Dirichlet series
| L(s) = 1 | − 3.31e5·11-s − 1.40e8·31-s + 2.95e7·41-s + 3.57e9·61-s + 1.85e9·71-s − 1.54e9·81-s − 1.89e10·101-s − 1.48e11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 2.05·11-s − 4.91·31-s + 0.255·41-s + 4.23·61-s + 1.02·71-s − 4/9·81-s − 1.80·101-s − 5.73·121-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+5)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 3^{16} \cdot 5^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.03548\times 10^{36}\) |
| Root analytic conductor: | \(13.8060\) |
| Motivic weight: | \(10\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 3^{16} \cdot 5^{32} ,\ ( \ : [5]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{11}{2})\) | \(\approx\) | \(0.0003519601633\) |
| \(L(\frac12)\) | \(\approx\) | \(0.0003519601633\) |
| \(L(6)\) | 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 | \( ( 1 + p^{18} T^{4} )^{4} \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - 197292832374494404 T^{4} + \)\(16\!\cdots\!10\)\( T^{8} - \)\(80\!\cdots\!16\)\( p^{4} T^{12} + \)\(37\!\cdots\!19\)\( p^{8} T^{16} - \)\(80\!\cdots\!16\)\( p^{44} T^{20} + \)\(16\!\cdots\!10\)\( p^{80} T^{24} - 197292832374494404 p^{120} T^{28} + p^{160} T^{32} \) |
| 11 | \( ( 1 + 82776 T + 54314315020 T^{2} + 3977091470105064 T^{3} + \)\(14\!\cdots\!34\)\( T^{4} + 3977091470105064 p^{10} T^{5} + 54314315020 p^{20} T^{6} + 82776 p^{30} T^{7} + p^{40} T^{8} )^{4} \) | |
| 13 | \( 1 + \)\(19\!\cdots\!08\)\( T^{4} + \)\(29\!\cdots\!78\)\( T^{8} + \)\(28\!\cdots\!56\)\( T^{12} - \)\(49\!\cdots\!05\)\( T^{16} + \)\(28\!\cdots\!56\)\( p^{40} T^{20} + \)\(29\!\cdots\!78\)\( p^{80} T^{24} + \)\(19\!\cdots\!08\)\( p^{120} T^{28} + p^{160} T^{32} \) | |
| 17 | \( 1 - \)\(18\!\cdots\!64\)\( T^{4} + \)\(17\!\cdots\!40\)\( T^{8} - \)\(11\!\cdots\!76\)\( T^{12} + \)\(53\!\cdots\!14\)\( T^{16} - \)\(11\!\cdots\!76\)\( p^{40} T^{20} + \)\(17\!\cdots\!40\)\( p^{80} T^{24} - \)\(18\!\cdots\!64\)\( p^{120} T^{28} + p^{160} T^{32} \) | |
| 19 | \( ( 1 - 12159415166308 T^{2} + \)\(13\!\cdots\!78\)\( T^{4} - \)\(86\!\cdots\!56\)\( T^{6} + \)\(63\!\cdots\!95\)\( T^{8} - \)\(86\!\cdots\!56\)\( p^{20} T^{10} + \)\(13\!\cdots\!78\)\( p^{40} T^{12} - 12159415166308 p^{60} T^{14} + p^{80} T^{16} )^{2} \) | |
| 23 | \( 1 - \)\(46\!\cdots\!08\)\( T^{4} + \)\(11\!\cdots\!28\)\( T^{8} - \)\(21\!\cdots\!56\)\( T^{12} + \)\(35\!\cdots\!70\)\( T^{16} - \)\(21\!\cdots\!56\)\( p^{40} T^{20} + \)\(11\!\cdots\!28\)\( p^{80} T^{24} - \)\(46\!\cdots\!08\)\( p^{120} T^{28} + p^{160} T^{32} \) | |
| 29 | \( ( 1 - 2702520384392792 T^{2} + \)\(33\!\cdots\!28\)\( T^{4} - \)\(24\!\cdots\!44\)\( T^{6} + \)\(12\!\cdots\!70\)\( T^{8} - \)\(24\!\cdots\!44\)\( p^{20} T^{10} + \)\(33\!\cdots\!28\)\( p^{40} T^{12} - 2702520384392792 p^{60} T^{14} + p^{80} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 35201204 T + 1416319277340010 T^{2} + \)\(24\!\cdots\!16\)\( T^{3} + \)\(54\!\cdots\!19\)\( T^{4} + \)\(24\!\cdots\!16\)\( p^{10} T^{5} + 1416319277340010 p^{20} T^{6} + 35201204 p^{30} T^{7} + p^{40} T^{8} )^{4} \) | |
