Dirichlet series
| L(s) = 1 | + 4·4-s + 12·11-s + 6·13-s + 6·16-s − 36·17-s − 6·23-s + 16·25-s − 6·29-s + 6·31-s + 4·37-s + 6·41-s − 2·43-s + 48·44-s − 18·47-s + 24·52-s + 30·59-s − 60·61-s + 14·67-s − 144·68-s − 16·79-s − 48·89-s − 24·92-s + 6·97-s + 64·100-s + 48·101-s − 42·103-s − 8·109-s + ⋯ |
| L(s) = 1 | + 2·4-s + 3.61·11-s + 1.66·13-s + 3/2·16-s − 8.73·17-s − 1.25·23-s + 16/5·25-s − 1.11·29-s + 1.07·31-s + 0.657·37-s + 0.937·41-s − 0.304·43-s + 7.23·44-s − 2.62·47-s + 3.32·52-s + 3.90·59-s − 7.68·61-s + 1.71·67-s − 17.4·68-s − 1.80·79-s − 5.08·89-s − 2.50·92-s + 0.609·97-s + 32/5·100-s + 4.77·101-s − 4.13·103-s − 0.766·109-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{48} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{48} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{48} \cdot 7^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.57714\times 10^{21}\) |
| Root analytic conductor: | \(4.59656\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{48} \cdot 7^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(22.42560551\) |
| \(L(\frac12)\) | \(\approx\) | \(22.42560551\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 - 16 T^{2} + 24 T^{3} + 123 T^{4} - 66 p T^{5} - 98 T^{6} + 2124 T^{7} - 4594 T^{8} + 102 T^{9} + 38946 T^{10} - 84396 T^{11} - 70916 T^{12} + 641196 T^{13} - 735151 T^{14} - 1444464 T^{15} + 6891966 T^{16} - 1444464 p T^{17} - 735151 p^{2} T^{18} + 641196 p^{3} T^{19} - 70916 p^{4} T^{20} - 84396 p^{5} T^{21} + 38946 p^{6} T^{22} + 102 p^{7} T^{23} - 4594 p^{8} T^{24} + 2124 p^{9} T^{25} - 98 p^{10} T^{26} - 66 p^{12} T^{27} + 123 p^{12} T^{28} + 24 p^{13} T^{29} - 16 p^{14} T^{30} + p^{16} T^{32} \) |
| 11 | \( 1 - 12 T + 106 T^{2} - 696 T^{3} + 3735 T^{4} - 17808 T^{5} + 77432 T^{6} - 326142 T^{7} + 1351880 T^{8} - 5458080 T^{9} + 1935918 p T^{10} - 79655676 T^{11} + 287397262 T^{12} - 1023482544 T^{13} + 3603683689 T^{14} - 12488798178 T^{15} + 42242848866 T^{16} - 12488798178 p T^{17} + 3603683689 p^{2} T^{18} - 1023482544 p^{3} T^{19} + 287397262 p^{4} T^{20} - 79655676 p^{5} T^{21} + 1935918 p^{7} T^{22} - 5458080 p^{7} T^{23} + 1351880 p^{8} T^{24} - 326142 p^{9} T^{25} + 77432 p^{10} T^{26} - 17808 p^{11} T^{27} + 3735 p^{12} T^{28} - 696 p^{13} T^{29} + 106 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 - 6 T + 47 T^{2} - 210 T^{3} + 888 T^{4} - 2658 T^{5} + 9181 T^{6} - 12564 T^{7} + 21641 T^{8} + 214770 T^{9} - 1221396 T^{10} + 6263526 T^{11} - 19011605 T^{12} + 72302586 T^{13} - 179114935 T^{14} + 57128796 p T^{15} - 178472592 p T^{16} + 57128796 p^{2} T^{17} - 179114935 p^{2} T^{18} + 72302586 p^{3} T^{19} - 19011605 p^{4} T^{20} + 6263526 p^{5} T^{21} - 1221396 p^{6} T^{22} + 214770 p^{7} T^{23} + 21641 p^{8} T^{24} - 12564 p^{9} T^{25} + 9181 p^{10} T^{26} - 2658 p^{11} T^{27} + 888 p^{12} T^{28} - 210 p^{13} T^{29} + 47 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( ( 1 + 18 T + 229 T^{2} + 2082 T^{3} + 15814 T^{4} + 100110 T^{5} + 558907 T^{6} + 160350 p T^{7} + 11952946 T^{8} + 160350 p^{2} T^{9} + 558907 p^{2} T^{10} + 