Dirichlet series
| L(s) = 1 | − 32·5-s − 320·13-s − 384·17-s + 512·25-s − 928·29-s − 640·37-s + 1.32e3·49-s − 64·53-s − 1.02e3·61-s + 1.02e4·65-s + 1.16e3·81-s + 1.22e4·85-s + 4.48e3·97-s + 5.05e3·101-s + 1.12e4·109-s + 6.24e3·113-s − 4.32e3·125-s + 127-s + 131-s + 137-s + 139-s + 2.96e4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2.86·5-s − 6.82·13-s − 5.47·17-s + 4.09·25-s − 5.94·29-s − 2.84·37-s + 3.87·49-s − 0.165·53-s − 2.14·61-s + 19.5·65-s + 1.59·81-s + 15.6·85-s + 4.68·97-s + 4.98·101-s + 9.87·109-s + 5.19·113-s − 3.09·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 17.0·145-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{128}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{128}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{128}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.34000\times 10^{18}\) |
| Root analytic conductor: | \(3.88644\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{128} ,\ ( \ : [3/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.07277217489\) |
| \(L(\frac12)\) | \(\approx\) | \(0.07277217489\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 1160 T^{4} + 297076 p T^{8} - 301261880 T^{12} + 41043944326 T^{16} - 301261880 p^{12} T^{20} + 297076 p^{25} T^{24} - 1160 p^{36} T^{28} + p^{48} T^{32} \) |
| 5 | \( ( 1 + 16 T + 128 T^{2} + 112 T^{3} + 1788 T^{4} - 31088 T^{5} - 1152 p^{4} T^{6} - 1499408 p^{2} T^{7} - 516782426 T^{8} - 1499408 p^{5} T^{9} - 1152 p^{10} T^{10} - 31088 p^{9} T^{11} + 1788 p^{12} T^{12} + 112 p^{15} T^{13} + 128 p^{18} T^{14} + 16 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 7 | \( ( 1 - 664 T^{2} + 411836 T^{4} - 170193192 T^{6} + 69392575814 T^{8} - 170193192 p^{6} T^{10} + 411836 p^{12} T^{12} - 664 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 11 | \( 1 + 5901816 T^{4} + 1572113601556 p T^{8} + 34016553599329466952 T^{12} + \)\(57\!\cdots\!46\)\( T^{16} + 34016553599329466952 p^{12} T^{20} + 1572113601556 p^{25} T^{24} + 5901816 p^{36} T^{28} + p^{48} T^{32} \) | |
| 13 | \( ( 1 + 160 T + 12800 T^{2} + 741216 T^{3} + 31567036 T^{4} + 850855328 T^{5} + 6779371008 T^{6} - 1082165028512 T^{7} - 83380520911386 T^{8} - 1082165028512 p^{3} T^{9} + 6779371008 p^{6} T^{10} + 850855328 p^{9} T^{11} + 31567036 p^{12} T^{12} + 741216 p^{15} T^{13} + 12800 p^{18} T^{14} + 160 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 17 | \( ( 1 + 96 T + 16172 T^{2} + 1303200 T^{3} + 112442022 T^{4} + 1303200 p^{3} T^{5} + 16172 p^{6} T^{6} + 96 p^{9} T^{7} + p^{12} T^{8} )^{4} \) | |
| 19 | \( 1 - 22971784 T^{4} + 2140008576438364 T^{8} + \)\(40\!\cdots\!76\)\( T^{12} + \)\(89\!\cdots\!50\)\( T^{16} + \)\(40\!\cdots\!76\)\( p^{12} T^{20} + 2140008576438364 p^{24} T^{24} - 22971784 p^{36} T^{28} + p^{48} T^{32} \) | |