| 37 | \( 1 - \)\(10\!\cdots\!08\)\( T^{4} + \)\(53\!\cdots\!28\)\( T^{8} - \)\(19\!\cdots\!56\)\( T^{12} + \)\(50\!\cdots\!70\)\( T^{16} - \)\(19\!\cdots\!56\)\( p^{40} T^{20} + \)\(53\!\cdots\!28\)\( p^{80} T^{24} - \)\(10\!\cdots\!08\)\( p^{120} T^{28} + p^{160} T^{32} \) | |
| 41 | \( ( 1 - 7388400 T + 20886066228918004 T^{2} - \)\(16\!\cdots\!00\)\( T^{3} + \)\(28\!\cdots\!06\)\( T^{4} - \)\(16\!\cdots\!00\)\( p^{10} T^{5} + 20886066228918004 p^{20} T^{6} - 7388400 p^{30} T^{7} + p^{40} T^{8} )^{4} \) | |
| 43 | \( 1 - \)\(84\!\cdots\!08\)\( T^{4} + \)\(75\!\cdots\!78\)\( T^{8} - \)\(39\!\cdots\!56\)\( T^{12} + \)\(23\!\cdots\!95\)\( T^{16} - \)\(39\!\cdots\!56\)\( p^{40} T^{20} + \)\(75\!\cdots\!78\)\( p^{80} T^{24} - \)\(84\!\cdots\!08\)\( p^{120} T^{28} + p^{160} T^{32} \) | |
| 47 | \( 1 - \)\(14\!\cdots\!44\)\( T^{4} + \)\(11\!\cdots\!80\)\( T^{8} - \)\(53\!\cdots\!56\)\( T^{12} + \)\(17\!\cdots\!74\)\( T^{16} - \)\(53\!\cdots\!56\)\( p^{40} T^{20} + \)\(11\!\cdots\!80\)\( p^{80} T^{24} - \)\(14\!\cdots\!44\)\( p^{120} T^{28} + p^{160} T^{32} \) | |
| 53 | \( 1 + \)\(35\!\cdots\!56\)\( T^{4} + \)\(10\!\cdots\!80\)\( T^{8} - \)\(28\!\cdots\!76\)\( p^{4} T^{12} - \)\(15\!\cdots\!66\)\( p^{8} T^{16} - \)\(28\!\cdots\!76\)\( p^{44} T^{20} + \)\(10\!\cdots\!80\)\( p^{80} T^{24} + \)\(35\!\cdots\!56\)\( p^{120} T^{28} + p^{160} T^{32} \) | |
| 59 | \( ( 1 - 2948603619524520632 T^{2} + \)\(39\!\cdots\!88\)\( T^{4} - \)\(33\!\cdots\!44\)\( T^{6} + \)\(19\!\cdots\!70\)\( T^{8} - \)\(33\!\cdots\!44\)\( p^{20} T^{10} + \)\(39\!\cdots\!88\)\( p^{40} T^{12} - 2948603619524520632 p^{60} T^{14} + p^{80} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 894215708 T + 1013888684766468778 T^{2} - \)\(91\!\cdots\!56\)\( T^{3} + \)\(71\!\cdots\!95\)\( T^{4} - \)\(91\!\cdots\!56\)\( p^{10} T^{5} + 1013888684766468778 p^{20} T^{6} - 894215708 p^{30} T^{7} + p^{40} T^{8} )^{4} \) | |
| 67 | \( 1 + \)\(12\!\cdots\!76\)\( T^{4} + \)\(71\!\cdots\!70\)\( T^{8} + \)\(22\!\cdots\!64\)\( T^{12} + \)\(64\!\cdots\!59\)\( T^{16} + \)\(22\!\cdots\!64\)\( p^{40} T^{20} + \)\(71\!\cdots\!70\)\( p^{80} T^{24} + \)\(12\!\cdots\!76\)\( p^{120} T^{28} + p^{160} T^{32} \) | |
| 71 | \( ( 1 - 463298160 T + 3205719637261167604 T^{2} + \)\(45\!\cdots\!20\)\( T^{3} - \)\(21\!\cdots\!94\)\( T^{4} + \)\(45\!\cdots\!20\)\( p^{10} T^{5} + 3205719637261167604 p^{20} T^{6} - 463298160 p^{30} T^{7} + p^{40} T^{8} )^{4} \) | |
| 73 | \( 1 + \)\(34\!\cdots\!36\)\( T^{4} + \)\(41\!\cdots\!40\)\( T^{8} - \)\(13\!\cdots\!76\)\( T^{12} - \)\(43\!\cdots\!86\)\( T^{16} - \)\(13\!\cdots\!76\)\( p^{40} T^{20} + \)\(41\!\cdots\!40\)\( p^{80} T^{24} + \)\(34\!\cdots\!36\)\( p^{120} T^{28} + p^{160} T^{32} \) | |
