Dirichlet series
| L(s) = 1 | − 2·2-s − 4·5-s + 8·10-s + 2·16-s + 24·17-s − 12·19-s + 8·25-s − 8·27-s − 32·29-s + 16·31-s + 8·32-s − 48·34-s − 16·37-s + 24·38-s − 64·43-s + 24·47-s − 16·50-s + 8·53-s + 16·54-s + 64·58-s − 28·59-s + 28·61-s − 32·62-s − 24·64-s + 32·74-s + 24·79-s − 8·80-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.78·5-s + 2.52·10-s + 1/2·16-s + 5.82·17-s − 2.75·19-s + 8/5·25-s − 1.53·27-s − 5.94·29-s + 2.87·31-s + 1.41·32-s − 8.23·34-s − 2.63·37-s + 3.89·38-s − 9.75·43-s + 3.50·47-s − 2.26·50-s + 1.09·53-s + 2.17·54-s + 8.40·58-s − 3.64·59-s + 3.58·61-s − 4.06·62-s − 3·64-s + 3.71·74-s + 2.70·79-s − 0.894·80-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{64} \cdot 7^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.56532\times 10^{12}\) |
| Root analytic conductor: | \(2.50205\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{64} \cdot 7^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.046859221\) |
| \(L(\frac12)\) | \(\approx\) | \(1.046859221\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + 7 p T^{4} + p^{4} T^{5} + p^{5} T^{6} + 5 p^{3} T^{7} + 13 p^{2} T^{8} + 5 p^{4} T^{9} + p^{7} T^{10} + p^{7} T^{11} + 7 p^{5} T^{12} + p^{8} T^{13} + p^{8} T^{14} + p^{8} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 \) | |
| good | 3 | \( 1 + 8 T^{3} + 16 T^{4} + 4 T^{5} + 32 T^{6} + 80 T^{7} + 142 T^{8} - 32 p T^{9} + 136 T^{10} + 44 p T^{11} - 896 T^{12} - 980 p T^{13} - 2848 T^{14} - 10496 T^{15} - 14765 T^{16} - 10496 p T^{17} - 2848 p^{2} T^{18} - 980 p^{4} T^{19} - 896 p^{4} T^{20} + 44 p^{6} T^{21} + 136 p^{6} T^{22} - 32 p^{8} T^{23} + 142 p^{8} T^{24} + 80 p^{9} T^{25} + 32 p^{10} T^{26} + 4 p^{11} T^{27} + 16 p^{12} T^{28} + 8 p^{13} T^{29} + p^{16} T^{32} \) |
| 5 | \( 1 + 4 T + 8 T^{2} + 16 T^{3} - 24 T^{5} + 32 T^{6} + 52 T^{7} + 638 T^{8} + 32 p^{2} T^{9} - 784 T^{10} - 1992 T^{11} - 1024 p^{2} T^{12} - 54648 T^{13} - 83928 T^{14} - 103712 T^{15} + 151491 T^{16} - 103712 p T^{17} - 83928 p^{2} T^{18} - 54648 p^{3} T^{19} - 1024 p^{6} T^{20} - 1992 p^{5} T^{21} - 784 p^{6} T^{22} + 32 p^{9} T^{23} + 638 p^{8} T^{24} + 52 p^{9} T^{25} + 32 p^{10} T^{26} - 24 p^{11} T^{27} + 16 p^{13} T^{29} + 8 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 11 | \( 1 + 64 T^{3} + 28 T^{4} + 288 T^{5} + 2048 T^{6} + 64 p T^{7} + 31722 T^{8} + 28928 T^{9} + 29184 T^{10} + 1215136 T^{11} - 975504 T^{12} + 5336608 T^{13} + 20563968 T^{14} - 12794368 T^{15} + 269763635 T^{16} - 12794368 p T^{17} + 20563968 p^{2} T^{18} + 5336608 p^{3} T^{19} - 975504 p^{4} T^{20} + 1215136 p^{5} T^{21} + 29184 p^{6} T^{22} + 28928 p^{7} T^{23} + 31722 p^{8} T^{24} + 64 p^{10} T^{25} + 2048 p^{10} T^{26} + 288 p^{11} T^{27} + 28 p^{12} T^{28} + 64 p^{13} T^{29} + p^{16} T^{32} \) | |
| 13 | \( ( 1 + 4 T^{3} - 336 T^{4} - 44 T^{5} + 8 T^{6} - 576 T^{7} + 70370 T^{8} - 576 p T^{9} + 8 p^{2} T^{10} - 44 p^{3} T^{11} - 336 p^{4} T^{12} + 4 p^{5} T^{13} + p^{8} T^{16} )^{2} \) | |
