Dirichlet series
| L(s) = 1 | − 8·3-s − 6·4-s + 28·9-s + 3·11-s + 48·12-s − 2·13-s + 20·16-s − 8·17-s − 21·23-s + 47·25-s − 48·27-s + 29·29-s − 3·32-s − 24·33-s − 168·36-s − 18·37-s + 16·39-s + 12·41-s − 3·43-s − 18·44-s − 160·48-s − 27·49-s + 64·51-s + 12·52-s − 26·53-s − 3·59-s + 29·61-s + ⋯ |
| L(s) = 1 | − 4.61·3-s − 3·4-s + 28/3·9-s + 0.904·11-s + 13.8·12-s − 0.554·13-s + 5·16-s − 1.94·17-s − 4.37·23-s + 47/5·25-s − 9.23·27-s + 5.38·29-s − 0.530·32-s − 4.17·33-s − 28·36-s − 2.95·37-s + 2.56·39-s + 1.87·41-s − 0.457·43-s − 2.71·44-s − 23.0·48-s − 3.85·49-s + 8.96·51-s + 1.66·52-s − 3.57·53-s − 0.390·59-s + 3.71·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{16} \cdot 13^{16} \cdot 17^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.80758\times 10^{11}\) |
| Root analytic conductor: | \(2.30088\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{16} \cdot 13^{16} \cdot 17^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2858501988\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2858501988\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T + T^{2} )^{8} \) |
| 13 | \( 1 + 2 T + 27 T^{2} + 51 T^{3} + 568 T^{4} + 1006 T^{5} + 480 p T^{6} + 13977 T^{7} + 90009 T^{8} + 13977 p T^{9} + 480 p^{3} T^{10} + 1006 p^{3} T^{11} + 568 p^{4} T^{12} + 51 p^{5} T^{13} + 27 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( ( 1 + T + T^{2} )^{8} \) | |
| good | 2 | \( 1 + 3 p T^{2} + p^{4} T^{4} + 3 T^{5} + 9 p T^{6} + 9 p T^{7} - 9 p T^{8} + 3 p^{4} T^{9} - 117 T^{10} + 39 T^{11} - 205 T^{12} - 99 p T^{13} - 9 p^{3} T^{14} - 231 p^{2} T^{15} + 65 p^{2} T^{16} - 231 p^{3} T^{17} - 9 p^{5} T^{18} - 99 p^{4} T^{19} - 205 p^{4} T^{20} + 39 p^{5} T^{21} - 117 p^{6} T^{22} + 3 p^{11} T^{23} - 9 p^{9} T^{24} + 9 p^{10} T^{25} + 9 p^{11} T^{26} + 3 p^{11} T^{27} + p^{16} T^{28} + 3 p^{15} T^{30} + p^{16} T^{32} \) |
| 5 | \( 1 - 47 T^{2} + 1037 T^{4} - 14309 T^{6} + 139621 T^{8} - 1043276 T^{10} + 51239 p^{3} T^{12} - 34702101 T^{14} + 176657526 T^{16} - 34702101 p^{2} T^{18} + 51239 p^{7} T^{20} - 1043276 p^{6} T^{22} + 139621 p^{8} T^{24} - 14309 p^{10} T^{26} + 1037 p^{12} T^{28} - 47 p^{14} T^{30} + p^{16} T^{32} \) | |
| 7 | \( 1 + 27 T^{2} + 419 T^{4} - 198 T^{5} + 562 p T^{6} - 3909 T^{7} + 21356 T^{8} - 6309 p T^{9} + 16370 T^{10} - 26409 p T^{11} - 1009676 T^{12} + 161460 p T^{13} - 1707954 p T^{14} + 501534 p^{2} T^{15} - 284282 p^{3} T^{16} + 501534 p^{3} T^{17} - 1707954 p^{3} T^{18} + 161460 p^{4} T^{19} - 1009676 p^{4} T^{20} - 26409 p^{6} T^{21} + 16370 p^{6} T^{22} - 6309 p^{8} T^{23} + 21356 p^{8} T^{24} - 3909 p^{9} T^{25} + 562 p^{11} T^{26} - 198 p^{11} T^{27} + 419 p^{12} T^{28} + 27 p^{14} T^{30} + p^{16} T^{32} \) | |
