Dirichlet series
| L(s) = 1 | − 16·2-s + 2·3-s + 136·4-s + 8·5-s − 32·6-s + 4·7-s − 816·8-s − 9-s − 128·10-s + 11-s + 272·12-s + 2·13-s − 64·14-s + 16·15-s + 3.87e3·16-s + 11·17-s + 16·18-s − 2·19-s + 1.08e3·20-s + 8·21-s − 16·22-s + 11·23-s − 1.63e3·24-s + 28·25-s − 32·26-s − 7·27-s + 544·28-s + ⋯ |
| L(s) = 1 | − 11.3·2-s + 1.15·3-s + 68·4-s + 3.57·5-s − 13.0·6-s + 1.51·7-s − 288.·8-s − 1/3·9-s − 40.4·10-s + 0.301·11-s + 78.5·12-s + 0.554·13-s − 17.1·14-s + 4.13·15-s + 969·16-s + 2.66·17-s + 3.77·18-s − 0.458·19-s + 243.·20-s + 1.74·21-s − 3.41·22-s + 2.29·23-s − 333.·24-s + 28/5·25-s − 6.27·26-s − 1.34·27-s + 102.·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.68221\times 10^{11}\) |
| Root analytic conductor: | \(2.24289\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.275399849\) |
| \(L(\frac12)\) | \(\approx\) | \(1.275399849\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{16} \) |
| 3 | \( 1 - 2 T + 5 T^{2} - 5 T^{3} + 8 T^{4} - 8 T^{5} - 20 T^{6} + 17 p T^{7} - 61 p T^{8} + 17 p^{2} T^{9} - 20 p^{2} T^{10} - 8 p^{3} T^{11} + 8 p^{4} T^{12} - 5 p^{5} T^{13} + 5 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 5 | \( ( 1 - T + T^{2} )^{8} \) | |
| 7 | \( 1 - 4 T + 15 T^{2} - 46 T^{3} + 85 T^{4} - 234 T^{5} + 295 T^{6} + 50 p T^{7} - 39 T^{8} + 50 p^{2} T^{9} + 295 p^{2} T^{10} - 234 p^{3} T^{11} + 85 p^{4} T^{12} - 46 p^{5} T^{13} + 15 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 11 | \( 1 - T - 35 T^{2} + 182 T^{3} + 577 T^{4} - 5289 T^{5} + 8314 T^{6} + 80986 T^{7} - 380701 T^{8} + 165840 T^{9} + 6417590 T^{10} - 21138520 T^{11} - 17857183 T^{12} + 372829025 T^{13} - 755353711 T^{14} - 1808389533 T^{15} + 15175368627 T^{16} - 1808389533 p T^{17} - 755353711 p^{2} T^{18} + 372829025 p^{3} T^{19} - 17857183 p^{4} T^{20} - 21138520 p^{5} T^{21} + 6417590 p^{6} T^{22} + 165840 p^{7} T^{23} - 380701 p^{8} T^{24} + 80986 p^{9} T^{25} + 8314 p^{10} T^{26} - 5289 p^{11} T^{27} + 577 p^{12} T^{28} + 182 p^{13} T^{29} - 35 p^{14} T^{30} - p^{15} T^{31} + p^{16} T^{32} \) |
| 13 | \( 1 - 2 T - 50 T^{2} + 148 T^{3} + 1054 T^{4} - 3995 T^{5} - 11532 T^{6} + 43443 T^{7} + 117342 T^{8} + 20138 T^{9} - 2880550 T^{10} - 3340678 T^{11} + 58331516 T^{12} - 10068559 T^{13} - 672470458 T^{14} + 306068259 T^{15} + 6775158987 T^{16} + 306068259 p T^{17} - 672470458 p^{2} T^{18} - 10068559 p^{3} T^{19} + 58331516 p^{4} T^{20} - 3340678 p^{5} T^{21} - 2880550 p^{6} T^{22} + 20138 p^{7} T^{23} + 117342 p^{8} T^{24} + 43443 p^{9} T^{25} - 11532 p^{10} T^{26} - 3995 p^{11} T^{27} + 1054 p^{12} T^{28} + 148 p^{13} T^{29} - 50 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 - 11 T + 36 T^{2} + 123 T^{3} - 1898 T^{4} + 9669 T^{5} - 17977 T^{6} - 83870 T^{7} + 834166 T^{8} - 3577974 T^{9} + 6674756 T^{10} + 20040864 T^{11} - 226666002 T^{12} + 969820747 T^{13} - 1237622341 T^{14} - 9875829222 T^{15} + 67383134907 