Dirichlet series
| L(s) = 1 | + 3.11e5·4-s + 9.15e9·16-s + 2.36e11·19-s + 1.25e12·25-s − 1.25e13·31-s + 1.64e15·49-s − 6.45e15·61-s − 7.25e15·64-s + 7.37e16·76-s − 2.21e15·79-s + 3.91e17·100-s − 1.01e18·109-s − 1.45e18·121-s − 3.90e18·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.60e19·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2.38·4-s + 0.532·16-s + 3.19·19-s + 1.64·25-s − 2.63·31-s + 7.07·49-s − 4.31·61-s − 3.22·64-s + 7.59·76-s − 0.164·79-s + 3.91·100-s − 4.87·109-s − 2.88·121-s − 6.26·124-s + 5.32·169-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(18-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+17/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{32} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.56084\times 10^{30}\) |
| Root analytic conductor: | \(9.08019\) |
| Motivic weight: | \(17\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{32} \cdot 5^{16} ,\ ( \ : [17/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(9)\) | \(\approx\) | \(1.009958887\) |
| \(L(\frac12)\) | \(\approx\) | \(1.009958887\) |
| \(L(\frac{19}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 250748400712 p T^{2} + 3146722591010101760 p^{7} T^{4} - \)\(53\!\cdots\!00\)\( p^{13} T^{6} + \)\(14\!\cdots\!36\)\( p^{26} T^{8} - \)\(53\!\cdots\!00\)\( p^{47} T^{10} + 3146722591010101760 p^{75} T^{12} - 250748400712 p^{103} T^{14} + p^{136} T^{16} \) | |
| good | 2 | \( ( 1 - 38999 p^{2} T^{2} + 498831253 p^{6} T^{4} - 113498182591 p^{15} T^{6} + 116153439773005 p^{22} T^{8} - 113498182591 p^{49} T^{10} + 498831253 p^{74} T^{12} - 38999 p^{104} T^{14} + p^{136} T^{16} )^{2} \) |
| 7 | \( ( 1 - 117567524758808 p T^{2} + \)\(41\!\cdots\!96\)\( p T^{4} - \)\(35\!\cdots\!44\)\( p^{5} T^{6} + \)\(19\!\cdots\!70\)\( p^{8} T^{8} - \)\(35\!\cdots\!44\)\( p^{39} T^{10} + \)\(41\!\cdots\!96\)\( p^{69} T^{12} - 117567524758808 p^{103} T^{14} + p^{136} T^{16} )^{2} \) | |
| 11 | \( ( 1 + 729677131009972168 T^{2} + \)\(22\!\cdots\!68\)\( p T^{4} + \)\(93\!\cdots\!36\)\( p^{3} T^{6} + \)\(45\!\cdots\!70\)\( p^{5} T^{8} + \)\(93\!\cdots\!36\)\( p^{37} T^{10} + \)\(22\!\cdots\!68\)\( p^{69} T^{12} + 729677131009972168 p^{102} T^{14} + p^{136} T^{16} )^{2} \) | |
| 13 | \( ( 1 - 23040527147537973464 T^{2} + \)\(28\!\cdots\!84\)\( p T^{4} - \)\(20\!\cdots\!76\)\( p^{3} T^{6} + \)\(11\!\cdots\!90\)\( p^{5} T^{8} - \)\(20\!\cdots\!76\)\( p^{37} T^{10} + \)\(28\!\cdots\!84\)\( p^{69} T^{12} - 23040527147537973464 p^{102} T^{14} + p^{136} T^{16} )^{2} \) | |
| 17 | \( ( 1 - \)\(18\!\cdots\!96\)\( T^{2} + \)\(34\!\cdots\!72\)\( T^{4} - \)\(37\!\cdots\!48\)\( T^{6} + \)\(37\!\cdots\!70\)\( T^{8} - \)\(37\!\cdots\!48\)\( p^{34} T^{10} + \)\(34\!\cdots\!72\)\( p^{68} T^{12} - \)\(18\!\cdots\!96\)\( p^{102} T^{14} + p^{136} T^{16} )^{2} \) | |
| 19 | \( ( 1 - 59069874584 T + \)\(18\!\cdots\!52\)\( T^{2} - \)\(83\!\cdots\!72\)\( T^{3} + \)\(14\!\cdots\!70\)\( T^{4} - \)\(83\!\cdots\!72\)\( p^{17} T^{5} + \)\(18\!\cdots\!52\)\( p^{34} T^{6} - 59069874584 p^{51} T^{7} + p^{68} T^{8} )^{4} \) | |
| 23 | \( ( 1 - \)\(31\!\cdots\!44\)\( T^{2} + \)\(96\!\cdots\!12\)\( T^{4} - \)\(18\!\cdots\!12\)\( T^{6} + \)\(30\!\cdots\!70\)\( T^{8} - \)\(18\!\cdots\!12\)\( p^{34} T^{10} + \)\(96\!\cdots\!12\)\( p^{68} T^{12} - \)\(31\!\cdots\!44\)\( p^{102} T^{14} + p^{136} T^{16} )^{2} \) | |