| 11 | \( 1 - 3 T + 51 T^{2} - 144 T^{3} + 1356 T^{4} - 4416 T^{5} + 26649 T^{6} - 840 p^{2} T^{7} + 39557 p T^{8} - 1833366 T^{9} + 6481380 T^{10} - 27013422 T^{11} + 90985041 T^{12} - 340144239 T^{13} + 1187197134 T^{14} - 3917231580 T^{15} + 13923997671 T^{16} - 3917231580 p T^{17} + 1187197134 p^{2} T^{18} - 340144239 p^{3} T^{19} + 90985041 p^{4} T^{20} - 27013422 p^{5} T^{21} + 6481380 p^{6} T^{22} - 1833366 p^{7} T^{23} + 39557 p^{9} T^{24} - 840 p^{11} T^{25} + 26649 p^{10} T^{26} - 4416 p^{11} T^{27} + 1356 p^{12} T^{28} - 144 p^{13} T^{29} + 51 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 + 60 T^{2} + 1354 T^{4} + 2094 T^{5} + 10227 T^{6} + 114894 T^{7} - 27078 T^{8} + 2190516 T^{9} + 4362384 T^{10} + 12637062 T^{11} + 214465892 T^{12} + 68267979 T^{13} + 3666504270 T^{14} + 7493835168 T^{15} + 48596854748 T^{16} + 7493835168 p T^{17} + 3666504270 p^{2} T^{18} + 68267979 p^{3} T^{19} + 214465892 p^{4} T^{20} + 12637062 p^{5} T^{21} + 4362384 p^{6} T^{22} + 2190516 p^{7} T^{23} - 27078 p^{8} T^{24} + 114894 p^{9} T^{25} + 10227 p^{10} T^{26} + 2094 p^{11} T^{27} + 1354 p^{12} T^{28} + 60 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 + 21 T + 132 T^{2} - 181 T^{3} - 4267 T^{4} + 5473 T^{5} + 175518 T^{6} + 303837 T^{7} - 2782544 T^{8} - 9597673 T^{9} + 27129288 T^{10} + 316258377 T^{11} + 928733850 T^{12} - 8114486651 T^{13} - 69389743063 T^{14} + 38814107702 T^{15} + 1704033356038 T^{16} + 38814107702 p T^{17} - 69389743063 p^{2} T^{18} - 8114486651 p^{3} T^{19} + 928733850 p^{4} T^{20} + 316258377 p^{5} T^{21} + 27129288 p^{6} T^{22} - 9597673 p^{7} T^{23} - 2782544 p^{8} T^{24} + 303837 p^{9} T^{25} + 175518 p^{10} T^{26} + 5473 p^{11} T^{27} - 4267 p^{12} T^{28} - 181 p^{13} T^{29} + 132 p^{14} T^{30} + 21 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 - p T + 324 T^{2} - 2273 T^{3} + 21222 T^{4} - 199068 T^{5} + 1124565 T^{6} - 5925365 T^{7} + 47699176 T^{8} - 258004386 T^{9} + 841077554 T^{10} - 5243165000 T^{11} + 31066721744 T^{12} - 74920339639 T^{13} + 267601941924 T^{14} - 1849361739459 T^{15} + 7814073375154 T^{16} - 1849361739459 p T^{17} + 267601941924 p^{2} T^{18} - 74920339639 p^{3} T^{19} + 31066721744 p^{4} T^{20} - 5243165000 p^{5} T^{21} + 841077554 p^{6} T^{22} - 258004386 p^{7} T^{23} + 47699176 p^{8} T^{24} - 5925365 p^{9} T^{25} + 1124565 p^{10} T^{26} - 199068 p^{11} T^{27} + 21222 p^{12} T^{28} - 2273 p^{13} T^{29} + 324 p^{14} T^{30} - p^{16} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 - 198 T^{2} + 21495 T^{4} - 1637772 T^{6} + 97284991 T^{8} - 153872808 p T^{10} + 200245078377 T^{12} - 7375866056787 T^{14} + 241774479905418 T^{16} - 7375866056787 p^{2} T^{18} + 200245078377 p^{4} T^{20} - 153872808 p^{7} T^{22} + 97284991 p^{8} T^{24} - 1637772 p^{10} T^{26} + 21495 p^{12} T^{28} - 198 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( 1 + 18 T + 294 T^{2} + 3348 T^{3} + 33820 T^{4} + 286803 T^{5} + 2191347 T^{6} + 14814456 T^{7} + 91138539 T^{8} + 504686754 T^{9} + 2551014942 T^{10} + 12027376818 T^{11} + 53103297809 T^{12} + 253449447420 T^{13} + 1255755223317 T^{14} + 7542189313293 T^{15} + 43491075813575 T^{16} + 7542189313293 p T^{17} + 1255755223317 p^{2} T^{18} + 253449447420 p^{3} T^{19} + 53103297809 p^{4} T^{20} + 12027376818 p^{5} T^{21} + 2551014942 p^{6} T^{22} + 504686754 p^{7} T^{23} + 91138539 p^{8} T^{24} + 14814456 p^{9} T^{25} + 2191347 p^{10} T^{26} + 286803 p^{11} T^{27} + 