Dirichlet series
| L(s) = 1 | + 4·2-s + 6·3-s + 12·4-s + 4·5-s + 24·6-s + 32·8-s + 18·9-s + 16·10-s − 6·11-s + 72·12-s + 24·15-s + 78·16-s + 16·17-s + 72·18-s − 14·19-s + 48·20-s − 24·22-s − 14·23-s + 192·24-s + 20·25-s + 32·27-s + 96·30-s + 4·31-s + 172·32-s − 36·33-s + 64·34-s + 216·36-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 3.46·3-s + 6·4-s + 1.78·5-s + 9.79·6-s + 11.3·8-s + 6·9-s + 5.05·10-s − 1.80·11-s + 20.7·12-s + 6.19·15-s + 39/2·16-s + 3.88·17-s + 16.9·18-s − 3.21·19-s + 10.7·20-s − 5.11·22-s − 2.91·23-s + 39.1·24-s + 4·25-s + 6.15·27-s + 17.5·30-s + 0.718·31-s + 30.4·32-s − 6.26·33-s + 10.9·34-s + 36·36-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 13^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 13^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(5^{16} \cdot 13^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.84561\times 10^{13}\) |
| Root analytic conductor: | \(2.59756\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 5^{16} \cdot 13^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(294.2657561\) |
| \(L(\frac12)\) | \(\approx\) | \(294.2657561\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 - 2 T - 4 T^{2} - 6 T^{3} + 62 T^{4} - 6 p T^{5} - 4 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 \) | |
| good | 2 | \( ( 1 - p T + T^{4} + p^{2} T^{5} + p^{2} T^{6} - 3 p^{2} T^{7} + T^{8} - 3 p^{3} T^{9} + p^{4} T^{10} + p^{5} T^{11} + p^{4} T^{12} - p^{8} T^{15} + p^{8} T^{16} )^{2} \) |
| 3 | \( 1 - 2 p T + 2 p^{2} T^{2} - 32 T^{3} + 28 T^{4} + 10 T^{5} - 52 T^{6} + 10 T^{7} + 134 T^{8} - 212 T^{9} + 46 T^{10} + 22 T^{11} + 520 p T^{12} - 5650 T^{13} + 8726 T^{14} - 4924 T^{15} - 1973 T^{16} - 4924 p T^{17} + 8726 p^{2} T^{18} - 5650 p^{3} T^{19} + 520 p^{5} T^{20} + 22 p^{5} T^{21} + 46 p^{6} T^{22} - 212 p^{7} T^{23} + 134 p^{8} T^{24} + 10 p^{9} T^{25} - 52 p^{10} T^{26} + 10 p^{11} T^{27} + 28 p^{12} T^{28} - 32 p^{13} T^{29} + 2 p^{16} T^{30} - 2 p^{16} T^{31} + p^{16} T^{32} \) | |
| 7 | \( 1 + 16 T^{2} + 60 T^{4} - 512 T^{6} - 5974 T^{8} - 39712 T^{10} - 142224 T^{12} + 2112272 T^{14} + 30435283 T^{16} + 2112272 p^{2} T^{18} - 142224 p^{4} T^{20} - 39712 p^{6} T^{22} - 5974 p^{8} T^{24} - 512 p^{10} T^{26} + 60 p^{12} T^{28} + 16 p^{14} T^{30} + p^{16} T^{32} \) | |
| 11 | \( 1 + 6 T + 18 T^{2} - 40 T^{3} - 600 T^{4} - 2526 T^{5} - 3556 T^{6} + 22510 T^{7} + 15322 p T^{8} + 552396 T^{9} + 436958 T^{10} - 5343466 T^{11} - 32474760 T^{12} - 91226474 T^{13} - 29566618 T^{14} + 932638676 T^{15} + 4649305435 T^{16} + 932638676 p T^{17} - 29566618 p^{2} T^{18} - 91226474 p^{3} T^{19} - 32474760 p^{4} T^{20} - 5343466 p^{5} T^{21} + 436958 p^{6} T^{22} + 552396 p^{7} T^{23} + 15322 p^{9} T^{24} + 22510 p^{9} T^{25} - 3556 p^{10} T^{26} - 2526 p^{11} T^{27} - 600 p^{12} T^{28} - 40 p^{13} T^{29} + 18 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 - 16 T + 128 T^{2} - 432 T^{3} - 1132 T^{4} + 20952 T^{5} - 97024 T^{6} + 53248 T^{7} + 1852842 T^{8} - 10420480 T^{9} + 16222752 T^{10} + 103654120 T^{11} - 750744240 T^{12} + 1968484600 T^{13} + 1043004544 T^{14} - 33662122176 T^{15} + 177619117779 T^{16} - 33662122176 p T^{17} + 