| 17 | \( ( 1 - 12 T + 36 T^{2} + 8 T^{3} + 738 T^{4} - 7052 T^{5} + 19504 T^{6} - 61860 T^{7} + 356163 T^{8} - 61860 p T^{9} + 19504 p^{2} T^{10} - 7052 p^{3} T^{11} + 738 p^{4} T^{12} + 8 p^{5} T^{13} + 36 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 + 12 T + 72 T^{2} + 80 T^{3} - 1360 T^{4} - 9912 T^{5} - 17824 T^{6} + 79060 T^{7} + 476062 T^{8} - 1531968 T^{9} - 21452048 T^{10} - 76046008 T^{11} + 122228480 T^{12} + 2012716152 T^{13} + 6889115048 T^{14} - 3798134592 T^{15} - 92850987965 T^{16} - 3798134592 p T^{17} + 6889115048 p^{2} T^{18} + 2012716152 p^{3} T^{19} + 122228480 p^{4} T^{20} - 76046008 p^{5} T^{21} - 21452048 p^{6} T^{22} - 1531968 p^{7} T^{23} + 476062 p^{8} T^{24} + 79060 p^{9} T^{25} - 17824 p^{10} T^{26} - 9912 p^{11} T^{27} - 1360 p^{12} T^{28} + 80 p^{13} T^{29} + 72 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( 1 + 104 T^{2} + 5660 T^{4} + 203952 T^{6} + 5136266 T^{8} + 77138552 T^{10} - 153385744 T^{12} - 48051835752 T^{14} - 1528122151917 T^{16} - 48051835752 p^{2} T^{18} - 153385744 p^{4} T^{20} + 77138552 p^{6} T^{22} + 5136266 p^{8} T^{24} + 203952 p^{10} T^{26} + 5660 p^{12} T^{28} + 104 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 + 16 T + 128 T^{2} + 688 T^{3} + 4924 T^{4} + 42960 T^{5} + 293760 T^{6} + 1412336 T^{7} + 6689574 T^{8} + 1412336 p T^{9} + 293760 p^{2} T^{10} + 42960 p^{3} T^{11} + 4924 p^{4} T^{12} + 688 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 - 8 T - 28 T^{2} + 384 T^{3} - 286 T^{4} - 560 T^{5} - 41296 T^{6} - 142584 T^{7} + 3267123 T^{8} - 142584 p T^{9} - 41296 p^{2} T^{10} - 560 p^{3} T^{11} - 286 p^{4} T^{12} + 384 p^{5} T^{13} - 28 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 + 16 T + 128 T^{2} + 736 T^{3} + 5092 T^{4} + 54576 T^{5} + 492288 T^{6} + 3502192 T^{7} + 23506282 T^{8} + 163005120 T^{9} + 1159430528 T^{10} + 8004987600 T^{11} + 52682505360 T^{12} + 342025730960 T^{13} + 2173882912640 T^{14} + 13646780305216 T^{15} + 84963554658739 T^{16} + 13646780305216 p T^{17} + 2173882912640 p^{2} T^{18} + 342025730960 p^{3} T^{19} + 52682505360 p^{4} T^{20} + 8004987600 p^{5} T^{21} + 1159430528 p^{6} T^{22} + 163005120 p^{7} T^{23} + 23506282 p^{8} T^{24} + 3502192 p^{9} T^{25} + 492288 p^{10} T^{26} + 54576 p^{11} T^{27} + 5092 p^{12} T^{28} + 736 p^{13} T^{29} + 128 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 40 T^{2} + 3564 T^{4} - 174616 T^{6} + 7396902 T^{8} - 174616 p^{2} T^{10} + 3564 p^{4} T^{12} - 40 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 + 32 T + 512 T^{2} + 5632 T^{3} + 45796 T^{4} + 275136 T^{5} + 1216512 T^{6} + 3773920 T^{7} + 12376614 T^{8} + 3773920 p T^{9} + 1216512 p^{2} T^{10} + 275136 p^{3} T^{11} + 45796 p^{4} T^{12} + 5632 p^{5} T^{13} + 512 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( ( 1 - 12 T - 36 T^{2} + 760 T^{3} + 2754 T^{4} - 40396 T^{5} - 79280 T^{6} + 20892 p T^{7} + 6477 p T^{8} + 20892 p^{2} T^{9} - 79280 p^{2} T^{10} - 40396 p^{3} T^{11} + 2754 p^{4} T^{12} + 760 p^{5} T^{13} - 36 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( 1 - 8 T + 