T^{16} - 9875829222 p T^{17} - 1237622341 p^{2} T^{18} + 969820747 p^{3} T^{19} - 226666002 p^{4} T^{20} + 20040864 p^{5} T^{21} + 6674756 p^{6} T^{22} - 3577974 p^{7} T^{23} + 834166 p^{8} T^{24} - 83870 p^{9} T^{25} - 17977 p^{10} T^{26} + 9669 p^{11} T^{27} - 1898 p^{12} T^{28} + 123 p^{13} T^{29} + 36 p^{14} T^{30} - 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 + 2 T - 54 T^{2} + 52 T^{3} + 1522 T^{4} - 5531 T^{5} - 18356 T^{6} + 142945 T^{7} - 156428 T^{8} - 2136646 T^{9} + 5666688 T^{10} + 23411554 T^{11} - 135081014 T^{12} - 508657467 T^{13} + 5117825840 T^{14} + 6906391627 T^{15} - 121849980107 T^{16} + 6906391627 p T^{17} + 5117825840 p^{2} T^{18} - 508657467 p^{3} T^{19} - 135081014 p^{4} T^{20} + 23411554 p^{5} T^{21} + 5666688 p^{6} T^{22} - 2136646 p^{7} T^{23} - 156428 p^{8} T^{24} + 142945 p^{9} T^{25} - 18356 p^{10} T^{26} - 5531 p^{11} T^{27} + 1522 p^{12} T^{28} + 52 p^{13} T^{29} - 54 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( 1 - 11 T - 8 T^{2} + 511 T^{3} - 1175 T^{4} - 318 p T^{5} + 31375 T^{6} - 146176 T^{7} + 863162 T^{8} + 3529440 T^{9} - 34166005 T^{10} + 8815708 T^{11} + 134850764 T^{12} - 1234120523 T^{13} + 19738400159 T^{14} + 2588217123 T^{15} - 586357209291 T^{16} + 2588217123 p T^{17} + 19738400159 p^{2} T^{18} - 1234120523 p^{3} T^{19} + 134850764 p^{4} T^{20} + 8815708 p^{5} T^{21} - 34166005 p^{6} T^{22} + 3529440 p^{7} T^{23} + 863162 p^{8} T^{24} - 146176 p^{9} T^{25} + 31375 p^{10} T^{26} - 318 p^{12} T^{27} - 1175 p^{12} T^{28} + 511 p^{13} T^{29} - 8 p^{14} T^{30} - 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 - 17 T + 46 T^{2} + 223 T^{3} + 3250 T^{4} - 18084 T^{5} - 204302 T^{6} + 750182 T^{7} + 3378518 T^{8} + 10195995 T^{9} - 165509581 T^{10} + 1120124 p T^{11} + 4544588759 T^{12} - 14362280831 T^{13} - 142821670546 T^{14} + 116125876596 T^{15} + 6133655884938 T^{16} + 116125876596 p T^{17} - 142821670546 p^{2} T^{18} - 14362280831 p^{3} T^{19} + 4544588759 p^{4} T^{20} + 1120124 p^{6} T^{21} - 165509581 p^{6} T^{22} + 10195995 p^{7} T^{23} + 3378518 p^{8} T^{24} + 750182 p^{9} T^{25} - 204302 p^{10} T^{26} - 18084 p^{11} T^{27} + 3250 p^{12} T^{28} + 223 p^{13} T^{29} + 46 p^{14} T^{30} - 17 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( ( 1 - 15 T + 294 T^{2} - 2963 T^{3} + 33518 T^{4} - 256035 T^{5} + 2103711 T^{6} - 12738397 T^{7} + 81705846 T^{8} - 12738397 p T^{9} + 2103711 p^{2} T^{10} - 256035 p^{3} T^{11} + 33518 p^{4} T^{12} - 2963 p^{5} T^{13} + 294 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 + 2 T - 109 T^{2} - 494 T^{3} + 4396 T^{4} + 37709 T^{5} + 5949 T^{6} - 1192233 T^{7} - 6568281 T^{8} + 428352 p T^{9} + 180379971 T^{10} - 663666882 T^{11} - 2633817279 T^{12} + 62063414607 T^{13} + 300032298102 T^{14} - 1464225969303 T^{15} - 17940373399443 T^{16} - 1464225969303 p T^{17} + 300032298102 p^{2} T^{18} + 62063414607 