100110 p^{3} T^{11} + 15814 p^{4} T^{12} + 2082 p^{5} T^{13} + 229 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 - 160 T^{2} + 12234 T^{4} - 592310 T^{6} + 20333351 T^{8} - 528417264 T^{10} + 11029384681 T^{12} - 201981615544 T^{14} + 3695387217759 T^{16} - 201981615544 p^{2} T^{18} + 11029384681 p^{4} T^{20} - 528417264 p^{6} T^{22} + 20333351 p^{8} T^{24} - 592310 p^{10} T^{26} + 12234 p^{12} T^{28} - 160 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 + 6 T + 130 T^{2} + 708 T^{3} + 8559 T^{4} + 45012 T^{5} + 399452 T^{6} + 2115900 T^{7} + 15122540 T^{8} + 81277422 T^{9} + 486352566 T^{10} + 2645846004 T^{11} + 13727139058 T^{12} + 75456395706 T^{13} + 352786829089 T^{14} + 1928667059838 T^{15} + 8415100540206 T^{16} + 1928667059838 p T^{17} + 352786829089 p^{2} T^{18} + 75456395706 p^{3} T^{19} + 13727139058 p^{4} T^{20} + 2645846004 p^{5} T^{21} + 486352566 p^{6} T^{22} + 81277422 p^{7} T^{23} + 15122540 p^{8} T^{24} + 2115900 p^{9} T^{25} + 399452 p^{10} T^{26} + 45012 p^{11} T^{27} + 8559 p^{12} T^{28} + 708 p^{13} T^{29} + 130 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 + 6 T + 196 T^{2} + 1104 T^{3} + 19989 T^{4} + 108858 T^{5} + 1435478 T^{6} + 7594740 T^{7} + 81322766 T^{8} + 416245488 T^{9} + 3844091772 T^{10} + 18872529852 T^{11} + 155981060614 T^{12} + 727660406082 T^{13} + 5509975202215 T^{14} + 24244157784798 T^{15} + 170557455776958 T^{16} + 24244157784798 p T^{17} + 5509975202215 p^{2} T^{18} + 727660406082 p^{3} T^{19} + 155981060614 p^{4} T^{20} + 18872529852 p^{5} T^{21} + 3844091772 p^{6} T^{22} + 416245488 p^{7} T^{23} + 81322766 p^{8} T^{24} + 7594740 p^{9} T^{25} + 1435478 p^{10} T^{26} + 108858 p^{11} T^{27} + 19989 p^{12} T^{28} + 1104 p^{13} T^{29} + 196 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 - 6 T + 164 T^{2} - 912 T^{3} + 14145 T^{4} - 82056 T^{5} + 882730 T^{6} - 5457276 T^{7} + 44537675 T^{8} - 289281588 T^{9} + 1950112812 T^{10} - 12838947000 T^{11} + 76837130221 T^{12} - 493000882878 T^{13} + 2756895706292 T^{14} - 16890347347524 T^{15} + 89911869890409 T^{16} - 16890347347524 p T^{17} + 2756895706292 p^{2} T^{18} - 493000882878 p^{3} T^{19} + 76837130221 p^{4} T^{20} - 12838947000 p^{5} T^{21} + 1950112812 p^{6} T^{22} - 289281588 p^{7} T^{23} + 44537675 p^{8} T^{24} - 5457276 p^{9} T^{25} + 882730 p^{10} T^{26} - 82056 p^{11} T^{27} + 14145 p^{12} T^{28} - 912 p^{13} T^{29} + 164 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( ( 1 - 2 T + 120 T^{2} - 448 T^{3} + 9650 T^{4} - 33378 T^{5} + 561256 T^{6} - 1817306 T^{7} + 23338683 T^{8} - 1817306 p T^{9} + 561256 p^{2} T^{10} - 33378 p^{3} T^{11} + 9650 p^{4} T^{12} - 448 p^{5} T^{13} + 120 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( 1 - 6 T - 223 T^{2} + 1686 T^{3} + 25980 T^{4} - 231654 T^{5} - 1971341 T^{6} + 20408106 T^{7} + 109216031 T^{8} - 1268388768 T^{9} - 4836103872 T^{10} + 1400930916 p T^{11} + 192545389345 T^{12} - 1824668193534 T^{13} - 7677147470143 T^{14} + 27964729912410 T^{15} + 313424888729076 T^{16} + 27964729912410 p T^{17} - 7677147470143 