1043004544 p^{2} T^{18} + 1968484600 p^{3} T^{19} - 750744240 p^{4} T^{20} + 103654120 p^{5} T^{21} + 16222752 p^{6} T^{22} - 10420480 p^{7} T^{23} + 1852842 p^{8} T^{24} + 53248 p^{9} T^{25} - 97024 p^{10} T^{26} + 20952 p^{11} T^{27} - 1132 p^{12} T^{28} - 432 p^{13} T^{29} + 128 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 + 14 T + 98 T^{2} + 48 T^{3} - 4544 T^{4} - 43602 T^{5} - 163964 T^{6} + 292102 T^{7} + 7878990 T^{8} + 46672172 T^{9} + 5305530 p T^{10} - 595770478 T^{11} - 6342984856 T^{12} - 1387907978 p T^{13} - 18577289106 T^{14} + 440564578996 T^{15} + 2972254601819 T^{16} + 440564578996 p T^{17} - 18577289106 p^{2} T^{18} - 1387907978 p^{4} T^{19} - 6342984856 p^{4} T^{20} - 595770478 p^{5} T^{21} + 5305530 p^{7} T^{22} + 46672172 p^{7} T^{23} + 7878990 p^{8} T^{24} + 292102 p^{9} T^{25} - 163964 p^{10} T^{26} - 43602 p^{11} T^{27} - 4544 p^{12} T^{28} + 48 p^{13} T^{29} + 98 p^{14} T^{30} + 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( 1 + 14 T + 98 T^{2} + 152 T^{3} - 2476 T^{4} - 22766 T^{5} - 64524 T^{6} + 210494 T^{7} + 2607990 T^{8} + 6251924 T^{9} - 1594654 p T^{10} - 324716834 T^{11} - 694584744 T^{12} + 3394652774 T^{13} + 23525635982 T^{14} + 32588230700 T^{15} - 142889041893 T^{16} + 32588230700 p T^{17} + 23525635982 p^{2} T^{18} + 3394652774 p^{3} T^{19} - 694584744 p^{4} T^{20} - 324716834 p^{5} T^{21} - 1594654 p^{7} T^{22} + 6251924 p^{7} T^{23} + 2607990 p^{8} T^{24} + 210494 p^{9} T^{25} - 64524 p^{10} T^{26} - 22766 p^{11} T^{27} - 2476 p^{12} T^{28} + 152 p^{13} T^{29} + 98 p^{14} T^{30} + 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 + 188 T^{2} + 18772 T^{4} + 1345096 T^{6} + 76975466 T^{8} + 3670217252 T^{10} + 149480425328 T^{12} + 5289950247844 T^{14} + 163870574618947 T^{16} + 5289950247844 p^{2} T^{18} + 149480425328 p^{4} T^{20} + 3670217252 p^{6} T^{22} + 76975466 p^{8} T^{24} + 1345096 p^{10} T^{26} + 18772 p^{12} T^{28} + 188 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( ( 1 - 2 T + 2 T^{2} - 30 T^{3} + 1184 T^{4} - 94 p T^{5} + 3910 T^{6} - 89742 T^{7} + 2057506 T^{8} - 89742 p T^{9} + 3910 p^{2} T^{10} - 94 p^{4} T^{11} + 1184 p^{4} T^{12} - 30 p^{5} T^{13} + 2 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 + 156 T^{2} + 10628 T^{4} + 514504 T^{6} + 26296394 T^{8} + 1339187588 T^{10} + 58243216048 T^{12} + 2307261613380 T^{14} + 87784315168771 T^{16} + 2307261613380 p^{2} T^{18} + 58243216048 p^{4} T^{20} + 1339187588 p^{6} T^{22} + 26296394 p^{8} T^{24} + 514504 p^{10} T^{26} + 10628 p^{12} T^{28} + 156 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( 1 + 16 T + 128 T^{2} + 560 T^{3} - 1372 T^{4} - 33912 T^{5} - 210176 T^{6} - 385120 T^{7} + 5510730 T^{8} + 62054592 T^{9} + 358483232 T^{10} + 503561528 T^{11} - 11875114224 T^{12} - 122279615096 T^{13} - 534639199104 T^{14} + 467412301952 T^{15} + 16721948821011 T^{16} + 467412301952 p T^{17} - 534639199104 p^{2} T^{18} - 122279615096 p^{3} T^{19} - 11875114224 p^{4} T^{20} + 503561528 p^{5} T^{21} + 358483232 p^{6} T^{22} + 62054592 p^{7} T^{23} + 5510730 p^{8} T^{24} - 385120 p^{9} T^{25} - 210176 p^{10} T^{26} - 33912 p^{11} T^{27} - 1372 p^{12} T^{28} + 560 