Dirichlet series
| L(s) = 1 | + 4·4-s + 12·5-s + 6·7-s + 4·9-s − 4·11-s + 6·16-s + 10·19-s + 48·20-s − 4·23-s + 57·25-s + 24·28-s + 6·31-s + 72·35-s + 16·36-s + 14·37-s − 32·41-s − 16·44-s + 48·45-s − 24·47-s + 15·49-s − 48·55-s − 28·61-s + 24·63-s − 16·67-s − 56·71-s + 44·73-s + 40·76-s + ⋯ |
| L(s) = 1 | + 2·4-s + 5.36·5-s + 2.26·7-s + 4/3·9-s − 1.20·11-s + 3/2·16-s + 2.29·19-s + 10.7·20-s − 0.834·23-s + 57/5·25-s + 4.53·28-s + 1.07·31-s + 12.1·35-s + 8/3·36-s + 2.30·37-s − 4.99·41-s − 2.41·44-s + 7.15·45-s − 3.50·47-s + 15/7·49-s − 6.47·55-s − 3.58·61-s + 3.02·63-s − 1.95·67-s − 6.64·71-s + 5.14·73-s + 4.58·76-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 7^{16} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.17680\times 10^{9}\) |
| Root analytic conductor: | \(1.92070\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{16} \cdot 7^{16} \cdot 11^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(85.66565778\) |
| \(L(\frac12)\) | \(\approx\) | \(85.66565778\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 - T^{2} + T^{4} )^{4} \) | |
| 7 | \( 1 - 6 T + 3 p T^{2} - 10 T^{3} - 137 T^{4} + 548 T^{5} - 284 T^{6} - 4016 T^{7} + 16958 T^{8} - 4016 p T^{9} - 284 p^{2} T^{10} + 548 p^{3} T^{11} - 137 p^{4} T^{12} - 10 p^{5} T^{13} + 3 p^{7} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 4 T - 9 T^{2} - 20 T^{3} + 421 T^{4} + 960 T^{5} - 254 p T^{6} + 2664 T^{7} + 71702 T^{8} + 2664 p T^{9} - 254 p^{3} T^{10} + 960 p^{3} T^{11} + 421 p^{4} T^{12} - 20 p^{5} T^{13} - 9 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 5 | \( 1 - 12 T + 87 T^{2} - 468 T^{3} + 2046 T^{4} - 7608 T^{5} + 24941 T^{6} - 74172 T^{7} + 205763 T^{8} - 546324 T^{9} + 1417364 T^{10} - 3630492 T^{11} + 9187996 T^{12} - 22829256 T^{13} + 11061364 p T^{14} - 129962868 T^{15} + 295561964 T^{16} - 129962868 p T^{17} + 11061364 p^{3} T^{18} - 22829256 p^{3} T^{19} + 9187996 p^{4} T^{20} - 3630492 p^{5} T^{21} + 1417364 p^{6} T^{22} - 546324 p^{7} T^{23} + 205763 p^{8} T^{24} - 74172 p^{9} T^{25} + 24941 p^{10} T^{26} - 7608 p^{11} T^{27} + 2046 p^{12} T^{28} - 468 p^{13} T^{29} + 87 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) |
| 13 | \( ( 1 + 40 T^{2} + 8 T^{3} + 576 T^{4} - 648 T^{5} + 3672 T^{6} - 30816 T^{7} + 19294 T^{8} - 30816 p T^{9} + 3672 p^{2} T^{10} - 648 p^{3} T^{11} + 576 p^{4} T^{12} + 8 p^{5} T^{13} + 40 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 - 83 T^{2} - 8 p T^{3} + 3476 T^{4} + 9688 T^{5} - 88483 T^{6} - 330960 T^{7} + 1572689 T^{8} + 363248 p T^{9} - 26163616 T^{10} - 52036944 T^{11} + 628563790 T^{12} - 195542384 T^{13} - 16654549258 T^{14} + 4829420752 T^{15} + 337953189016 T^{16} + 4829420752 p T^{17} - 16654549258 p^{2} T^{18} - 195542384 p^{3} T^{19} + 628563790 p^{4} T^{20} - 52036944 p^{5} T^{21} - 26163616 p^{6} T^{22} + 363248 p^{8} T^{23} + 1572689 p^{8} T^{24} - 330960 p^{9} T^{25} - 88483 p^{10} T^{26} + 9688 p^{11} T^{27} + 3476 p^{12} T^{28} - 8 p^{14} T^{29} - 83 