Dirichlet series
| L(s) = 1 | + 2-s − 4·3-s − 3·4-s + 3·5-s − 4·6-s + 6·9-s + 3·10-s − 5·11-s + 12·12-s − 10·13-s − 12·15-s + 5·16-s − 11·17-s + 6·18-s − 9·19-s − 9·20-s − 5·22-s + 16·23-s − 3·25-s − 10·26-s + 24·27-s − 18·29-s − 12·30-s − 11·31-s + 4·32-s + 20·33-s − 11·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 2.30·3-s − 3/2·4-s + 1.34·5-s − 1.63·6-s + 2·9-s + 0.948·10-s − 1.50·11-s + 3.46·12-s − 2.77·13-s − 3.09·15-s + 5/4·16-s − 2.66·17-s + 1.41·18-s − 2.06·19-s − 2.01·20-s − 1.06·22-s + 3.33·23-s − 3/5·25-s − 1.96·26-s + 4.61·27-s − 3.34·29-s − 2.19·30-s − 1.97·31-s + 0.707·32-s + 3.48·33-s − 1.88·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(7^{32} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.38628\times 10^{10}\) |
| Root analytic conductor: | \(2.07459\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 7^{32} \cdot 11^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.02260458365\) |
| \(L(\frac12)\) | \(\approx\) | \(0.02260458365\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + 5 T - T^{2} - 20 T^{3} + 50 T^{4} + 265 T^{5} + 1266 T^{6} - 1690 T^{7} - 28761 T^{8} - 1690 p T^{9} + 1266 p^{2} T^{10} + 265 p^{3} T^{11} + 50 p^{4} T^{12} - 20 p^{5} T^{13} - p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 2 | \( 1 - T + p^{2} T^{2} - 7 T^{3} + 7 p T^{4} - 17 p T^{5} + 3 p^{4} T^{6} - 115 T^{7} + 169 T^{8} - 299 T^{9} + 523 T^{10} - 379 p T^{11} + 1321 T^{12} - 925 p T^{13} + 2863 T^{14} - 541 p^{3} T^{15} + 5781 T^{16} - 541 p^{4} T^{17} + 2863 p^{2} T^{18} - 925 p^{4} T^{19} + 1321 p^{4} T^{20} - 379 p^{6} T^{21} + 523 p^{6} T^{22} - 299 p^{7} T^{23} + 169 p^{8} T^{24} - 115 p^{9} T^{25} + 3 p^{14} T^{26} - 17 p^{12} T^{27} + 7 p^{13} T^{28} - 7 p^{13} T^{29} + p^{16} T^{30} - p^{15} T^{31} + p^{16} T^{32} \) |
| 3 | \( ( 1 + 2 T + p T^{2} - 10 T^{3} - 10 p T^{4} - 52 T^{5} - T^{6} + 50 p T^{7} + 379 T^{8} + 50 p^{2} T^{9} - p^{2} T^{10} - 52 p^{3} T^{11} - 10 p^{5} T^{12} - 10 p^{5} T^{13} + p^{7} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 5 | \( 1 - 3 T + 12 T^{2} - 51 T^{3} + 138 T^{4} - 474 T^{5} + 1189 T^{6} - 3123 T^{7} + 8088 T^{8} - 16326 T^{9} + 38041 T^{10} - 69267 T^{11} + 129643 T^{12} - 221502 T^{13} + 294093 T^{14} - 572844 T^{15} + 907531 T^{16} - 572844 p T^{17} + 294093 p^{2} T^{18} - 221502 p^{3} T^{19} + 129643 p^{4} T^{20} - 69267 p^{5} T^{21} + 38041 p^{6} T^{22} - 16326 p^{7} T^{23} + 8088 p^{8} T^{24} - 3123 p^{9} T^{25} + 1189 p^{10} T^{26} - 474 p^{11} T^{27} + 138 p^{12} T^{28} - 51 p^{13} T^{29} + 12 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( ( 1 + 5 T + 35 T^{2} + 90 T^{3} + 47 p T^{4} + 670 T^{5} + 4380 T^{6} - 8865 T^{7} + 38151 T^{8} - 8865 p T^{9} + 4380 p^{2} T^{10} + 670 p^{3} T^{11} + 47 p^{5} T^{12} + 90 p^{5} T^{13} + 35 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 + 11 T + 82 T^{2} + 151 T^{3} - 1257 T^{4} - 14573 T^{5} - 43518 T^{6} + 93859 T^{7} + 1568504 T^{8} + 5547395 T^{9} - 2761766 T^{10} - 97790693 T^{11} - 344680251 T^{12} + 129416362 T^{13} + 3449705386 T^{14} + 9240382008 T^{15} + 336684367 T^{16} + 9240382008 p T^{17} + 3449705386 p^{2} T^{18} + 129416362 p^{3} T^{19} - 344680251 p^{4} T^{20} - 97790693 p^{5} T^{21} - 2761766 p^{6} T^{22} + 5547395 p^{7} T^{23} + 1568504 p^{8} T^{24} + 93859 p^{9} T^{25} - 43518 p^{10} T^{26} - 14573 p^{11} T^{27} - 1257 p^{12} T^{28} + 151 p^{13} T^{29} + 82 