p^{2} T^{18} - 1824668193534 p^{3} T^{19} + 192545389345 p^{4} T^{20} + 1400930916 p^{6} T^{21} - 4836103872 p^{6} T^{22} - 1268388768 p^{7} T^{23} + 109216031 p^{8} T^{24} + 20408106 p^{9} T^{25} - 1971341 p^{10} T^{26} - 231654 p^{11} T^{27} + 25980 p^{12} T^{28} + 1686 p^{13} T^{29} - 223 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 + 2 T - 209 T^{2} - 14 p T^{3} + 20774 T^{4} + 72052 T^{5} - 1457073 T^{6} - 4933350 T^{7} + 91147851 T^{8} + 235722522 T^{9} - 5466637218 T^{10} - 9144255228 T^{11} + 301177025103 T^{12} + 290829011802 T^{13} - 14773623661707 T^{14} - 4808085837138 T^{15} + 658882369500660 T^{16} - 4808085837138 p T^{17} - 14773623661707 p^{2} T^{18} + 290829011802 p^{3} T^{19} + 301177025103 p^{4} T^{20} - 9144255228 p^{5} T^{21} - 5466637218 p^{6} T^{22} + 235722522 p^{7} T^{23} + 91147851 p^{8} T^{24} - 4933350 p^{9} T^{25} - 1457073 p^{10} T^{26} + 72052 p^{11} T^{27} + 20774 p^{12} T^{28} - 14 p^{14} T^{29} - 209 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 + 18 T - 55 T^{2} - 1722 T^{3} + 12957 T^{4} + 149790 T^{5} - 1485620 T^{6} - 10153800 T^{7} + 96840002 T^{8} + 357022848 T^{9} - 6832958010 T^{10} - 20400467106 T^{11} + 370230517390 T^{12} + 853239339180 T^{13} - 20043242784151 T^{14} - 7467517424682 T^{15} + 1132351959046185 T^{16} - 7467517424682 p T^{17} - 20043242784151 p^{2} T^{18} + 853239339180 p^{3} T^{19} + 370230517390 p^{4} T^{20} - 20400467106 p^{5} T^{21} - 6832958010 p^{6} T^{22} + 357022848 p^{7} T^{23} + 96840002 p^{8} T^{24} - 10153800 p^{9} T^{25} - 1485620 p^{10} T^{26} + 149790 p^{11} T^{27} + 12957 p^{12} T^{28} - 1722 p^{13} T^{29} - 55 p^{14} T^{30} + 18 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 - 236 T^{2} + 26670 T^{4} - 2006170 T^{6} + 117433019 T^{8} - 6405482184 T^{10} + 397778936677 T^{12} - 26197224090524 T^{14} + 1521912137360247 T^{16} - 26197224090524 p^{2} T^{18} + 397778936677 p^{4} T^{20} - 6405482184 p^{6} T^{22} + 117433019 p^{8} T^{24} - 2006170 p^{10} T^{26} + 26670 p^{12} T^{28} - 236 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( 1 - 30 T + 200 T^{2} + 1272 T^{3} + 1905 T^{4} - 286356 T^{5} - 330074 T^{6} + 21958908 T^{7} + 129465491 T^{8} - 1641971784 T^{9} - 13037705556 T^{10} + 82617584088 T^{11} + 950143418149 T^{12} - 1952014889118 T^{13} - 70581518213368 T^{14} + 49596128462484 T^{15} + 4103530612541721 T^{16} + 49596128462484 p T^{17} - 70581518213368 p^{2} T^{18} - 1952014889118 p^{3} T^{19} + 950143418149 p^{4} T^{20} + 82617584088 p^{5} T^{21} - 13037705556 p^{6} T^{22} - 1641971784 p^{7} T^{23} + 129465491 p^{8} T^{24} + 21958908 p^{9} T^{25} - 330074 p^{10} T^{26} - 286356 p^{11} T^{27} + 1905 p^{12} T^{28} + 1272 p^{13} T^{29} + 200 p^{14} T^{30} - 30 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 + 60 T + 2036 T^{2} + 50160 T^{3} + 994527 T^{4} + 16748022 T^{5} + 247695826 T^{6} + 3290908266 T^{7} + 39930949007 T^{8} + 448092194298 T^{9} + 4697101745298 T^{10} + 46372941826116 T^{11} + 434191490396749 T^{12} + 3878280141400200 T^{13} + 33210009451501436 T^{14} + 