p^{13} T^{29} + 128 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 + 6 T + 18 T^{2} + 280 T^{3} + 244 T^{4} + 8770 T^{5} + 87428 T^{6} + 118350 T^{7} + 5420454 T^{8} - 3859084 T^{9} - 70460658 T^{10} + 1078305638 T^{11} - 1558909128 T^{12} + 57356902966 T^{13} + 190579503550 T^{14} - 839777888260 T^{15} + 16978968547563 T^{16} - 839777888260 p T^{17} + 190579503550 p^{2} T^{18} + 57356902966 p^{3} T^{19} - 1558909128 p^{4} T^{20} + 1078305638 p^{5} T^{21} - 70460658 p^{6} T^{22} - 3859084 p^{7} T^{23} + 5420454 p^{8} T^{24} + 118350 p^{9} T^{25} + 87428 p^{10} T^{26} + 8770 p^{11} T^{27} + 244 p^{12} T^{28} + 280 p^{13} T^{29} + 18 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( ( 1 - 216 T^{2} + 24532 T^{4} - 1862136 T^{6} + 101777478 T^{8} - 1862136 p^{2} T^{10} + 24532 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 53 | \( ( 1 + 24 T + 288 T^{2} + 2936 T^{3} + 30556 T^{4} + 284840 T^{5} + 2346080 T^{6} + 19062856 T^{7} + 147591846 T^{8} + 19062856 p T^{9} + 2346080 p^{2} T^{10} + 284840 p^{3} T^{11} + 30556 p^{4} T^{12} + 2936 p^{5} T^{13} + 288 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 + 22 T + 242 T^{2} + 1584 T^{3} + 9280 T^{4} + 116886 T^{5} + 1580260 T^{6} + 15081726 T^{7} + 115865038 T^{8} + 836301884 T^{9} + 6415111934 T^{10} + 52331337146 T^{11} + 471484093096 T^{12} + 4383025060634 T^{13} + 34072326729342 T^{14} + 211179616889604 T^{15} + 1391444749576411 T^{16} + 211179616889604 p T^{17} + 34072326729342 p^{2} T^{18} + 4383025060634 p^{3} T^{19} + 471484093096 p^{4} T^{20} + 52331337146 p^{5} T^{21} + 6415111934 p^{6} T^{22} + 836301884 p^{7} T^{23} + 115865038 p^{8} T^{24} + 15081726 p^{9} T^{25} + 1580260 p^{10} T^{26} + 116886 p^{11} T^{27} + 9280 p^{12} T^{28} + 1584 p^{13} T^{29} + 242 p^{14} T^{30} + 22 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( ( 1 + 10 T - 24 T^{2} + 604 T^{3} + 8138 T^{4} - 22330 T^{5} + 326760 T^{6} + 3363590 T^{7} - 15364937 T^{8} + 3363590 p T^{9} + 326760 p^{2} T^{10} - 22330 p^{3} T^{11} + 8138 p^{4} T^{12} + 604 p^{5} T^{13} - 24 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 6 T - 200 T^{2} + 612 T^{3} + 27114 T^{4} - 37266 T^{5} - 2570152 T^{6} + 1092762 T^{7} + 188755343 T^{8} + 1092762 p T^{9} - 2570152 p^{2} T^{10} - 37266 p^{3} T^{11} + 27114 p^{4} T^{12} + 612 p^{5} T^{13} - 200 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 - 10 T + 50 T^{2} + 16 p T^{3} - 136 p T^{4} + 72534 T^{5} + 402708 T^{6} - 830930 T^{7} + 45447198 T^{8} + 127534620 T^{9} + 2027496446 T^{10} + 23159799026 T^{11} + 116577924056 T^{12} + 2002275651970 T^{13} + 11329252900958 T^{14} + 77619902179636 T^{15} + 1607622307692683 T^{16} + 77619902179636 p T^{17} + 11329252900958 p^{2} T^{18} + 2002275651970 p^{3} T^{19} + 116577924056 p^{4} T^{20} + 23159799026 p^{5} T^{21} + 2027496446 p^{6} T^{22} + 127534620 p^{7} T^{23} + 45447198 p^{8} T^{24} - 830930 p^{9} T^{25} + 402708 p^{10} T^{26} + 72534 p^{11} T^{27} - 136 p^{13} T^{28} + 16 p^{14} T^{29} + 50 p^{14} T^{30} - 10 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( ( 1 + 2 T + 236 T^{2} + 382 T^{3} + 24538 T^{4} + 382 p T^{5} + 236 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 