Dirichlet series
| L(s) = 1 | + 6·5-s + 46·13-s − 12·17-s + 103·25-s + 42·29-s − 56·37-s − 84·41-s − 167·49-s − 72·53-s + 34·61-s + 276·65-s + 116·73-s − 72·85-s + 384·89-s − 148·97-s − 330·101-s − 392·109-s − 330·113-s − 524·121-s + 1.19e3·125-s + 127-s + 131-s + 137-s + 139-s + 252·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 6/5·5-s + 3.53·13-s − 0.705·17-s + 4.11·25-s + 1.44·29-s − 1.51·37-s − 2.04·41-s − 3.40·49-s − 1.35·53-s + 0.557·61-s + 4.24·65-s + 1.58·73-s − 0.847·85-s + 4.31·89-s − 1.52·97-s − 3.26·101-s − 3.59·109-s − 2.92·113-s − 4.33·121-s + 9.55·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 1.73·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{96} \cdot 3^{48}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.83526\times 10^{26}\) |
| Root analytic conductor: | \(6.86182\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{96} \cdot 3^{48} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(2.112314015\) |
| \(L(\frac12)\) | \(\approx\) | \(2.112314015\) |
| \(L(2)\) | 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 | \( 1 \) | |
| good | 5 | \( ( 1 - 3 T - 38 T^{2} - 201 T^{3} + 299 p T^{4} + 6984 T^{5} + 24112 T^{6} - 36276 p T^{7} - 866084 T^{8} - 36276 p^{3} T^{9} + 24112 p^{4} T^{10} + 6984 p^{6} T^{11} + 299 p^{9} T^{12} - 201 p^{10} T^{13} - 38 p^{12} T^{14} - 3 p^{14} T^{15} + p^{16} T^{16} )^{2} \) |
| 7 | \( 1 + 167 T^{2} + 1884 p T^{4} + 76561 p T^{6} + 4595825 T^{8} - 715412784 T^{10} - 48284089874 T^{12} - 2073288609886 T^{14} - 88693520972328 T^{16} - 2073288609886 p^{4} T^{18} - 48284089874 p^{8} T^{20} - 715412784 p^{12} T^{22} + 4595825 p^{16} T^{24} + 76561 p^{21} T^{26} + 1884 p^{25} T^{28} + 167 p^{28} T^{30} + p^{32} T^{32} \) | |
| 11 | \( 1 + 524 T^{2} + 140094 T^{4} + 24486904 T^{6} + 2972860745 T^{8} + 215973352008 T^{10} - 2384433396482 T^{12} - 3310557806176204 T^{14} - 541778612591523276 T^{16} - 3310557806176204 p^{4} T^{18} - 2384433396482 p^{8} T^{20} + 215973352008 p^{12} T^{22} + 2972860745 p^{16} T^{24} + 24486904 p^{20} T^{26} + 140094 p^{24} T^{28} + 524 p^{28} T^{30} + p^{32} T^{32} \) | |
| 13 | \( ( 1 - 23 T - 276 T^{2} + 5009 T^{3} + 162641 T^{4} - 1572720 T^{5} - 34351730 T^{6} + 33339262 T^{7} + 8566506504 T^{8} + 33339262 p^{2} T^{9} - 34351730 p^{4} T^{10} - 1572720 p^{6} T^{11} + 162641 p^{8} T^{12} + 5009 p^{10} T^{13} - 276 p^{12} T^{14} - 23 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 17 | \( ( 1 + 3 T + 2 p^{2} T^{2} + 6549 T^{3} + 169242 T^{4} + 6549 p^{2} T^{5} + 2 p^{6} T^{6} + 3 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
| 19 | \( ( 1 - 1673 T^{2} + 1559890 T^{4} - 937716023 T^{6} + 401371470970 T^{8} - 937716023 p^{4} T^{10} + 1559890 p^{8} T^{12} - 1673 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 23 | \( 1 + 2687 T^{2} + 161724 p T^{4} + 3461634655 T^{6} + 2442367009985 T^{8} + 1429403163693456 T^{10} + 759315200173466974 T^{12} + \)\(39\!\cdots\!18\)\( T^{14} + \)\(20\!\cdots\!24\)\( T^{16} + \)\(39\!\cdots\!18\)\( p^{4} T^{18} + 759315200173466974 p^{8} T^{20} + 1429403163693456 p^{12} T^{22} + 2442367009985 p^{16} T^{24} + 3461634655 p^{20} T^{26} + 161724 p^{25} T^{28} + 2687 p^{28} T^{30} + p^{32} T^{32} \) | |