| 29 | \( ( 1 + \)\(27\!\cdots\!72\)\( T^{2} + \)\(44\!\cdots\!68\)\( T^{4} + \)\(50\!\cdots\!24\)\( T^{6} + \)\(42\!\cdots\!70\)\( T^{8} + \)\(50\!\cdots\!24\)\( p^{34} T^{10} + \)\(44\!\cdots\!68\)\( p^{68} T^{12} + \)\(27\!\cdots\!72\)\( p^{102} T^{14} + p^{136} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 3125355600808 T + \)\(75\!\cdots\!68\)\( T^{2} + \)\(14\!\cdots\!96\)\( T^{3} + \)\(23\!\cdots\!70\)\( T^{4} + \)\(14\!\cdots\!96\)\( p^{17} T^{5} + \)\(75\!\cdots\!68\)\( p^{34} T^{6} + 3125355600808 p^{51} T^{7} + p^{68} T^{8} )^{4} \) | |
| 37 | \( ( 1 - \)\(16\!\cdots\!36\)\( T^{2} + \)\(12\!\cdots\!92\)\( T^{4} - \)\(66\!\cdots\!28\)\( T^{6} + \)\(29\!\cdots\!70\)\( T^{8} - \)\(66\!\cdots\!28\)\( p^{34} T^{10} + \)\(12\!\cdots\!92\)\( p^{68} T^{12} - \)\(16\!\cdots\!36\)\( p^{102} T^{14} + p^{136} T^{16} )^{2} \) | |
| 41 | \( ( 1 + \)\(28\!\cdots\!48\)\( T^{2} + \)\(59\!\cdots\!08\)\( T^{4} + \)\(31\!\cdots\!96\)\( T^{6} + \)\(10\!\cdots\!70\)\( T^{8} + \)\(31\!\cdots\!96\)\( p^{34} T^{10} + \)\(59\!\cdots\!08\)\( p^{68} T^{12} + \)\(28\!\cdots\!48\)\( p^{102} T^{14} + p^{136} T^{16} )^{2} \) | |
| 43 | \( ( 1 - \)\(59\!\cdots\!44\)\( T^{2} + \)\(19\!\cdots\!72\)\( T^{4} - \)\(14\!\cdots\!92\)\( T^{6} + \)\(25\!\cdots\!70\)\( T^{8} - \)\(14\!\cdots\!92\)\( p^{34} T^{10} + \)\(19\!\cdots\!72\)\( p^{68} T^{12} - \)\(59\!\cdots\!44\)\( p^{102} T^{14} + p^{136} T^{16} )^{2} \) | |
| 47 | \( ( 1 - \)\(15\!\cdots\!16\)\( T^{2} + \)\(11\!\cdots\!72\)\( T^{4} - \)\(54\!\cdots\!68\)\( T^{6} + \)\(17\!\cdots\!70\)\( T^{8} - \)\(54\!\cdots\!68\)\( p^{34} T^{10} + \)\(11\!\cdots\!72\)\( p^{68} T^{12} - \)\(15\!\cdots\!16\)\( p^{102} T^{14} + p^{136} T^{16} )^{2} \) | |
| 53 | \( ( 1 - \)\(83\!\cdots\!84\)\( T^{2} + \)\(35\!\cdots\!72\)\( T^{4} - \)\(10\!\cdots\!32\)\( T^{6} + \)\(22\!\cdots\!70\)\( T^{8} - \)\(10\!\cdots\!32\)\( p^{34} T^{10} + \)\(35\!\cdots\!72\)\( p^{68} T^{12} - \)\(83\!\cdots\!84\)\( p^{102} T^{14} + p^{136} T^{16} )^{2} \) | |
| 59 | \( ( 1 + \)\(44\!\cdots\!52\)\( T^{2} + \)\(13\!\cdots\!08\)\( T^{4} + \)\(26\!\cdots\!04\)\( T^{6} + \)\(67\!\cdots\!30\)\( p T^{8} + \)\(26\!\cdots\!04\)\( p^{34} T^{10} + \)\(13\!\cdots\!08\)\( p^{68} T^{12} + \)\(44\!\cdots\!52\)\( p^{102} T^{14} + p^{136} T^{16} )^{2} \) | |
| 61 | \( ( 1 + 1614817700100472 T + \)\(72\!\cdots\!28\)\( T^{2} + \)\(69\!\cdots\!64\)\( T^{3} + \)\(20\!\cdots\!70\)\( T^{4} + \)\(69\!\cdots\!64\)\( p^{17} T^{5} + \)\(72\!\cdots\!28\)\( p^{34} T^{6} + 1614817700100472 p^{51} T^{7} + p^{68} T^{8} )^{4} \) | |
| 67 | \( ( 1 - \)\(16\!\cdots\!16\)\( T^{2} + \)\(15\!\cdots\!12\)\( T^{4} - \)\(14\!\cdots\!48\)\( T^{6} + \)\(20\!\cdots\!70\)\( T^{8} - \)\(14\!\cdots\!48\)\( p^{34} T^{10} + \)\(15\!\cdots\!12\)\( p^{68} T^{12} - \)\(16\!\cdots\!16\)\( p^{102} T^{14} + p^{136} T^{16} )^{2} \) | |
| 71 | \( ( 1 + \)\(14\!\cdots\!28\)\( T^{2} + \)\(93\!\cdots\!68\)\( T^{4} + \)\(34\!\cdots\!76\)\( T^{6} + \)\(14\!\cdots\!70\)\( p T^{8} + \)\(34\!\cdots\!76\)\( p^{34} T^{10} + \)\(93\!\cdots\!68\)\( p^{68} T^{12} + \)\(14\!\cdots\!28\)\( p^{102} T^{14} + p^{136} T^{16} )^{2} \) | |