33820 p^{12} T^{28} + 3348 p^{13} T^{29} + 294 p^{14} T^{30} + 18 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 - 12 T + 225 T^{2} - 2124 T^{3} + 23261 T^{4} - 179139 T^{5} + 1426955 T^{6} - 8571924 T^{7} + 50034002 T^{8} - 179231004 T^{9} + 387851248 T^{10} + 5768652006 T^{11} - 66638639432 T^{12} + 707827872417 T^{13} - 4974445024731 T^{14} + 38389530251613 T^{15} - 232001429004968 T^{16} + 38389530251613 p T^{17} - 4974445024731 p^{2} T^{18} + 707827872417 p^{3} T^{19} - 66638639432 p^{4} T^{20} + 5768652006 p^{5} T^{21} + 387851248 p^{6} T^{22} - 179231004 p^{7} T^{23} + 50034002 p^{8} T^{24} - 8571924 p^{9} T^{25} + 1426955 p^{10} T^{26} - 179139 p^{11} T^{27} + 23261 p^{12} T^{28} - 2124 p^{13} T^{29} + 225 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 + 3 T - 160 T^{2} - 805 T^{3} + 11139 T^{4} + 87086 T^{5} - 8540 p T^{6} - 5832990 T^{7} - 1102684 T^{8} + 245929002 T^{9} + 702133974 T^{10} - 5123321992 T^{11} - 27901296946 T^{12} - 25055464278 T^{13} + 228457511025 T^{14} + 2204964621501 T^{15} + 12858725601678 T^{16} + 2204964621501 p T^{17} + 228457511025 p^{2} T^{18} - 25055464278 p^{3} T^{19} - 27901296946 p^{4} T^{20} - 5123321992 p^{5} T^{21} + 702133974 p^{6} T^{22} + 245929002 p^{7} T^{23} - 1102684 p^{8} T^{24} - 5832990 p^{9} T^{25} - 8540 p^{11} T^{26} + 87086 p^{11} T^{27} + 11139 p^{12} T^{28} - 805 p^{13} T^{29} - 160 p^{14} T^{30} + 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 - 363 T^{2} + 70393 T^{4} - 9470529 T^{6} + 978936033 T^{8} - 81889280721 T^{10} + 5711888942192 T^{12} - 338103914679027 T^{14} + 17144022013040714 T^{16} - 338103914679027 p^{2} T^{18} + 5711888942192 p^{4} T^{20} - 81889280721 p^{6} T^{22} + 978936033 p^{8} T^{24} - 9470529 p^{10} T^{26} + 70393 p^{12} T^{28} - 363 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( ( 1 + 13 T + 245 T^{2} + 2045 T^{3} + 22730 T^{4} + 138596 T^{5} + 1319370 T^{6} + 7093264 T^{7} + 69360936 T^{8} + 7093264 p T^{9} + 1319370 p^{2} T^{10} + 138596 p^{3} T^{11} + 22730 p^{4} T^{12} + 2045 p^{5} T^{13} + 245 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 + 3 T + 185 T^{2} + 546 T^{3} + 17549 T^{4} - 2610 T^{5} + 863018 T^{6} - 6770868 T^{7} + 5237179 T^{8} - 805687542 T^{9} - 1395742381 T^{10} - 44941562778 T^{11} + 43100400667 T^{12} - 628844207013 T^{13} + 19195483439325 T^{14} + 94094474435772 T^{15} + 1616846994932187 T^{16} + 94094474435772 p T^{17} + 19195483439325 p^{2} T^{18} - 628844207013 p^{3} T^{19} + 43100400667 p^{4} T^{20} - 44941562778 p^{5} T^{21} - 1395742381 p^{6} T^{22} - 805687542 p^{7} T^{23} + 5237179 p^{8} T^{24} - 6770868 p^{9} T^{25} + 863018 p^{10} T^{26} - 2610 p^{11} T^{27} + 17549 p^{12} T^{28} + 546 p^{13} T^{29} + 185 p^{14} T^{30} + 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 29 T + 167 T^{2} + 2718 T^{3} - 27271 T^{4} - 184112 T^{5} + 2203086 T^{6} + 12618955 T^{7} - 128723711 T^{8} - 713535808 T^{9} + 4188138152 T^{10} + 18871027902 T^{11} + 319196176053 T^{12} - 927387211339 T^{13} - 45687927594232 T^{14} + 43823749428128 T^{15} + 3085598647099676 T^{16} + 