| 23 | \( ( 1 - 30936 T^{2} + 649121468 T^{4} - 10376816461800 T^{6} + 129966114652954566 T^{8} - 10376816461800 p^{6} T^{10} + 649121468 p^{12} T^{12} - 30936 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 29 | \( ( 1 + 16 p T + 128 p^{2} T^{2} + 23637296 T^{3} + 3448317372 T^{4} + 94270261328 T^{5} - 48102186109056 T^{6} - 17931721451264464 T^{7} - 3974568172601700506 T^{8} - 17931721451264464 p^{3} T^{9} - 48102186109056 p^{6} T^{10} + 94270261328 p^{9} T^{11} + 3448317372 p^{12} T^{12} + 23637296 p^{15} T^{13} + 128 p^{20} T^{14} + 16 p^{22} T^{15} + p^{24} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 159352 T^{2} + 11411140124 T^{4} + 16390345362360 p T^{6} + 16871525430515085638 T^{8} + 16390345362360 p^{7} T^{10} + 11411140124 p^{12} T^{12} + 159352 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 37 | \( ( 1 + 320 T + 51200 T^{2} + 6777792 T^{3} - 1297284740 T^{4} - 272550913472 T^{5} + 2173918574592 T^{6} + 4997505283176128 T^{7} + 2915252458500872358 T^{8} + 4997505283176128 p^{3} T^{9} + 2173918574592 p^{6} T^{10} - 272550913472 p^{9} T^{11} - 1297284740 p^{12} T^{12} + 6777792 p^{15} T^{13} + 51200 p^{18} T^{14} + 320 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 41 | \( ( 1 - 67592 T^{2} + 5708933852 T^{4} - 680908987013304 T^{6} + 38812985998684751750 T^{8} - 680908987013304 p^{6} T^{10} + 5708933852 p^{12} T^{12} - 67592 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 43 | \( 1 + 504444152 T^{4} + 4529634122937615580 T^{8} - \)\(48\!\cdots\!64\)\( T^{12} + \)\(51\!\cdots\!10\)\( T^{16} - \)\(48\!\cdots\!64\)\( p^{12} T^{20} + 4529634122937615580 p^{24} T^{24} + 504444152 p^{36} T^{28} + p^{48} T^{32} \) | |
| 47 | \( ( 1 + 434424 T^{2} + 101469073820 T^{4} + 15759447458513352 T^{6} + \)\(18\!\cdots\!62\)\( T^{8} + 15759447458513352 p^{6} T^{10} + 101469073820 p^{12} T^{12} + 434424 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 53 | \( ( 1 + 32 T + 512 T^{2} - 117231904 T^{3} + 11175763452 T^{4} + 17530212878624 T^{5} + 7426904478961152 T^{6} - 828863419577029408 T^{7} - \)\(14\!\cdots\!42\)\( T^{8} - 828863419577029408 p^{3} T^{9} + 7426904478961152 p^{6} T^{10} + 17530212878624 p^{9} T^{11} + 11175763452 p^{12} T^{12} - 117231904 p^{15} T^{13} + 512 p^{18} T^{14} + 32 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 59 | \( 1 + 165614595320 T^{4} + \)\(15\!\cdots\!00\)\( T^{8} + \)\(10\!\cdots\!04\)\( T^{12} + \)\(51\!\cdots\!74\)\( T^{16} + \)\(10\!\cdots\!04\)\( p^{12} T^{20} + \)\(15\!\cdots\!00\)\( p^{24} T^{24} + 165614595320 p^{36} T^{28} + p^{48} T^{32} \) | |
| 61 | \( ( 1 + 512 T + 131072 T^{2} + 19517184 T^{3} + 85744520764 T^{4} + 68029175213824 T^{5} + 23782692119543808 T^{6} + 10597918132675589120 T^{7} + \)\(37\!\cdots\!14\)\( T^{8} + 10597918132675589120 p^{3} T^{9} + 23782692119543808 p^{6} T^{10} + 68029175213824 p^{9} T^{11} + 85744520764 p^{12} T^{12} + 19517184 p^{15} T^{13} + 131072 p^{18} T^{14} + 512 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 67 | \( 1 - 365679820168 T^{4} + \)\(59\!\cdots\!96\)\( T^{8} - \)\(61\!\cdots\!24\)\( T^{12} + \)\(53\!\cdots\!78\)\( T^{16} - \)\(61\!\cdots\!24\)\( p^{12} T^{20} + \)\(59\!\cdots\!96\)\( p^{24} T^{24} - 365679820168 p^{36} T^{28} + p^{48} T^{32} \) | |