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( 1 - 10 T - 45 T^{2} + 610 T^{3} + 1855 T^{4} - 21532 T^{5} - 86816 T^{6} + 626624 T^{7} + 3146255 T^{8} - 14837722 T^{9} - 93351433 T^{10} + 281206598 T^{11} + 2391274762 T^{12} - 204674058 p T^{13} - 54784474877 T^{14} + 26032287962 T^{15} + 1117775138348 T^{16} + 26032287962 p T^{17} - 54784474877 p^{2} T^{18} - 204674058 p^{4} T^{19} + 2391274762 p^{4} T^{20} + 281206598 p^{5} T^{21} - 93351433 p^{6} T^{22} - 14837722 p^{7} T^{23} + 3146255 p^{8} T^{24} + 626624 p^{9} T^{25} - 86816 p^{10} T^{26} - 21532 p^{11} T^{27} + 1855 p^{12} T^{28} + 610 p^{13} T^{29} - 45 p^{14} T^{30} - 10 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( 1 + 4 T - 41 T^{2} - 20 T^{3} + 650 T^{4} - 4928 T^{5} + 271 p T^{6} + 71332 T^{7} - 513481 T^{8} + 890932 T^{9} - 5650452 T^{10} - 16186748 T^{11} + 272810192 T^{12} - 1729443936 T^{13} + 7244408264 T^{14} + 39032598852 T^{15} - 342082835708 T^{16} + 39032598852 p T^{17} + 7244408264 p^{2} T^{18} - 1729443936 p^{3} T^{19} + 272810192 p^{4} T^{20} - 16186748 p^{5} T^{21} - 5650452 p^{6} T^{22} + 890932 p^{7} T^{23} - 513481 p^{8} T^{24} + 71332 p^{9} T^{25} + 271 p^{11} T^{26} - 4928 p^{11} T^{27} + 650 p^{12} T^{28} - 20 p^{13} T^{29} - 41 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 - 220 T^{2} + 25766 T^{4} - 2101752 T^{6} + 132425201 T^{8} - 6784185928 T^{10} + 290786240918 T^{12} - 10599475438788 T^{14} + 331431007100868 T^{16} - 10599475438788 p^{2} T^{18} + 290786240918 p^{4} T^{20} - 6784185928 p^{6} T^{22} + 132425201 p^{8} T^{24} - 2101752 p^{10} T^{26} + 25766 p^{12} T^{28} - 220 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( 1 - 6 T + 211 T^{2} - 1194 T^{3} + 23451 T^{4} - 122748 T^{5} + 1782572 T^{6} - 8647176 T^{7} + 103630655 T^{8} - 469064958 T^{9} + 4922296683 T^{10} - 20969566134 T^{11} + 199997869402 T^{12} - 808826403210 T^{13} + 7187956918807 T^{14} - 27748250298954 T^{15} + 233495470937700 T^{16} - 27748250298954 p T^{17} + 7187956918807 p^{2} T^{18} - 808826403210 p^{3} T^{19} + 199997869402 p^{4} T^{20} - 20969566134 p^{5} T^{21} + 4922296683 p^{6} T^{22} - 469064958 p^{7} T^{23} + 103630655 p^{8} T^{24} - 8647176 p^{9} T^{25} + 1782572 p^{10} T^{26} - 122748 p^{11} T^{27} + 23451 p^{12} T^{28} - 1194 p^{13} T^{29} + 211 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 - 14 T - 33 T^{2} + 1046 T^{3} + 1691 T^{4} - 53864 T^{5} - 86468 T^{6} + 2173700 T^{7} + 1235255 T^{8} - 65091530 T^{9} + 206049191 T^{10} + 759360790 T^{11} - 14671742086 T^{12} - 218657242 p T^{13} + 879590831371 T^{14} + 160835179910 T^{15} - 39314877822420 T^{16} + 160835179910 p T^{17} + 879590831371 p^{2} T^{18} - 218657242 p^{4} T^{19} - 14671742086 p^{4} T^{20} + 759360790 p^{5} T^{21} + 206049191 p^{6} T^{22} - 65091530 p^{7} T^{23} + 1235255 p^{8} T^{24} + 2173700 p^{9} T^{25} - 86468 p^{10} T^{26} - 53864 p^{11} T^{27} + 1691 p^{12} T^{28} + 1046 p^{13} T^{29} - 33 p^{14} T^{30} - 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( ( 