p^{14} T^{30} + 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 + 9 T + 2 p T^{2} + 45 T^{3} - 713 T^{4} - 9387 T^{5} - 55694 T^{6} - 172071 T^{7} + 626 p T^{8} + 3422331 T^{9} + 28508642 T^{10} + 136939077 T^{11} + 373421069 T^{12} + 123703272 T^{13} - 4687437236 T^{14} - 41957891316 T^{15} - 224298429341 T^{16} - 41957891316 p T^{17} - 4687437236 p^{2} T^{18} + 123703272 p^{3} T^{19} + 373421069 p^{4} T^{20} + 136939077 p^{5} T^{21} + 28508642 p^{6} T^{22} + 3422331 p^{7} T^{23} + 626 p^{9} T^{24} - 172071 p^{9} T^{25} - 55694 p^{10} T^{26} - 9387 p^{11} T^{27} - 713 p^{12} T^{28} + 45 p^{13} T^{29} + 2 p^{15} T^{30} + 9 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( ( 1 - 8 T - 19 T^{2} + 140 T^{3} + 1118 T^{4} - 1732 T^{5} - 36615 T^{6} + 59134 T^{7} + 444387 T^{8} + 59134 p T^{9} - 36615 p^{2} T^{10} - 1732 p^{3} T^{11} + 1118 p^{4} T^{12} + 140 p^{5} T^{13} - 19 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( ( 1 + 9 T - 22 T^{2} - 429 T^{3} - 762 T^{4} + 11754 T^{5} + 81940 T^{6} - 191700 T^{7} - 3727369 T^{8} - 191700 p T^{9} + 81940 p^{2} T^{10} + 11754 p^{3} T^{11} - 762 p^{4} T^{12} - 429 p^{5} T^{13} - 22 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( 1 + 11 T + 87 T^{2} + 8 T^{3} - 4067 T^{4} - 51562 T^{5} - 279707 T^{6} - 605448 T^{7} + 7071411 T^{8} + 79682271 T^{9} + 453595654 T^{10} + 966291599 T^{11} - 4382753906 T^{12} - 56567005227 T^{13} - 210031083446 T^{14} - 225648686906 T^{15} + 2581204556074 T^{16} - 225648686906 p T^{17} - 210031083446 p^{2} T^{18} - 56567005227 p^{3} T^{19} - 4382753906 p^{4} T^{20} + 966291599 p^{5} T^{21} + 453595654 p^{6} T^{22} + 79682271 p^{7} T^{23} + 7071411 p^{8} T^{24} - 605448 p^{9} T^{25} - 279707 p^{10} T^{26} - 51562 p^{11} T^{27} - 4067 p^{12} T^{28} + 8 p^{13} T^{29} + 87 p^{14} T^{30} + 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 + 6 T + 99 T^{2} + 662 T^{3} + 5739 T^{4} + 29514 T^{5} + 237308 T^{6} + 716560 T^{7} + 4809854 T^{8} + 17329244 T^{9} + 22995413 T^{10} + 480402038 T^{11} + 2695419596 T^{12} + 38023386470 T^{13} + 235277510708 T^{14} + 3056978579838 T^{15} + 9263636009781 T^{16} + 3056978579838 p T^{17} + 235277510708 p^{2} T^{18} + 38023386470 p^{3} T^{19} + 2695419596 p^{4} T^{20} + 480402038 p^{5} T^{21} + 22995413 p^{6} T^{22} + 17329244 p^{7} T^{23} + 4809854 p^{8} T^{24} + 716560 p^{9} T^{25} + 237308 p^{10} T^{26} + 29514 p^{11} T^{27} + 5739 p^{12} T^{28} + 662 p^{13} T^{29} + 99 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 22 T + 160 T^{2} - 414 T^{3} + 2641 T^{4} - 46342 T^{5} + 334288 T^{6} - 25916 p T^{7} + 2274033 T^{8} - 25916 p^{2} T^{9} + 334288 p^{2} T^{10} - 46342 p^{3} T^{11} + 2641 p^{4} T^{12} - 414 p^{5} T^{13} + 160 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{8} \) | |
| 47 | \( 1 - 7 T + 149 T^{2} - 2128 T^{3} + 19612 T^{4} - 240919 T^{5} + 2462607 T^{6} - 21813848 T^{7} + 211617499 T^{8} - 1862688919 T^{9} + 15071552909 T^{10} - 126763363055 T^{11} + 1005269592991 T^{12} - 7378260675254 T^{13} + 56266129109578 T^{14} - 404849575321212 T^{15} + 2719197771655735 T^{16} - 404849575321212 p T^{17} + 56266129109578 p^{2} T^{18} - 7378260675254 p^{3} T^{19} + 1005269592991 p^{4} T^{20} - 126763363055 p^{5} T^{21} + 15071552909 p^{6} T^{22} - 1862688919 p^{7} T^{23} + 211617499 p^{8} T^{24} - 21813848 p^{9} T^{25} + 2462607 p^{10} T^{26} - 240919 p^{11} T^{27} + 19612 p^{12} T^{28} - 