Dirichlet series
| L(s) = 1 | + 128·4-s − 964·7-s − 4.54e3·13-s + 6.14e3·16-s − 4.73e4·19-s − 4.63e4·25-s − 1.23e5·28-s − 7.70e4·31-s − 2.26e4·37-s + 2.26e5·43-s + 2.86e5·49-s − 5.81e5·52-s + 3.27e5·61-s − 1.71e6·67-s − 4.37e6·73-s − 6.06e6·76-s + 1.32e6·79-s + 4.37e6·91-s + 2.20e6·97-s − 5.92e6·100-s + 4.49e4·103-s + 8.94e5·109-s − 5.92e6·112-s − 2.06e6·121-s − 9.86e6·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 2·4-s − 2.81·7-s − 2.06·13-s + 3/2·16-s − 6.90·19-s − 2.96·25-s − 5.62·28-s − 2.58·31-s − 0.448·37-s + 2.85·43-s + 2.43·49-s − 4.13·52-s + 1.44·61-s − 5.69·67-s − 11.2·73-s − 13.8·76-s + 2.69·79-s + 5.80·91-s + 2.41·97-s − 5.92·100-s + 0.0411·103-s + 0.690·109-s − 4.21·112-s − 1.16·121-s − 5.17·124-s + 19.4·133-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{64}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{64}\right)^{s/2} \, \Gamma_{\C}(s+3)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{64}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.38522\times 10^{25}\) |
| Root analytic conductor: | \(6.10481\) |
| Motivic weight: | \(6\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{64} ,\ ( \ : [3]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{7}{2})\) | \(\approx\) | \(0.2917077796\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2917077796\) |
| \(L(4)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{5} T^{2} + p^{10} T^{4} )^{4} \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 + 46304 T^{2} + 98758746 p T^{4} + 1586304468544 T^{6} + 299743467090430721 T^{8} + 10982181459683683008 p^{4} T^{10} + 84123293632893911938 p^{8} T^{12} + \)\(38\!\cdots\!08\)\( p^{12} T^{14} + \)\(21\!\cdots\!96\)\( p^{16} T^{16} + \)\(38\!\cdots\!08\)\( p^{24} T^{18} + 84123293632893911938 p^{32} T^{20} + 10982181459683683008 p^{40} T^{22} + 299743467090430721 p^{48} T^{24} + 1586304468544 p^{60} T^{26} + 98758746 p^{73} T^{28} + 46304 p^{84} T^{30} + p^{96} T^{32} \) |
| 7 | \( ( 1 + 482 T + 205386 T^{2} - 21072680 T^{3} - 25918532218 T^{4} - 15310701236238 T^{5} - 426589181618072 T^{6} + 1170622518950587130 T^{7} + \)\(86\!\cdots\!87\)\( T^{8} + 1170622518950587130 p^{6} T^{9} - 426589181618072 p^{12} T^{10} - 15310701236238 p^{18} T^{11} - 25918532218 p^{24} T^{12} - 21072680 p^{30} T^{13} + 205386 p^{36} T^{14} + 482 p^{42} T^{15} + p^{48} T^{16} )^{2} \) | |
| 11 | \( 1 + 2067740 T^{2} + 1254024903096 T^{4} + 403939779814914232 T^{6} + \)\(61\!\cdots\!82\)\( T^{8} + \)\(30\!\cdots\!80\)\( T^{10} + \)\(58\!\cdots\!68\)\( T^{12} + \)\(61\!\cdots\!84\)\( T^{14} + \)\(59\!\cdots\!23\)\( T^{16} + \)\(61\!\cdots\!84\)\( p^{12} T^{18} + \)\(58\!\cdots\!68\)\( p^{24} T^{20} + \)\(30\!\cdots\!80\)\( p^{36} T^{22} + \)\(61\!\cdots\!82\)\( p^{48} T^{24} + 403939779814914232 p^{60} T^{26} + 1254024903096 p^{72} T^{28} + 2067740 p^{84} T^{30} + p^{96} T^{32} \) | |