32 T^{2} + 560 T^{3} - 9500 T^{4} + 64728 T^{5} - 57024 T^{6} - 3863896 T^{7} + 37730858 T^{8} - 216978272 T^{9} - 82214048 T^{10} + 10850527976 T^{11} - 111858364656 T^{12} + 551098329192 T^{13} + 37784252000 T^{14} - 27217026460960 T^{15} + 330020633863411 T^{16} - 27217026460960 p T^{17} + 37784252000 p^{2} T^{18} + 551098329192 p^{3} T^{19} - 111858364656 p^{4} T^{20} + 10850527976 p^{5} T^{21} - 82214048 p^{6} T^{22} - 216978272 p^{7} T^{23} + 37730858 p^{8} T^{24} - 3863896 p^{9} T^{25} - 57024 p^{10} T^{26} + 64728 p^{11} T^{27} - 9500 p^{12} T^{28} + 560 p^{13} T^{29} + 32 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 + 28 T + 392 T^{2} + 2016 T^{3} - 14128 T^{4} - 318864 T^{5} - 1357888 T^{6} + 15097572 T^{7} + 246003998 T^{8} + 988538432 T^{9} - 6664902384 T^{10} - 92152879952 T^{11} - 136222980224 T^{12} + 4524418830400 T^{13} + 32946087037192 T^{14} - 45881408275584 T^{15} - 1806770011595325 T^{16} - 45881408275584 p T^{17} + 32946087037192 p^{2} T^{18} + 4524418830400 p^{3} T^{19} - 136222980224 p^{4} T^{20} - 92152879952 p^{5} T^{21} - 6664902384 p^{6} T^{22} + 988538432 p^{7} T^{23} + 246003998 p^{8} T^{24} + 15097572 p^{9} T^{25} - 1357888 p^{10} T^{26} - 318864 p^{11} T^{27} - 14128 p^{12} T^{28} + 2016 p^{13} T^{29} + 392 p^{14} T^{30} + 28 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 28 T + 392 T^{2} - 880 T^{3} - 55520 T^{4} + 1092456 T^{5} - 8437728 T^{6} - 14398348 T^{7} + 1204434622 T^{8} - 14058048096 T^{9} + 62424918512 T^{10} + 529763952504 T^{11} - 11314478267904 T^{12} + 88275555985288 T^{13} - 128251932967384 T^{14} - 4645012180351648 T^{15} + 57006478590016003 T^{16} - 4645012180351648 p T^{17} - 128251932967384 p^{2} T^{18} + 88275555985288 p^{3} T^{19} - 11314478267904 p^{4} T^{20} + 529763952504 p^{5} T^{21} + 62424918512 p^{6} T^{22} - 14058048096 p^{7} T^{23} + 1204434622 p^{8} T^{24} - 14398348 p^{9} T^{25} - 8437728 p^{10} T^{26} + 1092456 p^{11} T^{27} - 55520 p^{12} T^{28} - 880 p^{13} T^{29} + 392 p^{14} T^{30} - 28 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 - 2048 T^{3} - 1508 T^{4} - 58368 T^{5} + 2097152 T^{6} + 2951168 T^{7} + 111706026 T^{8} - 1350295552 T^{9} - 1178075136 T^{10} - 103255061504 T^{11} + 685727681904 T^{12} - 1125117649920 T^{13} + 58594059878400 T^{14} - 394799369027584 T^{15} + 1281559065363699 T^{16} - 394799369027584 p T^{17} + 58594059878400 p^{2} T^{18} - 1125117649920 p^{3} T^{19} + 685727681904 p^{4} T^{20} - 103255061504 p^{5} T^{21} - 1178075136 p^{6} T^{22} - 1350295552 p^{7} T^{23} + 111706026 p^{8} T^{24} + 2951168 p^{9} T^{25} + 2097152 p^{10} T^{26} - 58368 p^{11} T^{27} - 1508 p^{12} T^{28} - 2048 p^{13} T^{29} + p^{16} T^{32} \) | |