1 + 16 T + 330 T^{2} + 3640 T^{3} + 42745 T^{4} + 362832 T^{5} + 3123194 T^{6} + 21761352 T^{7} + 152229812 T^{8} + 21761352 p T^{9} + 3123194 p^{2} T^{10} + 362832 p^{3} T^{11} + 42745 p^{4} T^{12} + 3640 p^{5} T^{13} + 330 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( 1 - 370 T^{2} + 69205 T^{4} - 8714850 T^{6} + 829915302 T^{8} - 63533346462 T^{10} + 4048663635627 T^{12} - 219017168933710 T^{14} + 10156745084830162 T^{16} - 219017168933710 p^{2} T^{18} + 4048663635627 p^{4} T^{20} - 63533346462 p^{6} T^{22} + 829915302 p^{8} T^{24} - 8714850 p^{10} T^{26} + 69205 p^{12} T^{28} - 370 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( 1 + 24 T + 319 T^{2} + 3048 T^{3} + 23178 T^{4} + 148668 T^{5} + 513441 T^{6} - 2932224 T^{7} - 69320617 T^{8} - 712534932 T^{9} - 5366273204 T^{10} - 34042967316 T^{11} - 142354134896 T^{12} + 52571251464 T^{13} + 7563961844008 T^{14} + 83457296949492 T^{15} + 624119873811908 T^{16} + 83457296949492 p T^{17} + 7563961844008 p^{2} T^{18} + 52571251464 p^{3} T^{19} - 142354134896 p^{4} T^{20} - 34042967316 p^{5} T^{21} - 5366273204 p^{6} T^{22} - 712534932 p^{7} T^{23} - 69320617 p^{8} T^{24} - 2932224 p^{9} T^{25} + 513441 p^{10} T^{26} + 148668 p^{11} T^{27} + 23178 p^{12} T^{28} + 3048 p^{13} T^{29} + 319 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 - 189 T^{2} - 232 T^{3} + 17439 T^{4} + 43888 T^{5} - 854100 T^{6} - 3942720 T^{7} + 233363 p T^{8} + 287201456 T^{9} + 1198176951 T^{10} - 19075388376 T^{11} - 112969340570 T^{12} + 1015766984472 T^{13} + 6717800645927 T^{14} - 24348070253360 T^{15} - 355137904117416 T^{16} - 24348070253360 p T^{17} + 6717800645927 p^{2} T^{18} + 1015766984472 p^{3} T^{19} - 112969340570 p^{4} T^{20} - 19075388376 p^{5} T^{21} + 1198176951 p^{6} T^{22} + 287201456 p^{7} T^{23} + 233363 p^{9} T^{24} - 3942720 p^{9} T^{25} - 854100 p^{10} T^{26} + 43888 p^{11} T^{27} + 17439 p^{12} T^{28} - 232 p^{13} T^{29} - 189 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( 1 + 206 T^{2} + 21291 T^{4} + 23424 T^{5} + 1254962 T^{6} + 4357248 T^{7} + 31229625 T^{8} + 402292608 T^{9} - 1442494564 T^{10} + 17256515712 T^{11} - 185711486594 T^{12} - 179381210496 T^{13} - 9893086439768 T^{14} - 73366314523008 T^{15} - 490118909154734 T^{16} - 73366314523008 p T^{17} - 9893086439768 p^{2} T^{18} - 179381210496 p^{3} T^{19} - 185711486594 p^{4} T^{20} + 17256515712 p^{5} T^{21} - 1442494564 p^{6} T^{22} + 402292608 p^{7} T^{23} + 31229625 p^{8} T^{24} + 4357248 p^{9} T^{25} + 1254962 p^{10} T^{26} + 23424 p^{11} T^{27} + 21291 p^{12} T^{28} + 206 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( 1 + 28 T + 132 T^{2} - 3320 T^{3} - 30797 T^{4} + 260388 T^{5} + 3451164 T^{6} - 15800648 T^{7} - 264155963 T^{8} + 1067754516 T^{9} + 18055205512 T^{10} - 1181688332 p T^{11} - 1214227077978 T^{12} + 4021253438196 T^{13} + 85128658602288 T^{14} - 82671310258548 T^{15} - 5243239011871242 T^{16} - 82671310258548 p T^{17} + 