43823749428128 p T^{17} - 45687927594232 p^{2} T^{18} - 927387211339 p^{3} T^{19} + 319196176053 p^{4} T^{20} + 18871027902 p^{5} T^{21} + 4188138152 p^{6} T^{22} - 713535808 p^{7} T^{23} - 128723711 p^{8} T^{24} + 12618955 p^{9} T^{25} + 2203086 p^{10} T^{26} - 184112 p^{11} T^{27} - 27271 p^{12} T^{28} + 2718 p^{13} T^{29} + 167 p^{14} T^{30} - 29 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 - 33 T + 799 T^{2} - 14388 T^{3} + 217608 T^{4} - 2834028 T^{5} + 33091828 T^{6} - 353619063 T^{7} + 3548983325 T^{8} - 34042845471 T^{9} + 317288288957 T^{10} - 2903230189194 T^{11} + 26145068429250 T^{12} - 232213710659784 T^{13} + 2021169244332005 T^{14} - 17221270880101239 T^{15} + 142790984594329294 T^{16} - 17221270880101239 p T^{17} + 2021169244332005 p^{2} T^{18} - 232213710659784 p^{3} T^{19} + 26145068429250 p^{4} T^{20} - 2903230189194 p^{5} T^{21} + 317288288957 p^{6} T^{22} - 34042845471 p^{7} T^{23} + 3548983325 p^{8} T^{24} - 353619063 p^{9} T^{25} + 33091828 p^{10} T^{26} - 2834028 p^{11} T^{27} + 217608 p^{12} T^{28} - 14388 p^{13} T^{29} + 799 p^{14} T^{30} - 33 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 - 27 T + 626 T^{2} - 10341 T^{3} + 148136 T^{4} - 1933623 T^{5} + 23264989 T^{6} - 274398300 T^{7} + 3076493494 T^{8} - 33289035336 T^{9} + 345666731178 T^{10} - 3431618523396 T^{11} + 33140364837730 T^{12} - 309095037793143 T^{13} + 2807972284096467 T^{14} - 24796736273930994 T^{15} + 211772501143165107 T^{16} - 24796736273930994 p T^{17} + 2807972284096467 p^{2} T^{18} - 309095037793143 p^{3} T^{19} + 33140364837730 p^{4} T^{20} - 3431618523396 p^{5} T^{21} + 345666731178 p^{6} T^{22} - 33289035336 p^{7} T^{23} + 3076493494 p^{8} T^{24} - 274398300 p^{9} T^{25} + 23264989 p^{10} T^{26} - 1933623 p^{11} T^{27} + 148136 p^{12} T^{28} - 10341 p^{13} T^{29} + 626 p^{14} T^{30} - 27 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 - 11 p T^{2} + 307857 T^{4} - 75525224 T^{6} + 13408754528 T^{8} - 1846205826244 T^{10} + 206018478996516 T^{12} - 19179697133303590 T^{14} + 1515420643832717074 T^{16} - 19179697133303590 p^{2} T^{18} + 206018478996516 p^{4} T^{20} - 1846205826244 p^{6} T^{22} + 13408754528 p^{8} T^{24} - 75525224 p^{10} T^{26} + 307857 p^{12} T^{28} - 11 p^{15} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 + 7 T + 164 T^{2} + 1073 T^{3} + 16438 T^{4} + 33144 T^{5} + 889990 T^{6} - 2840001 T^{7} + 37128438 T^{8} - 2840001 p T^{9} + 889990 p^{2} T^{10} + 33144 p^{3} T^{11} + 16438 p^{4} T^{12} + 1073 p^{5} T^{13} + 164 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 - 1013 T^{2} + 491738 T^{4} - 152485497 T^{6} + 33968590992 T^{8} - 5792302581579 T^{10} + 785681506939807 T^{12} - 86804348052448771 T^{14} + 7914372200830595396 T^{16} - 86804348052448771 p^{2} T^{18} + 785681506939807 p^{4} T^{20} - 5792302581579 p^{6} T^{22} + 33968590992 p^{8} T^{24} - 152485497 p^{10} T^{26} + 491738 p^{12} T^{28} - 1013 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( 1 + 3 T + 503 T^{2} + 1500 T^{3} + 136016 T^{4} + 377073 T^{5} + 24679223 T^{6} + 59696877 T^{7} + 