p^{3} T^{19} - 2633817279 p^{4} T^{20} - 663666882 p^{5} T^{21} + 180379971 p^{6} T^{22} + 428352 p^{8} T^{23} - 6568281 p^{8} T^{24} - 1192233 p^{9} T^{25} + 5949 p^{10} T^{26} + 37709 p^{11} T^{27} + 4396 p^{12} T^{28} - 494 p^{13} T^{29} - 109 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 - 7 T - 58 T^{2} + 821 T^{3} - 5081 T^{4} + 3487 T^{5} + 202690 T^{6} - 2313484 T^{7} + 22320617 T^{8} - 84574319 T^{9} - 449228446 T^{10} + 8174132915 T^{11} - 61865466443 T^{12} + 229869969509 T^{13} + 311827818587 T^{14} - 13738704403974 T^{15} + 126160448679906 T^{16} - 13738704403974 p T^{17} + 311827818587 p^{2} T^{18} + 229869969509 p^{3} T^{19} - 61865466443 p^{4} T^{20} + 8174132915 p^{5} T^{21} - 449228446 p^{6} T^{22} - 84574319 p^{7} T^{23} + 22320617 p^{8} T^{24} - 2313484 p^{9} T^{25} + 202690 p^{10} T^{26} + 3487 p^{11} T^{27} - 5081 p^{12} T^{28} + 821 p^{13} T^{29} - 58 p^{14} T^{30} - 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 + 13 T - 110 T^{2} - 1567 T^{3} + 10823 T^{4} + 109505 T^{5} - 883584 T^{6} - 5592870 T^{7} + 52864479 T^{8} + 206005533 T^{9} - 2766579456 T^{10} - 7286801151 T^{11} + 131529658809 T^{12} + 225678713547 T^{13} - 6233727820281 T^{14} - 3123476792046 T^{15} + 288754313221062 T^{16} - 3123476792046 p T^{17} - 6233727820281 p^{2} T^{18} + 225678713547 p^{3} T^{19} + 131529658809 p^{4} T^{20} - 7286801151 p^{5} T^{21} - 2766579456 p^{6} T^{22} + 206005533 p^{7} T^{23} + 52864479 p^{8} T^{24} - 5592870 p^{9} T^{25} - 883584 p^{10} T^{26} + 109505 p^{11} T^{27} + 10823 p^{12} T^{28} - 1567 p^{13} T^{29} - 110 p^{14} T^{30} + 13 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( ( 1 - 5 T + 167 T^{2} - 679 T^{3} + 12865 T^{4} - 55056 T^{5} + 721843 T^{6} - 3490742 T^{7} + 35510129 T^{8} - 3490742 p T^{9} + 721843 p^{2} T^{10} - 55056 p^{3} T^{11} + 12865 p^{4} T^{12} - 679 p^{5} T^{13} + 167 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( 1 - 18 T - 25 T^{2} + 3094 T^{3} - 20382 T^{4} - 155341 T^{5} + 2463947 T^{6} - 2962263 T^{7} - 127200023 T^{8} + 760818902 T^{9} + 3333886647 T^{10} - 61667632700 T^{11} + 106780502839 T^{12} + 3773454519213 T^{13} - 28393687643164 T^{14} - 98984202945069 T^{15} + 2205152767325745 T^{16} - 98984202945069 p T^{17} - 28393687643164 p^{2} T^{18} + 3773454519213 p^{3} T^{19} + 106780502839 p^{4} T^{20} - 61667632700 p^{5} T^{21} + 3333886647 p^{6} T^{22} + 760818902 p^{7} T^{23} - 127200023 p^{8} T^{24} - 2962263 p^{9} T^{25} + 2463947 p^{10} T^{26} - 155341 p^{11} T^{27} - 20382 p^{12} T^{28} + 3094 p^{13} T^{29} - 25 p^{14} T^{30} - 18 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( ( 1 + T + 317 T^{2} + 659 T^{3} + 48346 T^{4} + 123360 T^{5} + 4790140 T^{6} + 11791354 T^{7} + 334628690 T^{8} + 11791354 p T^{9} + 4790140 p^{2} T^{10} + 123360 p^{3} T^{11} + 48346 p^{4} T^{12} + 659 p^{5} T^{13} + 317 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 