| 29 | \( ( 1 - 21 T - 2432 T^{2} + 19167 T^{3} + 4062037 T^{4} - 228384 p T^{5} - 4703569190 T^{6} + 3469324686 T^{7} + 4129311885376 T^{8} + 3469324686 p^{2} T^{9} - 4703569190 p^{4} T^{10} - 228384 p^{7} T^{11} + 4062037 p^{8} T^{12} + 19167 p^{10} T^{13} - 2432 p^{12} T^{14} - 21 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 31 | \( 1 + 4715 T^{2} + 10729584 T^{4} + 17585852527 T^{6} + 25170831884477 T^{8} + 32193274685973408 T^{10} + 37136398859226646834 T^{12} + \)\(40\!\cdots\!98\)\( T^{14} + \)\(41\!\cdots\!28\)\( T^{16} + \)\(40\!\cdots\!98\)\( p^{4} T^{18} + 37136398859226646834 p^{8} T^{20} + 32193274685973408 p^{12} T^{22} + 25170831884477 p^{16} T^{24} + 17585852527 p^{20} T^{26} + 10729584 p^{24} T^{28} + 4715 p^{28} T^{30} + p^{32} T^{32} \) | |
| 37 | \( ( 1 + 14 T + 3040 T^{2} + 66050 T^{3} + 123814 p T^{4} + 66050 p^{2} T^{5} + 3040 p^{4} T^{6} + 14 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
| 41 | \( ( 1 + 42 T - 4424 T^{2} - 125040 T^{3} + 14943835 T^{4} + 232367040 T^{5} - 36025798940 T^{6} - 132403432026 T^{7} + 71285616374608 T^{8} - 132403432026 p^{2} T^{9} - 36025798940 p^{4} T^{10} + 232367040 p^{6} T^{11} + 14943835 p^{8} T^{12} - 125040 p^{10} T^{13} - 4424 p^{12} T^{14} + 42 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 43 | \( 1 + 10292 T^{2} + 52819302 T^{4} + 202099368136 T^{6} + 665872758097265 T^{8} + 1877680374529631208 T^{10} + \)\(45\!\cdots\!90\)\( T^{12} + \)\(10\!\cdots\!64\)\( T^{14} + \)\(19\!\cdots\!28\)\( T^{16} + \)\(10\!\cdots\!64\)\( p^{4} T^{18} + \)\(45\!\cdots\!90\)\( p^{8} T^{20} + 1877680374529631208 p^{12} T^{22} + 665872758097265 p^{16} T^{24} + 202099368136 p^{20} T^{26} + 52819302 p^{24} T^{28} + 10292 p^{28} T^{30} + p^{32} T^{32} \) | |
| 47 | \( 1 + 12983 T^{2} + 90223140 T^{4} + 428939892679 T^{6} + 33021025933951 p T^{8} + 4556649670813575888 T^{10} + \)\(11\!\cdots\!54\)\( T^{12} + \)\(26\!\cdots\!06\)\( T^{14} + \)\(58\!\cdots\!04\)\( T^{16} + \)\(26\!\cdots\!06\)\( p^{4} T^{18} + \)\(11\!\cdots\!54\)\( p^{8} T^{20} + 4556649670813575888 p^{12} T^{22} + 33021025933951 p^{17} T^{24} + 428939892679 p^{20} T^{26} + 90223140 p^{24} T^{28} + 12983 p^{28} T^{30} + p^{32} T^{32} \) | |
| 53 | \( ( 1 + 18 T + 10016 T^{2} + 126558 T^{3} + 40472766 T^{4} + 126558 p^{2} T^{5} + 10016 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
| 59 | \( 1 + 18092 T^{2} + 2981418 p T^{4} + 1133035156984 T^{6} + 5240167912552121 T^{8} + 17175194950853605128 T^{10} + \)\(35\!\cdots\!18\)\( T^{12} + \)\(21\!\cdots\!16\)\( T^{14} - \)\(83\!\cdots\!68\)\( T^{16} + \)\(21\!\cdots\!16\)\( p^{4} T^{18} + \)\(35\!\cdots\!18\)\( p^{8} T^{20} + 17175194950853605128 p^{12} T^{22} + 5240167912552121 p^{16} T^{24} + 1133035156984 p^{20} T^{26} + 2981418 p^{25} T^{28} + 18092 p^{28} T^{30} + p^{32} T^{32} \) | |
| 61 | \( ( 1 - 17 T - 3582 T^{2} + 125549 T^{3} - 18305593 T^{4} + 350178696 T^{5} - 12273736208 T^{6} - 2060229775052 T^{7} + 599123836260396 T^{8} - 2060229775052 p^{2} T^{9} - 12273736208 p^{4} T^{10} + 350178696 p^{6} T^{11} - 18305593 p^{8} T^{12} + 125549 p^{10} T^{13} - 3582 p^{12} T^{14} - 17 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 67 | \( 1 + 25148 T^{2} + 330341646 T^{4} + 3017899283800 T^{6} + 21559886998912025 T^{8} + \)\(12\!\cdots\!60\)\( T^{10} + \)\(66\!\cdots\!50\)\( T^{12} + \)\(31\!\cdots\!76\)\( T^{14} + \)\(14\!\cdots\!68\)\( T^{16} + \)\(31\!\cdots\!76\)\( p^{4} T^{18} + \)\(66\!\cdots\!50\)\( p^{8} T^{20} + \)\(12\!\cdots\!60\)\( p^{12} T^{22} + 21559886998912025 p^{16} T^{24} + 3017899283800 p^{20} T^{26} + 330341646 p^{24} T^{28} + 25148 p^{28} T^{30} + p^{32} T^{32} \) | |