3298209868 T^{8} + 6168107592 T^{9} + 335715671948 T^{10} + 350703730698 T^{11} + 26863506991096 T^{12} - 5177763290436 T^{13} + 1831682187616656 T^{14} - 3141628815459255 T^{15} + 138403693316109771 T^{16} - 3141628815459255 p T^{17} + 1831682187616656 p^{2} T^{18} - 5177763290436 p^{3} T^{19} + 26863506991096 p^{4} T^{20} + 350703730698 p^{5} T^{21} + 335715671948 p^{6} T^{22} + 6168107592 p^{7} T^{23} + 3298209868 p^{8} T^{24} + 59696877 p^{9} T^{25} + 24679223 p^{10} T^{26} + 377073 p^{11} T^{27} + 136016 p^{12} T^{28} + 1500 p^{13} T^{29} + 503 p^{14} T^{30} + 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 - 6 T + 392 T^{2} - 2280 T^{3} + 73160 T^{4} - 320028 T^{5} + 8333183 T^{6} - 30667896 T^{7} + 717419410 T^{8} - 4759452300 T^{9} + 71544111308 T^{10} - 909338170446 T^{11} + 7540477547416 T^{12} - 126816366433275 T^{13} + 616441865292360 T^{14} - 13189888455019728 T^{15} + 50047644129159414 T^{16} - 13189888455019728 p T^{17} + 616441865292360 p^{2} T^{18} - 126816366433275 p^{3} T^{19} + 7540477547416 p^{4} T^{20} - 909338170446 p^{5} T^{21} + 71544111308 p^{6} T^{22} - 4759452300 p^{7} T^{23} + 717419410 p^{8} T^{24} - 30667896 p^{9} T^{25} + 8333183 p^{10} T^{26} - 320028 p^{11} T^{27} + 73160 p^{12} T^{28} - 2280 p^{13} T^{29} + 392 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−2.95166852911250558647152966913, −2.71126391848698565626773651209, −2.66002283847143993063590720835, −2.61418884024243244603527187308, −2.48143325437364158561961818134, −2.41185057399762455128941867109, −2.34108627445528977029282873639, −2.12342077619450135194607106175, −2.08206586899034588363820179326, −2.01313368049985444746050668426, −1.82513635932488409978311187280, −1.75215651114802353467123641728, −1.74040988986346415170847634307, −1.46857647633549404693292782199, −1.38248082628224796230467724878, −1.26073319399385979583486152262, −1.16383343653697807480166854471, −0.901411947616363004329762991851, −0.883932928008140010337946086410, −0.883066928800285025186214397191, −0.817458714453481051439595342588, −0.51446943933569928447887651209, −0.51033045551186814624517045599, −0.31501045886817452001989936615, −0.19303029003303137678029766393, 0.19303029003303137678029766393, 0.31501045886817452001989936615, 0.51033045551186814624517045599, 0.51446943933569928447887651209, 0.817458714453481051439595342588, 0.883066928800285025186214397191, 0.883932928008140010337946086410, 0.901411947616363004329762991851, 1.16383343653697807480166854471, 1.26073319399385979583486152262, 1.38248082628224796230467724878, 1.46857647633549404693292782199, 1.74040988986346415170847634307, 1.75215651114802353467123641728, 1.82513635932488409978311187280, 2.01313368049985444746050668426, 2.08206586899034588363820179326, 2.12342077619450135194607106175, 2.34108627445528977029282873639, 2.41185057399762455128941867109, 2.48143325437364158561961818134, 2.61418884024243244603527187308, 2.66002283847143993063590720835, 2.71126391848698565626773651209, 2.95166852911250558647152966913
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.