2128 p^{13} T^{29} + 149 p^{14} T^{30} - 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 + 2 T + 177 T^{2} + 1582 T^{3} + 21921 T^{4} + 231770 T^{5} + 2684050 T^{6} + 23678026 T^{7} + 239791912 T^{8} + 2114692878 T^{9} + 17637992799 T^{10} + 151508122160 T^{11} + 1184498641232 T^{12} + 9380769650632 T^{13} + 69924170534704 T^{14} + 544239771304768 T^{15} + 3774658249521589 T^{16} + 544239771304768 p T^{17} + 69924170534704 p^{2} T^{18} + 9380769650632 p^{3} T^{19} + 1184498641232 p^{4} T^{20} + 151508122160 p^{5} T^{21} + 17637992799 p^{6} T^{22} + 2114692878 p^{7} T^{23} + 239791912 p^{8} T^{24} + 23678026 p^{9} T^{25} + 2684050 p^{10} T^{26} + 231770 p^{11} T^{27} + 21921 p^{12} T^{28} + 1582 p^{13} T^{29} + 177 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 - 25 T + 419 T^{2} - 5200 T^{3} + 58281 T^{4} - 598550 T^{5} + 6209893 T^{6} - 62948850 T^{7} + 627371027 T^{8} - 5955201125 T^{9} + 54113866884 T^{10} - 468835214275 T^{11} + 3967819915452 T^{12} - 33125257031325 T^{13} + 272714366831568 T^{14} - 2199975935884200 T^{15} + 17198372786595766 T^{16} - 2199975935884200 p T^{17} + 272714366831568 p^{2} T^{18} - 33125257031325 p^{3} T^{19} + 3967819915452 p^{4} T^{20} - 468835214275 p^{5} T^{21} + 54113866884 p^{6} T^{22} - 5955201125 p^{7} T^{23} + 627371027 p^{8} T^{24} - 62948850 p^{9} T^{25} + 6209893 p^{10} T^{26} - 598550 p^{11} T^{27} + 58281 p^{12} T^{28} - 5200 p^{13} T^{29} + 419 p^{14} T^{30} - 25 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 7 T + 157 T^{2} - 2810 T^{3} + 23827 T^{4} - 352454 T^{5} + 4248359 T^{6} - 37300018 T^{7} + 438042179 T^{8} - 4457193753 T^{9} + 37214201488 T^{10} - 377860830849 T^{11} + 3428826064944 T^{12} - 26601329050551 T^{13} + 245019222253856 T^{14} - 2011514545995138 T^{15} + 14453308043412074 T^{16} - 2011514545995138 p T^{17} + 245019222253856 p^{2} T^{18} - 26601329050551 p^{3} T^{19} + 3428826064944 p^{4} T^{20} - 377860830849 p^{5} T^{21} + 37214201488 p^{6} T^{22} - 4457193753 p^{7} T^{23} + 438042179 p^{8} T^{24} - 37300018 p^{9} T^{25} + 4248359 p^{10} T^{26} - 352454 p^{11} T^{27} + 23827 p^{12} T^{28} - 2810 p^{13} T^{29} + 157 p^{14} T^{30} - 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( ( 1 - 15 T - 110 T^{2} + 1095 T^{3} + 30612 T^{4} - 158730 T^{5} - 2667925 T^{6} + 1012950 T^{7} + 259471373 T^{8} + 1012950 p T^{9} - 2667925 p^{2} T^{10} - 158730 p^{3} T^{11} + 30612 p^{4} T^{12} + 1095 p^{5} T^{13} - 110 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( ( 1 + 14 T - 9 T^{2} - 147 T^{3} + 6927 T^{4} + 52059 T^{5} + 728307 T^{6} + 4047848 T^{7} - 14238189 T^{8} + 4047848 p T^{9} + 728307 p^{2} T^{10} + 52059 p^{3} T^{11} + 6927 p^{4} T^{12} - 147 p^{5} T^{13} - 9 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 - 3 T + 11 T^{2} + 696 T^{3} - 1251 T^{4} - 98282 T^{5} + 1060537 T^{6} - 2326410 T^{7} - 65290511 T^{8} + 308702063 T^{9} + 6492765102 T^{10} - 92451983031 T^{11} + 281336636896 T^{12} + 4375330012525 T^{13} - 27106436766918 T^{14} - 268331027179726 T^{15} + 5635646597034826 T^{16} - 268331027179726 p T^{17} - 27106436766918 p^{2} T^{18} + 4375330012525 p^{3} T^{19} + 281336636896 p^{4} T^{20} - 92451983031 p^{5} T^{21} + 6492765102 p^{6} T^{22} + 308702063 p^{7} T^{23} - 65290511 p^{8} T^{24} - 2326410 p^{9} T^{25} + 1060537 p^{10} T^{26} - 98282 p^{11} T^{27} - 1251 p^{12} T^{28} + 696 p^{13} T^{29} + 11 