Dirichlet series
| L(s) = 1 | − 16·5-s + 16·9-s + 4·11-s + 136·25-s − 16·31-s + 4·43-s − 256·45-s − 4·49-s − 64·55-s + 8·61-s − 20·67-s + 122·81-s + 64·99-s − 44·107-s − 16·113-s − 64·121-s − 816·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 256·155-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 7.15·5-s + 16/3·9-s + 1.20·11-s + 27.1·25-s − 2.87·31-s + 0.609·43-s − 38.1·45-s − 4/7·49-s − 8.62·55-s + 1.02·61-s − 2.44·67-s + 13.5·81-s + 6.43·99-s − 4.25·107-s − 1.50·113-s − 5.81·121-s − 72.9·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 20.5·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{80} \cdot 5^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.67464\times 10^{15}\) |
| Root analytic conductor: | \(2.99052\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{80} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1653120878\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1653120878\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T )^{16} \) | |
| 7 | \( 1 + 4 T^{2} + 36 T^{3} + 4 T^{4} + 204 T^{5} + 100 p T^{6} + 904 T^{7} + 5814 T^{8} + 904 p T^{9} + 100 p^{3} T^{10} + 204 p^{3} T^{11} + 4 p^{4} T^{12} + 36 p^{5} T^{13} + 4 p^{6} T^{14} + p^{8} T^{16} \) | |
| good | 3 | \( 1 - 16 T^{2} + 134 T^{4} - 772 T^{6} + 1171 p T^{8} - 4648 p T^{10} + 51310 T^{12} - 176516 T^{14} + 185548 p T^{16} - 176516 p^{2} T^{18} + 51310 p^{4} T^{20} - 4648 p^{7} T^{22} + 1171 p^{9} T^{24} - 772 p^{10} T^{26} + 134 p^{12} T^{28} - 16 p^{14} T^{30} + p^{16} T^{32} \) |
| 11 | \( ( 1 - 2 T + 38 T^{2} - 102 T^{3} + 849 T^{4} - 2360 T^{5} + 13338 T^{6} - 36764 T^{7} + 162252 T^{8} - 36764 p T^{9} + 13338 p^{2} T^{10} - 2360 p^{3} T^{11} + 849 p^{4} T^{12} - 102 p^{5} T^{13} + 38 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 13 | \( ( 1 + 42 T^{2} - 12 T^{3} + 1109 T^{4} - 552 T^{5} + 21146 T^{6} - 10156 T^{7} + 312412 T^{8} - 10156 p T^{9} + 21146 p^{2} T^{10} - 552 p^{3} T^{11} + 1109 p^{4} T^{12} - 12 p^{5} T^{13} + 42 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 - 8 p T^{2} + 9398 T^{4} - 441684 T^{6} + 15867977 T^{8} - 462761176 T^{10} + 11338206366 T^{12} - 238218042700 T^{14} + 4340632328516 T^{16} - 238218042700 p^{2} T^{18} + 11338206366 p^{4} T^{20} - 462761176 p^{6} T^{22} + 15867977 p^{8} T^{24} - 441684 p^{10} T^{26} + 9398 p^{12} T^{28} - 8 p^{15} T^{30} + p^{16} T^{32} \) | |
| 19 | \( 1 - 120 T^{2} + 7112 T^{4} - 14584 p T^{6} + 7791548 T^{8} - 163541048 T^{10} + 2611826040 T^{12} - 33956352872 T^{14} + 500871514950 T^{16} - 33956352872 p^{2} T^{18} + 2611826040 p^{4} T^{20} - 163541048 p^{6} T^{22} + 7791548 p^{8} T^{24} - 14584 p^{11} T^{26} + 7112 p^{12} T^{28} - 120 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 - 208 T^{2} + 22056 T^{4} - 1571904 T^{6} + 84053084 T^{8} - 3572959584 T^{10} + 124922536472 T^{12} - 3666816166032 T^{14} + 91371160875334 T^{16} - 3666816166032 p^{2} T^{18} + 124922536472 p^{4} T^{20} - 3572959584 p^{6} T^{22} + 84053084 p^{8} T^{24} - 1571904 p^{10} T^{26} + 22056 p^{12} T^{28} - 208 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( 1 - 292 T^{2} + 40806 T^{4} - 3658824 T^{6} + 238823825 T^{8} - 12229044600 T^{10} + 516201061814 T^{12} - 18541519099068 T^{14} + 576457234322116 T^{16} - 18541519099068 p^{2} T^{18} + 516201061814 p^{4} T^{20} - 12229044600 p^{6} T^{22} + 238823825 p^{8} T^{24} - 3658824 p^{10} T^{26} + 40806 p^{12} T^{28} - 292 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( ( 1 + 8 T + 156 T^{2} + 