Dirichlet series
| L(s) = 1 | − 12·3-s − 12·5-s − 14·7-s + 54·9-s + 40·11-s − 60·13-s + 144·15-s − 98·17-s + 124·19-s + 168·21-s + 104·23-s − 486·25-s − 194·29-s − 52·31-s − 480·33-s + 168·35-s − 102·37-s + 720·39-s + 1.05e3·41-s + 450·43-s − 648·45-s + 192·47-s + 249·49-s + 1.17e3·51-s + 524·53-s − 480·55-s − 1.48e3·57-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1.07·5-s − 0.755·7-s + 2·9-s + 1.09·11-s − 1.28·13-s + 2.47·15-s − 1.39·17-s + 1.49·19-s + 1.74·21-s + 0.942·23-s − 3.88·25-s − 1.24·29-s − 0.301·31-s − 2.53·33-s + 0.811·35-s − 0.453·37-s + 2.95·39-s + 4.01·41-s + 1.59·43-s − 2.14·45-s + 0.595·47-s + 0.725·49-s + 3.22·51-s + 1.35·53-s − 1.17·55-s − 3.45·57-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 3^{8} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.37602\times 10^{12}\) |
| Root analytic conductor: | \(6.06771\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 3^{8} \cdot 13^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.1339558316\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1339558316\) |
| \(L(\frac{5}{2})\) | 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 | \( ( 1 + p T + p^{2} T^{2} )^{4} \) | |
| 13 | \( 1 + 60 T + 506 T^{2} - 6720 p T^{3} - 20229 p^{2} T^{4} - 6720 p^{4} T^{5} + 506 p^{6} T^{6} + 60 p^{9} T^{7} + p^{12} T^{8} \) | |
| good | 5 | \( ( 1 + 6 T + 297 T^{2} + 2094 T^{3} + 49084 T^{4} + 2094 p^{3} T^{5} + 297 p^{6} T^{6} + 6 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 7 | \( 1 + 2 p T - 53 T^{2} - 22 T^{3} - 104283 T^{4} - 2930016 T^{5} + 17952738 T^{6} + 924058068 T^{7} + 13714552090 T^{8} + 924058068 p^{3} T^{9} + 17952738 p^{6} T^{10} - 2930016 p^{9} T^{11} - 104283 p^{12} T^{12} - 22 p^{15} T^{13} - 53 p^{18} T^{14} + 2 p^{22} T^{15} + p^{24} T^{16} \) | |
| 11 | \( 1 - 40 T - 2468 T^{2} + 70512 T^{3} + 4570682 T^{4} - 44545624 T^{5} - 7946436624 T^{6} + 27402190264 T^{7} + 10473836567683 T^{8} + 27402190264 p^{3} T^{9} - 7946436624 p^{6} T^{10} - 44545624 p^{9} T^{11} + 4570682 p^{12} T^{12} + 70512 p^{15} T^{13} - 2468 p^{18} T^{14} - 40 p^{21} T^{15} + p^{24} T^{16} \) | |
| 17 | \( 1 + 98 T - 2793 T^{2} - 757474 T^{3} - 39070747 T^{4} - 1766559356 T^{5} - 5603508198 p T^{6} + 15598264298880 T^{7} + 2444458128770490 T^{8} + 15598264298880 p^{3} T^{9} - 5603508198 p^{7} T^{10} - 1766559356 p^{9} T^{11} - 39070747 p^{12} T^{12} - 757474 p^{15} T^{13} - 2793 p^{18} T^{14} + 98 p^{21} T^{15} + p^{24} T^{16} \) | |
| 19 | \( 1 - 124 T - 6128 T^{2} + 930872 T^{3} + 36764706 T^{4} - 766191780 T^{5} - 628347242016 T^{6} + 685991638980 T^{7} + 4819313934921283 T^{8} + 685991638980 p^{3} T^{9} - 628347242016 p^{6} T^{10} - 766191780 p^{9} T^{11} + 36764706 p^{12} T^{12} + 930872 p^{15} T^{13} - 6128 p^{18} T^{14} - 124 p^{21} T^{15} + p^{24} T^{16} \) | |
| 23 | \( 1 - 104 T - 31028 T^{2} + 2329392 T^{3} + 683910986 T^{4} - 1337123944 p T^{5} - 10789205119440 T^{6} + 166104063875672 T^{7} + 139878980554581235 T^{8} + 166104063875672 p^{3} T^{9} - 10789205119440 p^{6} T^{10} - 1337123944 p^{10} T^{11} + 683910986 p^{12} T^{12} + 2329392 p^{15} T^{13} - 31028 p^{18} T^{14} - 104 p^{21} T^{15} + p^{24} T^{16} \) | |