| 73 | \( ( 1 - \)\(28\!\cdots\!24\)\( T^{2} + \)\(37\!\cdots\!52\)\( T^{4} - \)\(31\!\cdots\!12\)\( T^{6} + \)\(18\!\cdots\!70\)\( T^{8} - \)\(31\!\cdots\!12\)\( p^{34} T^{10} + \)\(37\!\cdots\!52\)\( p^{68} T^{12} - \)\(28\!\cdots\!24\)\( p^{102} T^{14} + p^{136} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 553140460111144 T + \)\(48\!\cdots\!12\)\( T^{2} - \)\(12\!\cdots\!88\)\( T^{3} + \)\(11\!\cdots\!70\)\( T^{4} - \)\(12\!\cdots\!88\)\( p^{17} T^{5} + \)\(48\!\cdots\!12\)\( p^{34} T^{6} + 553140460111144 p^{51} T^{7} + p^{68} T^{8} )^{4} \) | |
| 83 | \( ( 1 - \)\(29\!\cdots\!64\)\( T^{2} + \)\(38\!\cdots\!52\)\( T^{4} - \)\(30\!\cdots\!52\)\( T^{6} + \)\(15\!\cdots\!70\)\( T^{8} - \)\(30\!\cdots\!52\)\( p^{34} T^{10} + \)\(38\!\cdots\!52\)\( p^{68} T^{12} - \)\(29\!\cdots\!64\)\( p^{102} T^{14} + p^{136} T^{16} )^{2} \) | |
| 89 | \( ( 1 + \)\(40\!\cdots\!32\)\( T^{2} + \)\(71\!\cdots\!48\)\( T^{4} + \)\(83\!\cdots\!84\)\( T^{6} + \)\(97\!\cdots\!70\)\( T^{8} + \)\(83\!\cdots\!84\)\( p^{34} T^{10} + \)\(71\!\cdots\!48\)\( p^{68} T^{12} + \)\(40\!\cdots\!32\)\( p^{102} T^{14} + p^{136} T^{16} )^{2} \) | |
| 97 | \( ( 1 - \)\(32\!\cdots\!96\)\( T^{2} + \)\(53\!\cdots\!32\)\( T^{4} - \)\(54\!\cdots\!68\)\( T^{6} + \)\(38\!\cdots\!70\)\( T^{8} - \)\(54\!\cdots\!68\)\( p^{34} T^{10} + \)\(53\!\cdots\!32\)\( p^{68} T^{12} - \)\(32\!\cdots\!96\)\( p^{102} T^{14} + p^{136} T^{16} )^{2} \) | |
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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
−2.15824830041332738425217365519, −2.15027299977493943532753378809, −2.08564960765429343686205301648, −1.95701526688779948247194905298, −1.93542236219923165931535769701, −1.80893270660164850874905838928, −1.59797978363526482786480621360, −1.58869864924857337985549851513, −1.40026390698250646890993975411, −1.35710112519926134469647211409, −1.24264103100453610309251418077, −1.21453030099202450730492789355, −1.13912963438515938012763169956, −1.13513586525521464259646753366, −1.01613724301411452152295408600, −0.840689410337071310910239128566, −0.826406445221305824412817225474, −0.70175700819897992819753803104, −0.67973813668015822993106390967, −0.48545233445951793163072186350, −0.35592625762368877394243404237, −0.32499441906161293623299747205, −0.27655544555037424649703310278, −0.10289222057400395109602165105, −0.02286077728230096357813403536, 0.02286077728230096357813403536, 0.10289222057400395109602165105, 0.27655544555037424649703310278, 0.32499441906161293623299747205, 0.35592625762368877394243404237, 0.48545233445951793163072186350, 0.67973813668015822993106390967, 0.70175700819897992819753803104, 0.826406445221305824412817225474, 0.840689410337071310910239128566, 1.01613724301411452152295408600, 1.13513586525521464259646753366, 1.13912963438515938012763169956, 1.21453030099202450730492789355, 1.24264103100453610309251418077, 1.35710112519926134469647211409, 1.40026390698250646890993975411, 1.58869864924857337985549851513, 1.59797978363526482786480621360, 1.80893270660164850874905838928, 1.93542236219923165931535769701, 1.95701526688779948247194905298, 2.08564960765429343686205301648, 2.15027299977493943532753378809, 2.15824830041332738425217365519
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.