1296 T^{3} + 13292 T^{4} + 96288 T^{5} + 741700 T^{6} + 4392984 T^{7} + 27984230 T^{8} + 4392984 p T^{9} + 741700 p^{2} T^{10} + 96288 p^{3} T^{11} + 13292 p^{4} T^{12} + 1296 p^{5} T^{13} + 156 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 - 340 T^{2} + 59936 T^{4} - 7152252 T^{6} + 641015212 T^{8} - 45497691444 T^{10} + 2635851611424 T^{12} - 126850111202108 T^{14} + 5118309159851494 T^{16} - 126850111202108 p^{2} T^{18} + 2635851611424 p^{4} T^{20} - 45497691444 p^{6} T^{22} + 641015212 p^{8} T^{24} - 7152252 p^{10} T^{26} + 59936 p^{12} T^{28} - 340 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( 1 - 232 T^{2} + 32104 T^{4} - 3202936 T^{6} + 255876156 T^{8} - 17043323304 T^{10} + 975399522136 T^{12} - 48558130743544 T^{14} + 2123654190398598 T^{16} - 48558130743544 p^{2} T^{18} + 975399522136 p^{4} T^{20} - 17043323304 p^{6} T^{22} + 255876156 p^{8} T^{24} - 3202936 p^{10} T^{26} + 32104 p^{12} T^{28} - 232 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( ( 1 - 2 T + 216 T^{2} - 270 T^{3} + 22980 T^{4} - 21190 T^{5} + 1612552 T^{6} - 1259098 T^{7} + 81139350 T^{8} - 1259098 p T^{9} + 1612552 p^{2} T^{10} - 21190 p^{3} T^{11} + 22980 p^{4} T^{12} - 270 p^{5} T^{13} + 216 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( ( 1 + 210 T^{2} + 424 T^{3} + 20501 T^{4} + 71348 T^{5} + 1376534 T^{6} + 5382808 T^{7} + 72656708 T^{8} + 5382808 p T^{9} + 1376534 p^{2} T^{10} + 71348 p^{3} T^{11} + 20501 p^{4} T^{12} + 424 p^{5} T^{13} + 210 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 53 | \( 1 - 444 T^{2} + 95440 T^{4} - 13625140 T^{6} + 1487417420 T^{8} - 133281679708 T^{10} + 10117851372400 T^{12} - 660669978323828 T^{14} + 37485039307099558 T^{16} - 660669978323828 p^{2} T^{18} + 10117851372400 p^{4} T^{20} - 133281679708 p^{6} T^{22} + 1487417420 p^{8} T^{24} - 13625140 p^{10} T^{26} + 95440 p^{12} T^{28} - 444 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( 1 - 544 T^{2} + 142872 T^{4} - 24347616 T^{6} + 3056211228 T^{8} - 304237741216 T^{10} + 25229659138856 T^{12} - 1801962119594848 T^{14} + 113058863038407174 T^{16} - 1801962119594848 p^{2} T^{18} + 25229659138856 p^{4} T^{20} - 304237741216 p^{6} T^{22} + 3056211228 p^{8} T^{24} - 24347616 p^{10} T^{26} + 142872 p^{12} T^{28} - 544 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 - 4 T + 220 T^{2} - 452 T^{3} + 21276 T^{4} - 14700 T^{5} + 1481892 T^{6} - 584396 T^{7} + 94645158 T^{8} - 584396 p T^{9} + 1481892 p^{2} T^{10} - 14700 p^{3} T^{11} + 21276 p^{4} T^{12} - 452 p^{5} T^{13} + 220 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( ( 1 + 10 T + 356 T^{2} + 3486 T^{3} + 61044 T^{4} + 581590 T^{5} + 6733628 T^{6} + 59512546 T^{7} + 528259766 T^{8} + 59512546 p T^{9} + 6733628 p^{2} T^{10} + 581590 p^{3} T^{11} + 61044 p^{4} T^{12} + 3486 p^{5} T^{13} + 356 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 - 644 T^{2} + 198624 T^{4} - 39221932 T^{6} + 5591888940 T^{8} - 617721349412 T^{10} + 55999222297184 T^{12} - 4424201931687980 T^{14} + 322698601587423398 T^{16} - 4424201931687980 p^{2} T^{18} + 55999222297184 p^{4} T^{20} - 617721349412 p^{6} T^{22} + 5591888940 p^{8} T^{24} - 39221932 p^{10} T^{26} + 198624 p^{12} T^{28} - 644 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( 1 - 556 T^{2} + 165360 T^{4} - 34160612 T^{6} + 5437568204 T^{8} - 702529893004 T^{10} + 75867773158224 T^{12} - 6964409971692516 T^{14} + 548248027302940262 T^{16} - 6964409971692516 p^{2} T^{18} + 75867773158224 p^{4} T^{20} - 