85128658602288 p^{2} T^{18} + 4021253438196 p^{3} T^{19} - 1214227077978 p^{4} T^{20} - 1181688332 p^{6} T^{21} + 18055205512 p^{6} T^{22} + 1067754516 p^{7} T^{23} - 264155963 p^{8} T^{24} - 15800648 p^{9} T^{25} + 3451164 p^{10} T^{26} + 260388 p^{11} T^{27} - 30797 p^{12} T^{28} - 3320 p^{13} T^{29} + 132 p^{14} T^{30} + 28 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 + 16 T - 186 T^{2} - 4088 T^{3} + 17155 T^{4} + 531672 T^{5} - 1174782 T^{6} - 45150464 T^{7} + 109860217 T^{8} + 2760233088 T^{9} - 14334854684 T^{10} - 130824947456 T^{11} + 1614095177022 T^{12} + 4879888122672 T^{13} - 143662297955568 T^{14} - 102109742962512 T^{15} + 10522512537183234 T^{16} - 102109742962512 p T^{17} - 143662297955568 p^{2} T^{18} + 4879888122672 p^{3} T^{19} + 1614095177022 p^{4} T^{20} - 130824947456 p^{5} T^{21} - 14334854684 p^{6} T^{22} + 2760233088 p^{7} T^{23} + 109860217 p^{8} T^{24} - 45150464 p^{9} T^{25} - 1174782 p^{10} T^{26} + 531672 p^{11} T^{27} + 17155 p^{12} T^{28} - 4088 p^{13} T^{29} - 186 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( ( 1 + 28 T + 665 T^{2} + 9556 T^{3} + 124346 T^{4} + 1164764 T^{5} + 10825387 T^{6} + 79684020 T^{7} + 712940082 T^{8} + 79684020 p T^{9} + 10825387 p^{2} T^{10} + 1164764 p^{3} T^{11} + 124346 p^{4} T^{12} + 9556 p^{5} T^{13} + 665 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 - 44 T + 732 T^{2} - 5000 T^{3} + 5052 T^{4} + 60564 T^{5} - 1170168 T^{6} + 30721884 T^{7} - 151939238 T^{8} - 1947541152 T^{9} + 15985812484 T^{10} - 23523632180 T^{11} + 1180237150256 T^{12} - 10207425497860 T^{13} - 40635532820892 T^{14} - 79264670270400 T^{15} + 8440882624589587 T^{16} - 79264670270400 p T^{17} - 40635532820892 p^{2} T^{18} - 10207425497860 p^{3} T^{19} + 1180237150256 p^{4} T^{20} - 23523632180 p^{5} T^{21} + 15985812484 p^{6} T^{22} - 1947541152 p^{7} T^{23} - 151939238 p^{8} T^{24} + 30721884 p^{9} T^{25} - 1170168 p^{10} T^{26} + 60564 p^{11} T^{27} + 5052 p^{12} T^{28} - 5000 p^{13} T^{29} + 732 p^{14} T^{30} - 44 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 30 T + 719 T^{2} - 12570 T^{3} + 189438 T^{4} - 2490402 T^{5} + 29620697 T^{6} - 314951958 T^{7} + 3015838143 T^{8} - 24654844776 T^{9} + 163821266684 T^{10} - 633865980192 T^{11} - 3527632354616 T^{12} + 117997166527224 T^{13} - 1739518467334544 T^{14} + 19828557434833632 T^{15} - 189428265761930420 T^{16} + 19828557434833632 p T^{17} - 1739518467334544 p^{2} T^{18} + 117997166527224 p^{3} T^{19} - 3527632354616 p^{4} T^{20} - 633865980192 p^{5} T^{21} + 163821266684 p^{6} T^{22} - 24654844776 p^{7} T^{23} + 3015838143 p^{8} T^{24} - 314951958 p^{9} T^{25} + 29620697 p^{10} T^{26} - 2490402 p^{11} T^{27} + 189438 p^{12} T^{28} - 12570 p^{13} T^{29} + 719 p^{14} T^{30} - 30 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( ( 1 + 4 T + 523 T^{2} + 1544 T^{3} + 123705 T^{4} + 263372 T^{5} + 17791866 T^{6} + 28394068 T^{7} + 1752220430 T^{8} + 28394068 p T^{9} + 17791866 p^{2} T^{10} + 263372 