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 9 T + 156 T^{2} - 861 T^{3} + 2506 T^{4} + 98394 T^{5} - 1985283 T^{6} + 19471377 T^{7} - 143824296 T^{8} + 9028584 p T^{9} + 7748031243 T^{10} - 94344490293 T^{11} + 1366072992785 T^{12} - 5625357609390 T^{13} + 11704956821451 T^{14} + 387894213236796 T^{15} - 6232535914916185 T^{16} + 387894213236796 p T^{17} + 11704956821451 p^{2} T^{18} - 5625357609390 p^{3} T^{19} + 1366072992785 p^{4} T^{20} - 94344490293 p^{5} T^{21} + 7748031243 p^{6} T^{22} + 9028584 p^{8} T^{23} - 143824296 p^{8} T^{24} + 19471377 p^{9} T^{25} - 1985283 p^{10} T^{26} + 98394 p^{11} T^{27} + 2506 p^{12} T^{28} - 861 p^{13} T^{29} + 156 p^{14} T^{30} - 9 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( ( 1 + 23 T + 98 T^{2} - 1745 T^{3} - 15470 T^{4} + 98632 T^{5} + 1750974 T^{6} - 1504700 T^{7} - 137136651 T^{8} - 1504700 p T^{9} + 1750974 p^{2} T^{10} + 98632 p^{3} T^{11} - 15470 p^{4} T^{12} - 1745 p^{5} T^{13} + 98 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 + 17 T - 23 T^{2} - 1534 T^{3} + 282 T^{4} + 47093 T^{5} - 689650 T^{6} - 2747686 T^{7} + 49542809 T^{8} - 2747686 p T^{9} - 689650 p^{2} T^{10} + 47093 p^{3} T^{11} + 282 p^{4} T^{12} - 1534 p^{5} T^{13} - 23 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( ( 1 + 30 T + 361 T^{2} + 2160 T^{3} + 10062 T^{4} + 83370 T^{5} - 577417 T^{6} - 32754150 T^{7} - 458148745 T^{8} - 32754150 p T^{9} - 577417 p^{2} T^{10} + 83370 p^{3} T^{11} + 10062 p^{4} T^{12} + 2160 p^{5} T^{13} + 361 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
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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
−2.79339141225435190876328042872, −2.73344355544314738311019382328, −2.69176810845821926821850226441, −2.64653888035360339250165104487, −2.62126628854938098941769799570, −2.55961237812147282896374830178, −2.53003733348728895260325066395, −2.39693178823830380205299030729, −2.30879919967879086078265191352, −2.14206530850284289128481118714, −2.13764635942259892076577912316, −1.90528593407225647632955287549, −1.86885124370133997709734611174, −1.71920227950621962189721105844, −1.71893340071186279030003002871, −1.67687613678664592924335244339, −1.35937732575795820611796251353, −1.08303561288070800139855714598, −1.01368648177773341261312339192, −0.893331891629461796850764631480, −0.78831146308015814067850852494, −0.71841696098465378766623501395, −0.65691979087499963968337557404, −0.34845751108907971996080621836, −0.02271222316587832260579304152, 0.02271222316587832260579304152, 0.34845751108907971996080621836, 0.65691979087499963968337557404, 0.71841696098465378766623501395, 0.78831146308015814067850852494, 0.893331891629461796850764631480, 1.01368648177773341261312339192, 1.08303561288070800139855714598, 1.35937732575795820611796251353, 1.67687613678664592924335244339, 1.71893340071186279030003002871, 1.71920227950621962189721105844, 1.86885124370133997709734611174, 1.90528593407225647632955287549, 2.13764635942259892076577912316, 2.14206530850284289128481118714, 2.30879919967879086078265191352, 2.39693178823830380205299030729, 2.53003733348728895260325066395, 2.55961237812147282896374830178, 2.62126628854938098941769799570, 2.64653888035360339250165104487, 2.69176810845821926821850226441, 2.73344355544314738311019382328, 2.79339141225435190876328042872
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.