| 71 | \( ( 1 - 72 T^{2} + 15132 T^{4} - 769400 T^{6} + 103336198 T^{8} - 769400 p^{2} T^{10} + 15132 p^{4} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 + 312 T^{2} + 55268 T^{4} + 6241808 T^{6} + 455982666 T^{8} + 13324541896 T^{10} - 1587641280496 T^{12} - 296939848172984 T^{14} - 27090025565051885 T^{16} - 296939848172984 p^{2} T^{18} - 1587641280496 p^{4} T^{20} + 13324541896 p^{6} T^{22} + 455982666 p^{8} T^{24} + 6241808 p^{10} T^{26} + 55268 p^{12} T^{28} + 312 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 - 12 T - 12 T^{2} + 392 T^{3} + 746 T^{4} - 25876 T^{5} + 594544 T^{6} - 1024876 T^{7} - 49948285 T^{8} - 1024876 p T^{9} + 594544 p^{2} T^{10} - 25876 p^{3} T^{11} + 746 p^{4} T^{12} + 392 p^{5} T^{13} - 12 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 + 4 T^{3} + 24944 T^{4} - 964 T^{5} + 8 T^{6} - 52496 T^{7} + 249837570 T^{8} - 52496 p T^{9} + 8 p^{2} T^{10} - 964 p^{3} T^{11} + 24944 p^{4} T^{12} + 4 p^{5} T^{13} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 + 280 T^{2} + 25444 T^{4} + 369232 T^{6} + 61693642 T^{8} + 37754988008 T^{10} + 54229589136 p T^{12} + 198590973245288 T^{14} + 261899859909523 T^{16} + 198590973245288 p^{2} T^{18} + 54229589136 p^{5} T^{20} + 37754988008 p^{6} T^{22} + 61693642 p^{8} T^{24} + 369232 p^{10} T^{26} + 25444 p^{12} T^{28} + 280 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( ( 1 - 16 T + 364 T^{2} - 4008 T^{3} + 52510 T^{4} - 4008 p T^{5} + 364 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
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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
−2.76908585189651110237638063281, −2.68165659304051860367199858160, −2.46860830927600097575610697654, −2.29908636703836337710636035180, −2.20271142103659188422088070326, −2.16864011178727554546666097605, −2.15907069867154111959251254194, −2.05568608714693906671949111368, −2.05043778373961596631419483145, −1.97401477191300654976668224326, −1.91824481197374694372744729072, −1.80144837465464268800172351550, −1.65354666267759970674101552297, −1.63597655571203896558250061329, −1.44557125903602089891961907465, −1.38089991177625148258270406978, −1.13421729419213883257552096824, −1.09080646393959547958695122767, −1.03176344966935513818500407259, −0.805239803229678138187522588629, −0.67410960467706994013364565261, −0.46067352066132051171115427465, −0.37109103235270689000116437297, −0.29949660605600630103365392833, −0.27765978605720780918795264830, 0.27765978605720780918795264830, 0.29949660605600630103365392833, 0.37109103235270689000116437297, 0.46067352066132051171115427465, 0.67410960467706994013364565261, 0.805239803229678138187522588629, 1.03176344966935513818500407259, 1.09080646393959547958695122767, 1.13421729419213883257552096824, 1.38089991177625148258270406978, 1.44557125903602089891961907465, 1.63597655571203896558250061329, 1.65354666267759970674101552297, 1.80144837465464268800172351550, 1.91824481197374694372744729072, 1.97401477191300654976668224326, 2.05043778373961596631419483145, 2.05568608714693906671949111368, 2.15907069867154111959251254194, 2.16864011178727554546666097605, 2.20271142103659188422088070326, 2.29908636703836337710636035180, 2.46860830927600097575610697654, 2.68165659304051860367199858160, 2.76908585189651110237638063281
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.