p^{3} T^{11} + 123705 p^{4} T^{12} + 1544 p^{5} T^{13} + 523 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 + 36 T + 968 T^{2} + 19296 T^{3} + 325668 T^{4} + 4609212 T^{5} + 56729552 T^{6} + 597370692 T^{7} + 5319065562 T^{8} + 37089600696 T^{9} + 149466922904 T^{10} - 879537441732 T^{11} - 29159705241296 T^{12} - 437657801841588 T^{13} - 5158540285986536 T^{14} - 54240214087792200 T^{15} - 525790322706413885 T^{16} - 54240214087792200 p T^{17} - 5158540285986536 p^{2} T^{18} - 437657801841588 p^{3} T^{19} - 29159705241296 p^{4} T^{20} - 879537441732 p^{5} T^{21} + 149466922904 p^{6} T^{22} + 37089600696 p^{7} T^{23} + 5319065562 p^{8} T^{24} + 597370692 p^{9} T^{25} + 56729552 p^{10} T^{26} + 4609212 p^{11} T^{27} + 325668 p^{12} T^{28} + 19296 p^{13} T^{29} + 968 p^{14} T^{30} + 36 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 - 608 T^{2} + 187588 T^{4} - 40328000 T^{6} + 6818823050 T^{8} - 967972100768 T^{10} + 120672937256208 T^{12} - 13517338511335328 T^{14} + 1373778662576096019 T^{16} - 13517338511335328 p^{2} T^{18} + 120672937256208 p^{4} T^{20} - 967972100768 p^{6} T^{22} + 6818823050 p^{8} T^{24} - 40328000 p^{10} T^{26} + 187588 p^{12} T^{28} - 608 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−3.00886280133530829134081322188, −2.97963833672219119071455242424, −2.83667443900854372000207353040, −2.71397765206563790399271145027, −2.65369244048533580370006682648, −2.50443971542512371821989616311, −2.39165465111588241835397365918, −2.27411970935197369960676325994, −2.12286420418478248789033528172, −2.09420809459570619048205245717, −2.07541084867313070786934285749, −1.94226760662342207976657616790, −1.91007684713703292971761979037, −1.88032023741707717066010688505, −1.85468659036377673700970345353, −1.68395959354423926997103661904, −1.48638439191670567042674116751, −1.47678592673389484306255600211, −1.38105287776608291993295119116, −1.35268240223151840511931003061, −1.26017861867522612164791279033, −1.19422142418489692507863836335, −0.66674791235821533085641364275, −0.61236163220150357328390656309, −0.30251181642814331507091943220, 0.30251181642814331507091943220, 0.61236163220150357328390656309, 0.66674791235821533085641364275, 1.19422142418489692507863836335, 1.26017861867522612164791279033, 1.35268240223151840511931003061, 1.38105287776608291993295119116, 1.47678592673389484306255600211, 1.48638439191670567042674116751, 1.68395959354423926997103661904, 1.85468659036377673700970345353, 1.88032023741707717066010688505, 1.91007684713703292971761979037, 1.94226760662342207976657616790, 2.07541084867313070786934285749, 2.09420809459570619048205245717, 2.12286420418478248789033528172, 2.27411970935197369960676325994, 2.39165465111588241835397365918, 2.50443971542512371821989616311, 2.65369244048533580370006682648, 2.71397765206563790399271145027, 2.83667443900854372000207353040, 2.97963833672219119071455242424, 3.00886280133530829134081322188
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.