27 T + 644 T^{2} - 9523 T^{3} + 127396 T^{4} - 1294067 T^{5} + 12537951 T^{6} - 101639815 T^{7} + 845706468 T^{8} - 101639815 p T^{9} + 12537951 p^{2} T^{10} - 1294067 p^{3} T^{11} + 127396 p^{4} T^{12} - 9523 p^{5} T^{13} + 644 p^{6} T^{14} - 27 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 10 T + 183 T^{2} - 1911 T^{3} + 28302 T^{4} - 239130 T^{5} + 2683324 T^{6} - 21197185 T^{7} + 215653584 T^{8} - 21197185 p T^{9} + 2683324 p^{2} T^{10} - 239130 p^{3} T^{11} + 28302 p^{4} T^{12} - 1911 p^{5} T^{13} + 183 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( ( 1 + 19 T + 400 T^{2} + 3773 T^{3} + 41473 T^{4} + 130458 T^{5} + 473407 T^{6} - 19694168 T^{7} - 122824232 T^{8} - 19694168 p T^{9} + 473407 p^{2} T^{10} + 130458 p^{3} T^{11} + 41473 p^{4} T^{12} + 3773 p^{5} T^{13} + 400 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 + 8 T - 390 T^{2} - 908 T^{3} + 103150 T^{4} - 105353 T^{5} - 16697378 T^{6} + 75246967 T^{7} + 1927206172 T^{8} - 14189615254 T^{9} - 139595267550 T^{10} + 1686118517626 T^{11} + 5218854241786 T^{12} - 119151906708597 T^{13} + 190749604993352 T^{14} + 3821131738775017 T^{15} - 31483648169073569 T^{16} + 3821131738775017 p T^{17} + 190749604993352 p^{2} T^{18} - 119151906708597 p^{3} T^{19} + 5218854241786 p^{4} T^{20} + 1686118517626 p^{5} T^{21} - 139595267550 p^{6} T^{22} - 14189615254 p^{7} T^{23} + 1927206172 p^{8} T^{24} + 75246967 p^{9} T^{25} - 16697378 p^{10} T^{26} - 105353 p^{11} T^{27} + 103150 p^{12} T^{28} - 908 p^{13} T^{29} - 390 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( ( 1 - 25 T + 807 T^{2} - 13099 T^{3} + 239467 T^{4} - 2871909 T^{5} + 37853854 T^{6} - 357554245 T^{7} + 46953672 p T^{8} - 357554245 p T^{9} + 37853854 p^{2} T^{10} - 2871909 p^{3} T^{11} + 239467 p^{4} T^{12} - 13099 p^{5} T^{13} + 807 p^{6} T^{14} - 25 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 - 2 T - 379 T^{2} + 46 T^{3} + 71929 T^{4} + 111425 T^{5} - 9247598 T^{6} - 27250484 T^{7} + 925626125 T^{8} + 3684425840 T^{9} - 76460286955 T^{10} - 351032380757 T^{11} + 5350114348114 T^{12} + 23636419050310 T^{13} - 345108599067805 T^{14} - 759472537118196 T^{15} + 25092808099894680 T^{16} - 759472537118196 p T^{17} - 345108599067805 p^{2} T^{18} + 23636419050310 p^{3} T^{19} + 5350114348114 p^{4} T^{20} - 351032380757 p^{5} T^{21} - 76460286955 p^{6} T^{22} + 3684425840 p^{7} T^{23} + 925626125 p^{8} T^{24} - 27250484 p^{9} T^{25} - 9247598 p^{10} T^{26} + 111425 p^{11} T^{27} + 71929 p^{12} T^{28} + 46 p^{13} T^{29} - 379 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( 1 + 6 T - 361 T^{2} - 3094 T^{3} + 62940 T^{4} + 748723 T^{5} - 6298663 T^{6} - 119355885 T^{7} + 250393933 T^{8} + 13841837530 T^{9} + 33272999661 T^{10} - 1196873353342 T^{11} - 8138474562173 T^{12} + 74723700233589 T^{13} + 1028573157441158 T^{14} - 2348330140909041 T^{15} - 100635305983812723 