| 13 | \( ( 1 + 2270 T - 7493802 T^{2} - 31598982812 T^{3} + 12728672963921 T^{4} + 169308194320547916 T^{5} + \)\(17\!\cdots\!94\)\( T^{6} - \)\(38\!\cdots\!78\)\( T^{7} - \)\(12\!\cdots\!00\)\( T^{8} - \)\(38\!\cdots\!78\)\( p^{6} T^{9} + \)\(17\!\cdots\!94\)\( p^{12} T^{10} + 169308194320547916 p^{18} T^{11} + 12728672963921 p^{24} T^{12} - 31598982812 p^{30} T^{13} - 7493802 p^{36} T^{14} + 2270 p^{42} T^{15} + p^{48} T^{16} )^{2} \) | |
| 17 | \( ( 1 - 141183800 T^{2} + 9651342187425574 T^{4} - \)\(41\!\cdots\!68\)\( T^{6} + \)\(11\!\cdots\!39\)\( T^{8} - \)\(41\!\cdots\!68\)\( p^{12} T^{10} + 9651342187425574 p^{24} T^{12} - 141183800 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
| 19 | \( ( 1 + 11842 T + 122570434 T^{2} + 382269231070 T^{3} + 2961485052850642 T^{4} + 382269231070 p^{6} T^{5} + 122570434 p^{12} T^{6} + 11842 p^{18} T^{7} + p^{24} T^{8} )^{4} \) | |
| 23 | \( 1 + 602193140 T^{2} + 153789361583661864 T^{4} + \)\(26\!\cdots\!20\)\( T^{6} + \)\(53\!\cdots\!90\)\( T^{8} + \)\(12\!\cdots\!40\)\( T^{10} + \)\(22\!\cdots\!36\)\( T^{12} + \)\(14\!\cdots\!40\)\( p T^{14} + \)\(46\!\cdots\!19\)\( T^{16} + \)\(14\!\cdots\!40\)\( p^{13} T^{18} + \)\(22\!\cdots\!36\)\( p^{24} T^{20} + \)\(12\!\cdots\!40\)\( p^{36} T^{22} + \)\(53\!\cdots\!90\)\( p^{48} T^{24} + \)\(26\!\cdots\!20\)\( p^{60} T^{26} + 153789361583661864 p^{72} T^{28} + 602193140 p^{84} T^{30} + p^{96} T^{32} \) | |
| 29 | \( 1 + 3072361832 T^{2} + 162804988118563218 p T^{4} + \)\(52\!\cdots\!04\)\( T^{6} + \)\(47\!\cdots\!05\)\( T^{8} + \)\(36\!\cdots\!44\)\( T^{10} + \)\(24\!\cdots\!86\)\( T^{12} + \)\(15\!\cdots\!56\)\( T^{14} + \)\(91\!\cdots\!44\)\( T^{16} + \)\(15\!\cdots\!56\)\( p^{12} T^{18} + \)\(24\!\cdots\!86\)\( p^{24} T^{20} + \)\(36\!\cdots\!44\)\( p^{36} T^{22} + \)\(47\!\cdots\!05\)\( p^{48} T^{24} + \)\(52\!\cdots\!04\)\( p^{60} T^{26} + 162804988118563218 p^{73} T^{28} + 3072361832 p^{84} T^{30} + p^{96} T^{32} \) | |
| 31 | \( ( 1 + 38528 T - 570157560 T^{2} + 32709208259200 T^{3} + 2046352697699897426 T^{4} - \)\(30\!\cdots\!72\)\( T^{5} + \)\(35\!\cdots\!08\)\( T^{6} + \)\(34\!\cdots\!44\)\( T^{7} - \)\(71\!\cdots\!21\)\( T^{8} + \)\(34\!\cdots\!44\)\( p^{6} T^{9} + \)\(35\!\cdots\!08\)\( p^{12} T^{10} - \)\(30\!\cdots\!72\)\( p^{18} T^{11} + 2046352697699897426 p^{24} T^{12} + 32709208259200 p^{30} T^{13} - 570157560 p^{36} T^{14} + 38528 p^{42} T^{15} + p^{48} T^{16} )^{2} \) | |