| 29 | \( 1 + 194 T - 27125 T^{2} - 1865874 T^{3} + 1099650497 T^{4} - 14338327924 T^{5} - 14789787237114 T^{6} - 276586056146240 T^{7} - 154105790501586134 T^{8} - 276586056146240 p^{3} T^{9} - 14789787237114 p^{6} T^{10} - 14338327924 p^{9} T^{11} + 1099650497 p^{12} T^{12} - 1865874 p^{15} T^{13} - 27125 p^{18} T^{14} + 194 p^{21} T^{15} + p^{24} T^{16} \) | |
| 31 | \( ( 1 + 26 T + 38189 T^{2} - 5395494 T^{3} + 828557428 T^{4} - 5395494 p^{3} T^{5} + 38189 p^{6} T^{6} + 26 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 37 | \( 1 + 102 T - 182877 T^{2} - 9931142 T^{3} + 21369360081 T^{4} + 662655005764 T^{5} - 1663037197595306 T^{6} - 11821658162464608 T^{7} + 98704436398107267850 T^{8} - 11821658162464608 p^{3} T^{9} - 1663037197595306 p^{6} T^{10} + 662655005764 p^{9} T^{11} + 21369360081 p^{12} T^{12} - 9931142 p^{15} T^{13} - 182877 p^{18} T^{14} + 102 p^{21} T^{15} + p^{24} T^{16} \) | |
| 41 | \( 1 - 1054 T + 477447 T^{2} - 145322050 T^{3} + 47162611541 T^{4} - 17254208855276 T^{5} + 5316483691339818 T^{6} - 1302078301207983120 T^{7} + \)\(31\!\cdots\!70\)\( T^{8} - 1302078301207983120 p^{3} T^{9} + 5316483691339818 p^{6} T^{10} - 17254208855276 p^{9} T^{11} + 47162611541 p^{12} T^{12} - 145322050 p^{15} T^{13} + 477447 p^{18} T^{14} - 1054 p^{21} T^{15} + p^{24} T^{16} \) | |
| 43 | \( 1 - 450 T - 74253 T^{2} + 26582090 T^{3} + 13587940893 T^{4} - 817309487800 T^{5} - 1521526876499654 T^{6} + 120403009441562220 T^{7} + 66607185174001367314 T^{8} + 120403009441562220 p^{3} T^{9} - 1521526876499654 p^{6} T^{10} - 817309487800 p^{9} T^{11} + 13587940893 p^{12} T^{12} + 26582090 p^{15} T^{13} - 74253 p^{18} T^{14} - 450 p^{21} T^{15} + p^{24} T^{16} \) | |
| 47 | \( ( 1 - 96 T - 19308 T^{2} - 124896 T^{3} + 17303792422 T^{4} - 124896 p^{3} T^{5} - 19308 p^{6} T^{6} - 96 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 53 | \( ( 1 - 262 T + 483789 T^{2} - 119532570 T^{3} + 100466115904 T^{4} - 119532570 p^{3} T^{5} + 483789 p^{6} T^{6} - 262 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 59 | \( 1 - 308 T - 462816 T^{2} + 26147176 T^{3} + 137934583394 T^{4} + 12328313957300 T^{5} - 28942563453319392 T^{6} - 1355850150077088468 T^{7} + \)\(50\!\cdots\!19\)\( T^{8} - 1355850150077088468 p^{3} T^{9} - 28942563453319392 p^{6} T^{10} + 12328313957300 p^{9} T^{11} + 137934583394 p^{12} T^{12} + 26147176 p^{15} T^{13} - 462816 p^{18} T^{14} - 308 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( 1 - 928 T - 179890 T^{2} + 201558976 T^{3} + 177696830049 T^{4} - 64129477359040 T^{5} - 54140507382887698 T^{6} + 2586015583756856352 T^{7} + \)\(17\!\cdots\!00\)\( T^{8} + 2586015583756856352 p^{3} T^{9} - 54140507382887698 p^{6} T^{10} - 64129477359040 p^{9} T^{11} + 177696830049 p^{12} T^{12} + 201558976 p^{15} T^{13} - 179890 p^{18} T^{14} - 928 p^{21} T^{15} + p^{24} T^{16} \) | |
| 67 | \( 1 + 1134 T + 67155 T^{2} - 454892710 T^{3} - 118569468723 T^{4} + 120072545143640 T^{5} + 65359583866727722 T^{6} - 895528247162159556 T^{7} - \)\(99\!\cdots\!82\)\( T^{8} - 895528247162159556 p^{3} T^{9} + 65359583866727722 p^{6} T^{10} + 120072545143640 p^{9} T^{11} - 118569468723 p^{12} T^{12} - 454892710 p^{15} T^{13} + 67155 p^{18} T^{14} + 1134 p^{21} T^{15} + p^{24} T^{16} \) | |
| 71 | \( 1 - 1064 T - 68948 T^{2} + 102489264 T^{3} + 203659452746 T^{4} - 2469291585560 T^{5} - 84651813258257232 T^{6} + 32716456196553983000 T^{7} - \)\(13\!\cdots\!45\)\( T^{8} + 32716456196553983000 p^{3} T^{9} - 84651813258257232 p^{6} T^{10} - 2469291585560 p^{9} T^{11} + 203659452746 p^{12} T^{12} + 102489264 p^{15} T^{13} - 68948 p^{18} T^{14} - 1064 p^{21} T^{15} + p^{24} T^{16} \) | |