T^{16} - 2348330140909041 p T^{17} + 1028573157441158 p^{2} T^{18} + 74723700233589 p^{3} T^{19} - 8138474562173 p^{4} T^{20} - 1196873353342 p^{5} T^{21} + 33272999661 p^{6} T^{22} + 13841837530 p^{7} T^{23} + 250393933 p^{8} T^{24} - 119355885 p^{9} T^{25} - 6298663 p^{10} T^{26} + 748723 p^{11} T^{27} + 62940 p^{12} T^{28} - 3094 p^{13} T^{29} - 361 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 - 26 T - 56 T^{2} + 2492 T^{3} + 90143 T^{4} - 817018 T^{5} - 11928114 T^{6} - 57652506 T^{7} + 2793757095 T^{8} + 11900406246 T^{9} - 106656506274 T^{10} - 4250712480600 T^{11} + 2907448664925 T^{12} + 284134319779998 T^{13} + 4080312688533132 T^{14} - 249527458766268 p T^{15} - 351229803717868731 T^{16} - 249527458766268 p^{2} T^{17} + 4080312688533132 p^{2} T^{18} + 284134319779998 p^{3} T^{19} + 2907448664925 p^{4} T^{20} - 4250712480600 p^{5} T^{21} - 106656506274 p^{6} T^{22} + 11900406246 p^{7} T^{23} + 2793757095 p^{8} T^{24} - 57652506 p^{9} T^{25} - 11928114 p^{10} T^{26} - 817018 p^{11} T^{27} + 90143 p^{12} T^{28} + 2492 p^{13} T^{29} - 56 p^{14} T^{30} - 26 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−2.62795830361231431315216521751, −2.53031063379358800228686114144, −2.43552873561276168320235241440, −2.38950179677767182420383218145, −2.22940446256406910489669197778, −2.21759646319098869761016061067, −2.19154741720087814963968006267, −2.15244198884246532912923334627, −1.95130698274305457789969861333, −1.84113802483322056501018123538, −1.81676894652539480540176291978, −1.76487442993915784144351759858, −1.74492879988632525760616594683, −1.49658977389916115675831477868, −1.35100470718697936698890405560, −1.34140063108091855632433071203, −1.11237675984058315613027233949, −1.08445874562826869742992800865, −1.04524203634278503185634118968, −0.859189649824870529058602089556, −0.819694694871868081607301204702, −0.77336241069871618124826443583, −0.73193651490986429127265168186, −0.48967435316127325420660519437, −0.46979706644094715782197522417, 0.46979706644094715782197522417, 0.48967435316127325420660519437, 0.73193651490986429127265168186, 0.77336241069871618124826443583, 0.819694694871868081607301204702, 0.859189649824870529058602089556, 1.04524203634278503185634118968, 1.08445874562826869742992800865, 1.11237675984058315613027233949, 1.34140063108091855632433071203, 1.35100470718697936698890405560, 1.49658977389916115675831477868, 1.74492879988632525760616594683, 1.76487442993915784144351759858, 1.81676894652539480540176291978, 1.84113802483322056501018123538, 1.95130698274305457789969861333, 2.15244198884246532912923334627, 2.19154741720087814963968006267, 2.21759646319098869761016061067, 2.22940446256406910489669197778, 2.38950179677767182420383218145, 2.43552873561276168320235241440, 2.53031063379358800228686114144, 2.62795830361231431315216521751
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.