} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + \)\(54\!\cdots\!20\)\( T + \)\(15\!\cdots\!56\)\( T^{2} + \)\(72\!\cdots\!40\)\( T^{3} + \)\(99\!\cdots\!26\)\( T^{4} + \)\(72\!\cdots\!40\)\( p^{23} T^{5} + \)\(15\!\cdots\!56\)\( p^{46} T^{6} + \)\(54\!\cdots\!20\)\( p^{69} T^{7} + p^{92} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - \)\(13\!\cdots\!12\)\( T + \)\(13\!\cdots\!40\)\( T^{2} + \)\(32\!\cdots\!32\)\( T^{3} - \)\(15\!\cdots\!94\)\( T^{4} + \)\(32\!\cdots\!32\)\( p^{23} T^{5} + \)\(13\!\cdots\!40\)\( p^{46} T^{6} - \)\(13\!\cdots\!12\)\( p^{69} T^{7} + p^{92} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - \)\(35\!\cdots\!88\)\( T + \)\(14\!\cdots\!72\)\( T^{2} - \)\(82\!\cdots\!56\)\( T^{3} + \)\(13\!\cdots\!14\)\( T^{4} - \)\(82\!\cdots\!56\)\( p^{23} T^{5} + \)\(14\!\cdots\!72\)\( p^{46} T^{6} - \)\(35\!\cdots\!88\)\( p^{69} T^{7} + p^{92} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - \)\(53\!\cdots\!96\)\( T + \)\(21\!\cdots\!48\)\( T^{2} - \)\(36\!\cdots\!20\)\( T^{3} + \)\(39\!\cdots\!26\)\( T^{4} - \)\(36\!\cdots\!20\)\( p^{23} T^{5} + \)\(21\!\cdots\!48\)\( p^{46} T^{6} - \)\(53\!\cdots\!96\)\( p^{69} T^{7} + p^{92} T^{8} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.60859447130441752726123108972, −6.86974713356843098482333440619, −6.64038858522089036739458901230, −6.35072316189268323545908903649, −6.16912285563166633507371063394, −6.12892566708436694527166856640, −5.52468152427286101718455071583, −5.44504623069873620591139586502, −5.41484302616050984608646264544, −4.48727608694434196105132338646, −4.42653071662556736403210811049, −4.24572488848107305832805345767, −4.21437941307635843328311247805, −3.61370116069756166756031769144, −3.21941168469094664698244925732, −3.13872276758387800379022223798, −2.62907648685036391164530230984, −2.40523351125067043884975555738, −1.94948512297158874487118621800, −1.72754411199806583472585841419, −1.64493568700714203015815811956, −1.31724005563808407425593649443, −0.58129753339913141418090571720, −0.55164985739212602119210687933, −0.35635474675008751923168397677, 0.35635474675008751923168397677, 0.55164985739212602119210687933, 0.58129753339913141418090571720, 1.31724005563808407425593649443, 1.64493568700714203015815811956, 1.72754411199806583472585841419, 1.94948512297158874487118621800, 2.40523351125067043884975555738, 2.62907648685036391164530230984, 3.13872276758387800379022223798, 3.21941168469094664698244925732, 3.61370116069756166756031769144, 4.21437941307635843328311247805, 4.24572488848107305832805345767, 4.42653071662556736403210811049, 4.48727608694434196105132338646, 5.41484302616050984608646264544, 5.44504623069873620591139586502, 5.52468152427286101718455071583, 6.12892566708436694527166856640, 6.16912285563166633507371063394, 6.35072316189268323545908903649, 6.64038858522089036739458901230, 6.86974713356843098482333440619, 7.60859447130441752726123108972