| 73 | \( ( 1 - 952 T + 768042 T^{2} - 3278032 p T^{3} + 174760946603 T^{4} - 3278032 p^{4} T^{5} + 768042 p^{6} T^{6} - 952 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 79 | \( ( 1 - 746 T + 2156493 T^{2} - 1122200666 T^{3} + 1640976331028 T^{4} - 1122200666 p^{3} T^{5} + 2156493 p^{6} T^{6} - 746 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 83 | \( ( 1 - 404 T + 21824 p T^{2} - 594579060 T^{3} + 1476544322126 T^{4} - 594579060 p^{3} T^{5} + 21824 p^{7} T^{6} - 404 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 89 | \( 1 + 1620 T + 618504 T^{2} - 80603544 T^{3} - 239417118814 T^{4} - 558776902734420 T^{5} - 275784072094961664 T^{6} + 80252233301530985244 T^{7} + \)\(88\!\cdots\!47\)\( T^{8} + 80252233301530985244 p^{3} T^{9} - 275784072094961664 p^{6} T^{10} - 558776902734420 p^{9} T^{11} - 239417118814 p^{12} T^{12} - 80603544 p^{15} T^{13} + 618504 p^{18} T^{14} + 1620 p^{21} T^{15} + p^{24} T^{16} \) | |
| 97 | \( 1 + 2166 T + 530811 T^{2} - 2884171238 T^{3} - 1500352722087 T^{4} + 4495692461175436 T^{5} + 5887077858288456094 T^{6} - \)\(75\!\cdots\!36\)\( T^{7} - \)\(51\!\cdots\!26\)\( T^{8} - \)\(75\!\cdots\!36\)\( p^{3} T^{9} + 5887077858288456094 p^{6} T^{10} + 4495692461175436 p^{9} T^{11} - 1500352722087 p^{12} T^{12} - 2884171238 p^{15} T^{13} + 530811 p^{18} T^{14} + 2166 p^{21} T^{15} + p^{24} T^{16} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.04200261651840635055887491749, −3.95463641498309529153793364098, −3.90664315494120613216583277190, −3.84792678172945992339654935512, −3.80152691504847297416604826183, −3.70364657329080320035244865153, −3.26475221085730458736898093705, −3.22628219779038945621431401184, −3.18052840888050789229762847086, −2.64776392577271267467931909954, −2.57817519631466111597379913911, −2.46657787030549427364445442545, −2.46473780422317324955975719071, −2.39114714793762086770552010928, −2.08383052743919806309953661448, −1.74982928580601839459824547853, −1.66789895899830751625607151409, −1.43761757133638017364565335710, −1.35038626603412759449635626298, −0.863061728695205179557549492575, −0.57520323370447319176564832965, −0.57399395517810304400052009244, −0.56747296023011533067270233222, −0.55504434542636311598478314247, −0.04560375974541999764423656593, 0.04560375974541999764423656593, 0.55504434542636311598478314247, 0.56747296023011533067270233222, 0.57399395517810304400052009244, 0.57520323370447319176564832965, 0.863061728695205179557549492575, 1.35038626603412759449635626298, 1.43761757133638017364565335710, 1.66789895899830751625607151409, 1.74982928580601839459824547853, 2.08383052743919806309953661448, 2.39114714793762086770552010928, 2.46473780422317324955975719071, 2.46657787030549427364445442545, 2.57817519631466111597379913911, 2.64776392577271267467931909954, 3.18052840888050789229762847086, 3.22628219779038945621431401184, 3.26475221085730458736898093705, 3.70364657329080320035244865153, 3.80152691504847297416604826183, 3.84792678172945992339654935512, 3.90664315494120613216583277190, 3.95463641498309529153793364098, 4.04200261651840635055887491749
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.