273658668372668376 T^{15} + 2175265354887641205 T^{16} + 273658668372668376 p T^{17} + 33210009451501436 p^{2} T^{18} + 3878280141400200 p^{3} T^{19} + 434191490396749 p^{4} T^{20} + 46372941826116 p^{5} T^{21} + 4697101745298 p^{6} T^{22} + 448092194298 p^{7} T^{23} + 39930949007 p^{8} T^{24} + 3290908266 p^{9} T^{25} + 247695826 p^{10} T^{26} + 16748022 p^{11} T^{27} + 994527 p^{12} T^{28} + 50160 p^{13} T^{29} + 2036 p^{14} T^{30} + 60 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 - 14 T - 239 T^{2} + 5102 T^{3} + 20453 T^{4} - 910354 T^{5} + 617796 T^{6} + 105135588 T^{7} - 361795050 T^{8} - 8998911156 T^{9} + 52479815982 T^{10} + 604126066254 T^{11} - 5379501807402 T^{12} - 30548371355232 T^{13} + 457126004374545 T^{14} + 769012940073186 T^{15} - 33126987823382439 T^{16} + 769012940073186 p T^{17} + 457126004374545 p^{2} T^{18} - 30548371355232 p^{3} T^{19} - 5379501807402 p^{4} T^{20} + 604126066254 p^{5} T^{21} + 52479815982 p^{6} T^{22} - 8998911156 p^{7} T^{23} - 361795050 p^{8} T^{24} + 105135588 p^{9} T^{25} + 617796 p^{10} T^{26} - 910354 p^{11} T^{27} + 20453 p^{12} T^{28} + 5102 p^{13} T^{29} - 239 p^{14} T^{30} - 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 - 650 T^{2} + 199389 T^{4} - 38930632 T^{6} + 5566228364 T^{8} - 640511863116 T^{10} + 63163988645884 T^{12} - 5475157404521894 T^{14} + 416179213677825948 T^{16} - 5475157404521894 p^{2} T^{18} + 63163988645884 p^{4} T^{20} - 640511863116 p^{6} T^{22} + 5566228364 p^{8} T^{24} - 38930632 p^{10} T^{26} + 199389 p^{12} T^{28} - 650 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( 1 - 868 T^{2} + 362562 T^{4} - 97046486 T^{6} + 18706634819 T^{8} - 2766649955820 T^{10} + 326423690571037 T^{12} - 31489561186704028 T^{14} + 2518726833968266143 T^{16} - 31489561186704028 p^{2} T^{18} + 326423690571037 p^{4} T^{20} - 2766649955820 p^{6} T^{22} + 18706634819 p^{8} T^{24} - 97046486 p^{10} T^{26} + 362562 p^{12} T^{28} - 868 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( 1 + 16 T - 227 T^{2} - 5356 T^{3} + 15491 T^{4} + 818138 T^{5} + 465174 T^{6} - 79955730 T^{7} - 161428446 T^{8} + 5129891178 T^{9} + 6184544046 T^{10} - 147661653486 T^{11} + 2275304241738 T^{12} - 3934261822074 T^{13} - 453211437102513 T^{14} + 289666157971230 T^{15} + 45106559543464353 T^{16} + 289666157971230 p T^{17} - 453211437102513 p^{2} T^{18} - 3934261822074 p^{3} T^{19} + 2275304241738 p^{4} T^{20} - 147661653486 p^{5} T^{21} + 6184544046 p^{6} T^{22} + 5129891178 p^{7} T^{23} - 161428446 p^{8} T^{24} - 79955730 p^{9} T^{25} + 465174 p^{10} T^{26} + 818138 p^{11} T^{27} + 15491 p^{12} T^{28} - 5356 p^{13} T^{29} - 227 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 - 487 T^{2} - 312 T^{3} + 123774 T^{4} + 132990 T^{5} - 22183883 T^{6} - 29138634 T^{7} + 3170469341 T^{8} + 4110229572 T^{9} - 386467088226 T^{10} - 393115428402 T^{11} + 41656592194789 T^{12} + 25282823866380 T^{13} - 4030130568645907 T^{14} - 780079655467782 T^{15} + 351851707607703156 T^{16} - 780079655467782 p T^{17} - 4030130568645907 p^{2} T^{18} + 25282823866380 