| 71 | \( ( 1 - 1119576 T^{2} + 845240011580 T^{4} - 458427553477745256 T^{6} + \)\(18\!\cdots\!26\)\( T^{8} - 458427553477745256 p^{6} T^{10} + 845240011580 p^{12} T^{12} - 1119576 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 1969416 T^{2} + 1896409843292 T^{4} - 1210300529203298232 T^{6} + \)\(55\!\cdots\!50\)\( T^{8} - 1210300529203298232 p^{6} T^{10} + 1896409843292 p^{12} T^{12} - 1969416 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 2878840 T^{2} + 3993378831260 T^{4} + 3465442907909496648 T^{6} + \)\(20\!\cdots\!86\)\( T^{8} + 3465442907909496648 p^{6} T^{10} + 3993378831260 p^{12} T^{12} + 2878840 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 83 | \( 1 - 1233954108552 T^{4} + \)\(80\!\cdots\!96\)\( T^{8} - \)\(38\!\cdots\!16\)\( T^{12} + \)\(14\!\cdots\!06\)\( T^{16} - \)\(38\!\cdots\!16\)\( p^{12} T^{20} + \)\(80\!\cdots\!96\)\( p^{24} T^{24} - 1233954108552 p^{36} T^{28} + p^{48} T^{32} \) | |
| 89 | \( ( 1 - 4182152 T^{2} + 8516836400348 T^{4} - 10742379172393448760 T^{6} + \)\(91\!\cdots\!02\)\( T^{8} - 10742379172393448760 p^{6} T^{10} + 8516836400348 p^{12} T^{12} - 4182152 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 1120 T + 2386796 T^{2} - 1138826400 T^{3} + 2110524053990 T^{4} - 1138826400 p^{3} T^{5} + 2386796 p^{6} T^{6} - 1120 p^{9} T^{7} + p^{12} T^{8} )^{4} \) | |
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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
−2.97162008160868762910401616520, −2.75539480013054312037601039537, −2.49458053256875641045450799404, −2.41193823187897368267295741310, −2.28193563856536553644789012857, −2.25078962568756083948469287821, −2.19627212537365762304911591183, −2.15275891386593314938116973819, −2.14680113918729003965444855695, −2.13415318112629730607024616211, −2.07589688644212401971213600791, −2.03113997831887285046587452348, −1.73449484561814443272082170163, −1.68484081108188756879997418378, −1.48494754478088548330922735132, −1.31131738951338892625920083896, −1.05690735094210317246904368486, −0.71377578789009009147706733815, −0.67637128698292245937642473943, −0.64325189778827088499250804371, −0.53502401354335996449206805709, −0.27408569135929623095737637633, −0.25543784054902197840791778881, −0.22294696092028951240059312376, −0.07129887482077655183209621520, 0.07129887482077655183209621520, 0.22294696092028951240059312376, 0.25543784054902197840791778881, 0.27408569135929623095737637633, 0.53502401354335996449206805709, 0.64325189778827088499250804371, 0.67637128698292245937642473943, 0.71377578789009009147706733815, 1.05690735094210317246904368486, 1.31131738951338892625920083896, 1.48494754478088548330922735132, 1.68484081108188756879997418378, 1.73449484561814443272082170163, 2.03113997831887285046587452348, 2.07589688644212401971213600791, 2.13415318112629730607024616211, 2.14680113918729003965444855695, 2.15275891386593314938116973819, 2.19627212537365762304911591183, 2.25078962568756083948469287821, 2.28193563856536553644789012857, 2.41193823187897368267295741310, 2.49458053256875641045450799404, 2.75539480013054312037601039537, 2.97162008160868762910401616520
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.