79 | \( ( 1 - 340 T^{2} + 65892 T^{4} - 8434412 T^{6} + 781259126 T^{8} - 8434412 p^{2} T^{10} + 65892 p^{4} T^{12} - 340 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 - 248 T^{2} + 43620 T^{4} - 5170264 T^{6} + 500303558 T^{8} - 5170264 p^{2} T^{10} + 43620 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 - 28 T + 392 T^{2} - 2936 T^{3} + 9188 T^{4} - 77724 T^{5} + 2884624 T^{6} - 53687972 T^{7} + 632626458 T^{8} - 5891667312 T^{9} + 51121583768 T^{10} - 439917631700 T^{11} + 3707064600240 T^{12} - 35867177700484 T^{13} + 447238362442776 T^{14} - 5784542431175120 T^{15} + 62434283320753155 T^{16} - 5784542431175120 p T^{17} + 447238362442776 p^{2} T^{18} - 35867177700484 p^{3} T^{19} + 3707064600240 p^{4} T^{20} - 439917631700 p^{5} T^{21} + 51121583768 p^{6} T^{22} - 5891667312 p^{7} T^{23} + 632626458 p^{8} T^{24} - 53687972 p^{9} T^{25} + 2884624 p^{10} T^{26} - 77724 p^{11} T^{27} + 9188 p^{12} T^{28} - 2936 p^{13} T^{29} + 392 p^{14} T^{30} - 28 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( ( 1 + 6 T - 224 T^{2} - 972 T^{3} + 24678 T^{4} + 28050 T^{5} - 3426136 T^{6} + 1019754 T^{7} + 433593719 T^{8} + 1019754 p T^{9} - 3426136 p^{2} T^{10} + 28050 p^{3} T^{11} + 24678 p^{4} T^{12} - 972 p^{5} T^{13} - 224 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
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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
−2.70332004788150405923961666261, −2.68189279796298010384482048585, −2.61649943599274630537810236997, −2.50978487376172813031466752604, −2.35007815700167123404487050332, −2.27317583496859273536553458414, −2.21744946673034502109114757979, −2.14400372835007300316463468640, −1.98886373093577531031249618824, −1.98583020872589817902171448809, −1.90290136545545046940058749604, −1.82749534583763335960859010328, −1.82470616220010993356134439253, −1.79003691922679019032039874711, −1.61279674096115341082445535263, −1.51647238840692445057927870626, −1.39568371833415070916172878014, −1.32148111055156189729399829651, −1.24585307637570218579077698944, −1.24048839637082620397762637777, −1.01721554117563155702802081890, −1.01030775519140181112494629166, −0.53466856732429202585638608544, −0.29673012991438619094023640574, −0.15895914331426169701483092850, 0.15895914331426169701483092850, 0.29673012991438619094023640574, 0.53466856732429202585638608544, 1.01030775519140181112494629166, 1.01721554117563155702802081890, 1.24048839637082620397762637777, 1.24585307637570218579077698944, 1.32148111055156189729399829651, 1.39568371833415070916172878014, 1.51647238840692445057927870626, 1.61279674096115341082445535263, 1.79003691922679019032039874711, 1.82470616220010993356134439253, 1.82749534583763335960859010328, 1.90290136545545046940058749604, 1.98583020872589817902171448809, 1.98886373093577531031249618824, 2.14400372835007300316463468640, 2.21744946673034502109114757979, 2.27317583496859273536553458414, 2.35007815700167123404487050332, 2.50978487376172813031466752604, 2.61649943599274630537810236997, 2.68189279796298010384482048585, 2.70332004788150405923961666261
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.