p^{3} T^{19} + 41656592194789 p^{4} T^{20} - 393115428402 p^{5} T^{21} - 386467088226 p^{6} T^{22} + 4110229572 p^{7} T^{23} + 3170469341 p^{8} T^{24} - 29138634 p^{9} T^{25} - 22183883 p^{10} T^{26} + 132990 p^{11} T^{27} + 123774 p^{12} T^{28} - 312 p^{13} T^{29} - 487 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 24 T + 586 T^{2} + 8850 T^{3} + 129799 T^{4} + 1587672 T^{5} + 18818377 T^{6} + 201921036 T^{7} + 2005586395 T^{8} + 201921036 p T^{9} + 18818377 p^{2} T^{10} + 1587672 p^{3} T^{11} + 129799 p^{4} T^{12} + 8850 p^{5} T^{13} + 586 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 - 6 T + 395 T^{2} - 2298 T^{3} + 68379 T^{4} - 571872 T^{5} + 8922340 T^{6} - 109039644 T^{7} + 1233764039 T^{8} - 14418445794 T^{9} + 171203907675 T^{10} - 1605167799666 T^{11} + 21002714173786 T^{12} - 194363495836338 T^{13} + 2133943680044999 T^{14} - 22669145285986746 T^{15} + 200280216945639852 T^{16} - 22669145285986746 p T^{17} + 2133943680044999 p^{2} T^{18} - 194363495836338 p^{3} T^{19} + 21002714173786 p^{4} T^{20} - 1605167799666 p^{5} T^{21} + 171203907675 p^{6} T^{22} - 14418445794 p^{7} T^{23} + 1233764039 p^{8} T^{24} - 109039644 p^{9} T^{25} + 8922340 p^{10} T^{26} - 571872 p^{11} T^{27} + 68379 p^{12} T^{28} - 2298 p^{13} T^{29} + 395 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−2.21109745514938240048527149195, −2.17015561049134002907113594950, −2.02502353949336219907902175738, −1.85263835357145232301796634589, −1.80575379437515925960661770448, −1.68897135297855767129434182352, −1.67237449472486625279760812364, −1.62673688776003336848355651797, −1.60502656401518003550531674886, −1.59409928359158496858379734988, −1.59167193094821900333948657553, −1.57943604829284642115484034654, −1.35790983184296411112198020882, −1.26116240922118748072791123922, −1.20869949135131539920316883346, −1.03495380020935980676716212865, −0.969110174273724350974627711791, −0.965746789316351674721834741205, −0.78710079551593638986395535848, −0.61862119694491507250989034533, −0.45357552346232670139221274578, −0.33827859858776540564058116957, −0.31590253289337403104240652563, −0.21056898691418579822208400192, −0.20695162977980528163057222717, 0.20695162977980528163057222717, 0.21056898691418579822208400192, 0.31590253289337403104240652563, 0.33827859858776540564058116957, 0.45357552346232670139221274578, 0.61862119694491507250989034533, 0.78710079551593638986395535848, 0.965746789316351674721834741205, 0.969110174273724350974627711791, 1.03495380020935980676716212865, 1.20869949135131539920316883346, 1.26116240922118748072791123922, 1.35790983184296411112198020882, 1.57943604829284642115484034654, 1.59167193094821900333948657553, 1.59409928359158496858379734988, 1.60502656401518003550531674886, 1.62673688776003336848355651797, 1.67237449472486625279760812364, 1.68897135297855767129434182352, 1.80575379437515925960661770448, 1.85263835357145232301796634589, 2.02502353949336219907902175738, 2.17015561049134002907113594950, 2.21109745514938240048527149195
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.