| 79 | \( ( 1 - 52587262505497101304 T^{2} + \)\(12\!\cdots\!60\)\( T^{4} - \)\(18\!\cdots\!16\)\( T^{6} + \)\(20\!\cdots\!94\)\( T^{8} - \)\(18\!\cdots\!16\)\( p^{20} T^{10} + \)\(12\!\cdots\!60\)\( p^{40} T^{12} - 52587262505497101304 p^{60} T^{14} + p^{80} T^{16} )^{2} \) | |
| 83 | \( 1 + \)\(13\!\cdots\!92\)\( T^{4} + \)\(15\!\cdots\!28\)\( T^{8} + \)\(25\!\cdots\!44\)\( T^{12} + \)\(11\!\cdots\!70\)\( T^{16} + \)\(25\!\cdots\!44\)\( p^{40} T^{20} + \)\(15\!\cdots\!28\)\( p^{80} T^{24} + \)\(13\!\cdots\!92\)\( p^{120} T^{28} + p^{160} T^{32} \) | |
| 89 | \( ( 1 - 97327905405050174408 T^{2} + \)\(62\!\cdots\!28\)\( T^{4} - \)\(26\!\cdots\!56\)\( T^{6} + \)\(92\!\cdots\!70\)\( T^{8} - \)\(26\!\cdots\!56\)\( p^{20} T^{10} + \)\(62\!\cdots\!28\)\( p^{40} T^{12} - 97327905405050174408 p^{60} T^{14} + p^{80} T^{16} )^{2} \) | |
| 97 | \( 1 + \)\(18\!\cdots\!92\)\( T^{4} + \)\(17\!\cdots\!78\)\( T^{8} + \)\(14\!\cdots\!44\)\( T^{12} + \)\(12\!\cdots\!95\)\( T^{16} + \)\(14\!\cdots\!44\)\( p^{40} T^{20} + \)\(17\!\cdots\!78\)\( p^{80} T^{24} + \)\(18\!\cdots\!92\)\( p^{120} T^{28} + p^{160} T^{32} \) | |
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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
−1.77399157050840749169448939855, −1.68870756677645734754313475189, −1.61846909695636509824643086209, −1.61306694460312354859928519503, −1.50384031707441918479476374739, −1.40210376971215513004586222796, −1.34454216797751286127406579960, −1.24444609141497052646315171457, −1.24173785114494554175147296270, −1.17651289213584859133700699904, −1.05252320846996518449768010455, −0.977023754214709853436064580166, −0.970191627561600650688329368094, −0.958677837887395316976400200072, −0.903405264452300844188332496236, −0.76905404561731283638452951763, −0.71691297522869950941342765177, −0.44605562088508565210871805378, −0.38831492963380222619969679053, −0.32010713503366265159692249853, −0.19622741202829308383502579333, −0.11291979207633946709021935201, −0.05761272290755584563131266527, −0.04735510250533229522833412561, −0.02060980301419660918878602185, 0.02060980301419660918878602185, 0.04735510250533229522833412561, 0.05761272290755584563131266527, 0.11291979207633946709021935201, 0.19622741202829308383502579333, 0.32010713503366265159692249853, 0.38831492963380222619969679053, 0.44605562088508565210871805378, 0.71691297522869950941342765177, 0.76905404561731283638452951763, 0.903405264452300844188332496236, 0.958677837887395316976400200072, 0.970191627561600650688329368094, 0.977023754214709853436064580166, 1.05252320846996518449768010455, 1.17651289213584859133700699904, 1.24173785114494554175147296270, 1.24444609141497052646315171457, 1.34454216797751286127406579960, 1.40210376971215513004586222796, 1.50384031707441918479476374739, 1.61306694460312354859928519503, 1.61846909695636509824643086209, 1.68870756677645734754313475189, 1.77399157050840749169448939855
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.