702529893004 p^{6} T^{22} + 5437568204 p^{8} T^{24} - 34160612 p^{10} T^{26} + 165360 p^{12} T^{28} - 556 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( 1 - 672 T^{2} + 239646 T^{4} - 58660812 T^{6} + 10893265057 T^{8} - 1614943604520 T^{10} + 196775937800454 T^{12} - 20041769506403180 T^{14} + 1721626596797889012 T^{16} - 20041769506403180 p^{2} T^{18} + 196775937800454 p^{4} T^{20} - 1614943604520 p^{6} T^{22} + 10893265057 p^{8} T^{24} - 58660812 p^{10} T^{26} + 239646 p^{12} T^{28} - 672 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( 1 - 764 T^{2} + 280944 T^{4} - 66837700 T^{6} + 11715405100 T^{8} - 1636191670636 T^{10} + 191589412200016 T^{12} - 19378096949526164 T^{14} + 1715972865423799398 T^{16} - 19378096949526164 p^{2} T^{18} + 191589412200016 p^{4} T^{20} - 1636191670636 p^{6} T^{22} + 11715405100 p^{8} T^{24} - 66837700 p^{10} T^{26} + 280944 p^{12} T^{28} - 764 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( 1 - 904 T^{2} + 398408 T^{4} - 114210008 T^{6} + 23987033852 T^{8} - 3943146167496 T^{10} + 529135284297848 T^{12} - 59592961731275736 T^{14} + 5726182845912077830 T^{16} - 59592961731275736 p^{2} T^{18} + 529135284297848 p^{4} T^{20} - 3943146167496 p^{6} T^{22} + 23987033852 p^{8} T^{24} - 114210008 p^{10} T^{26} + 398408 p^{12} T^{28} - 904 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( 1 - 664 T^{2} + 230726 T^{4} - 56036276 T^{6} + 10668809817 T^{8} - 1685656930712 T^{10} + 227831471257294 T^{12} - 26797828856727836 T^{14} + 2769020205902604324 T^{16} - 26797828856727836 p^{2} T^{18} + 227831471257294 p^{4} T^{20} - 1685656930712 p^{6} T^{22} + 10668809817 p^{8} T^{24} - 56036276 p^{10} T^{26} + 230726 p^{12} T^{28} - 664 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−2.64374014412449823369604284381, −2.62809313592916042453190702165, −2.43881208586371189270119880874, −2.41698675675498187091954260413, −2.15254157518453180757076059161, −2.10192112610336348079905815373, −2.06667116487886167684138649917, −2.04439432169986665036911698767, −1.70343868084902920611587348328, −1.63997868849849536012047378539, −1.57519029750201684495183102656, −1.53130466862444561871069703712, −1.44272426362794815449177917559, −1.38287507660435935777690649677, −1.31879835793098123063851290642, −1.26076143360306152201734627811, −1.22155211821892572429915485596, −0.997187102909741545647440440454, −0.963002119996702278466358610348, −0.923009021091535190446585234908, −0.59669716054952721708160606038, −0.40182354346054904262165218835, −0.32147641455116785659640311840, −0.30316440824889368243273170751, −0.05526172139875803334748769371, 0.05526172139875803334748769371, 0.30316440824889368243273170751, 0.32147641455116785659640311840, 0.40182354346054904262165218835, 0.59669716054952721708160606038, 0.923009021091535190446585234908, 0.963002119996702278466358610348, 0.997187102909741545647440440454, 1.22155211821892572429915485596, 1.26076143360306152201734627811, 1.31879835793098123063851290642, 1.38287507660435935777690649677, 1.44272426362794815449177917559, 1.53130466862444561871069703712, 1.57519029750201684495183102656, 1.63997868849849536012047378539, 1.70343868084902920611587348328, 2.04439432169986665036911698767, 2.06667116487886167684138649917, 2.10192112610336348079905815373, 2.15254157518453180757076059161, 2.41698675675498187091954260413, 2.43881208586371189270119880874, 2.62809313592916042453190702165, 2.64374014412449823369604284381
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.