| 71 | \( ( 1 - 12968 T^{2} + 137492380 T^{4} - 912038324888 T^{6} + 5454839368725190 T^{8} - 912038324888 p^{4} T^{10} + 137492380 p^{8} T^{12} - 12968 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 29 T + 7774 T^{2} - 620747 T^{3} + 46170850 T^{4} - 620747 p^{2} T^{5} + 7774 p^{4} T^{6} - 29 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
| 79 | \( 1 + 293 p T^{2} + 328058736 T^{4} + 3136165224559 T^{6} + 21315067551811709 T^{8} + 91269506056145418912 T^{10} + \)\(59\!\cdots\!06\)\( T^{12} - \)\(27\!\cdots\!10\)\( T^{14} - \)\(25\!\cdots\!44\)\( T^{16} - \)\(27\!\cdots\!10\)\( p^{4} T^{18} + \)\(59\!\cdots\!06\)\( p^{8} T^{20} + 91269506056145418912 p^{12} T^{22} + 21315067551811709 p^{16} T^{24} + 3136165224559 p^{20} T^{26} + 328058736 p^{24} T^{28} + 293 p^{29} T^{30} + p^{32} T^{32} \) | |
| 83 | \( 1 + 17795 T^{2} + 28452672 T^{4} - 239952438809 T^{6} + 12519030554664557 T^{8} + 90364909803409302048 T^{10} - \)\(32\!\cdots\!46\)\( T^{12} + \)\(11\!\cdots\!22\)\( T^{14} + \)\(52\!\cdots\!80\)\( T^{16} + \)\(11\!\cdots\!22\)\( p^{4} T^{18} - \)\(32\!\cdots\!46\)\( p^{8} T^{20} + 90364909803409302048 p^{12} T^{22} + 12519030554664557 p^{16} T^{24} - 239952438809 p^{20} T^{26} + 28452672 p^{24} T^{28} + 17795 p^{28} T^{30} + p^{32} T^{32} \) | |
| 89 | \( ( 1 - 96 T + 27716 T^{2} - 1940016 T^{3} + 308634966 T^{4} - 1940016 p^{2} T^{5} + 27716 p^{4} T^{6} - 96 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
| 97 | \( ( 1 + 74 T - 29868 T^{2} - 1540328 T^{3} + 607324823 T^{4} + 20751693936 T^{5} - 8122914917000 T^{6} - 74589814314322 T^{7} + 88320904907559480 T^{8} - 74589814314322 p^{2} T^{9} - 8122914917000 p^{4} T^{10} + 20751693936 p^{6} T^{11} + 607324823 p^{8} T^{12} - 1540328 p^{10} T^{13} - 29868 p^{12} T^{14} + 74 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
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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
−2.21254363821660602718311964442, −2.04600168649768467236288895028, −1.92184264970924322566090167002, −1.92125174292803829960118368902, −1.76911424712179096315048078091, −1.62328829075793675299727610206, −1.55593261791739602058338777015, −1.50354000524587270392901385842, −1.45224044012686706142442617258, −1.36587670546414484891444569351, −1.34792359121168511273762461265, −1.34614047518433391127153022281, −1.29183292968759141083709220295, −1.27794087154710350581077675613, −1.00229021932008264780434647622, −0.898317860190383048832897532570, −0.854140105463020107622608795919, −0.812561919857023008717115823358, −0.810394825176548256394342789308, −0.54585719847499886023594411804, −0.37201140736744352305994127879, −0.32373791750791630967821300347, −0.32004032463383253297136626806, −0.23551794440637781388705759848, −0.02550347047533722113401493656, 0.02550347047533722113401493656, 0.23551794440637781388705759848, 0.32004032463383253297136626806, 0.32373791750791630967821300347, 0.37201140736744352305994127879, 0.54585719847499886023594411804, 0.810394825176548256394342789308, 0.812561919857023008717115823358, 0.854140105463020107622608795919, 0.898317860190383048832897532570, 1.00229021932008264780434647622, 1.27794087154710350581077675613, 1.29183292968759141083709220295, 1.34614047518433391127153022281, 1.34792359121168511273762461265, 1.36587670546414484891444569351, 1.45224044012686706142442617258, 1.50354000524587270392901385842, 1.55593261791739602058338777015, 1.62328829075793675299727610206, 1.76911424712179096315048078091, 1.92125174292803829960118368902, 1.92184264970924322566090167002, 2.04600168649768467236288895028, 2.21254363821660602718311964442
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.