| 37 | \( ( 1 + 5674 T + 2614216798 T^{2} - 33031702931096 T^{3} + 7074013795851396475 T^{4} - 33031702931096 p^{6} T^{5} + 2614216798 p^{12} T^{6} + 5674 p^{18} T^{7} + p^{24} T^{8} )^{4} \) | |
| 41 | \( 1 + 24361730072 T^{2} + \)\(34\!\cdots\!04\)\( T^{4} + \)\(30\!\cdots\!60\)\( T^{6} + \)\(18\!\cdots\!94\)\( T^{8} + \)\(56\!\cdots\!36\)\( T^{10} - \)\(98\!\cdots\!20\)\( T^{12} - \)\(24\!\cdots\!88\)\( T^{14} - \)\(15\!\cdots\!97\)\( T^{16} - \)\(24\!\cdots\!88\)\( p^{12} T^{18} - \)\(98\!\cdots\!20\)\( p^{24} T^{20} + \)\(56\!\cdots\!36\)\( p^{36} T^{22} + \)\(18\!\cdots\!94\)\( p^{48} T^{24} + \)\(30\!\cdots\!60\)\( p^{60} T^{26} + \)\(34\!\cdots\!04\)\( p^{72} T^{28} + 24361730072 p^{84} T^{30} + p^{96} T^{32} \) | |
| 43 | \( ( 1 - 113302 T - 12535550190 T^{2} + 732261356165704 T^{3} + \)\(20\!\cdots\!86\)\( T^{4} - \)\(47\!\cdots\!46\)\( T^{5} - \)\(18\!\cdots\!56\)\( T^{6} + \)\(96\!\cdots\!02\)\( p^{2} T^{7} + \)\(12\!\cdots\!35\)\( T^{8} + \)\(96\!\cdots\!02\)\( p^{8} T^{9} - \)\(18\!\cdots\!56\)\( p^{12} T^{10} - \)\(47\!\cdots\!46\)\( p^{18} T^{11} + \)\(20\!\cdots\!86\)\( p^{24} T^{12} + 732261356165704 p^{30} T^{13} - 12535550190 p^{36} T^{14} - 113302 p^{42} T^{15} + p^{48} T^{16} )^{2} \) | |
| 47 | \( 1 + 45886260248 T^{2} + \)\(10\!\cdots\!56\)\( T^{4} + \)\(14\!\cdots\!16\)\( T^{6} + \)\(11\!\cdots\!18\)\( T^{8} + \)\(44\!\cdots\!12\)\( T^{10} + \)\(16\!\cdots\!68\)\( T^{12} + \)\(68\!\cdots\!04\)\( T^{14} + \)\(11\!\cdots\!83\)\( T^{16} + \)\(68\!\cdots\!04\)\( p^{12} T^{18} + \)\(16\!\cdots\!68\)\( p^{24} T^{20} + \)\(44\!\cdots\!12\)\( p^{36} T^{22} + \)\(11\!\cdots\!18\)\( p^{48} T^{24} + \)\(14\!\cdots\!16\)\( p^{60} T^{26} + \)\(10\!\cdots\!56\)\( p^{72} T^{28} + 45886260248 p^{84} T^{30} + p^{96} T^{32} \) | |
| 53 | \( ( 1 - 97731632312 T^{2} + \)\(51\!\cdots\!92\)\( T^{4} - \)\(18\!\cdots\!08\)\( T^{6} + \)\(47\!\cdots\!10\)\( T^{8} - \)\(18\!\cdots\!08\)\( p^{12} T^{10} + \)\(51\!\cdots\!92\)\( p^{24} T^{12} - 97731632312 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
| 59 | \( 1 + 68754072824 T^{2} + \)\(18\!\cdots\!92\)\( T^{4} - \)\(42\!\cdots\!28\)\( T^{6} - \)\(40\!\cdots\!30\)\( T^{8} + \)\(10\!\cdots\!96\)\( T^{10} + \)\(64\!\cdots\!92\)\( T^{12} + \)\(26\!\cdots\!60\)\( T^{14} + \)\(77\!\cdots\!55\)\( T^{16} + \)\(26\!\cdots\!60\)\( p^{12} T^{18} + \)\(64\!\cdots\!92\)\( p^{24} T^{20} + \)\(10\!\cdots\!96\)\( p^{36} T^{22} - \)\(40\!\cdots\!30\)\( p^{48} T^{24} - \)\(42\!\cdots\!28\)\( p^{60} T^{26} + \)\(18\!\cdots\!92\)\( p^{72} T^{28} + 68754072824 p^{84} T^{30} + p^{96} T^{32} \) | |
| 61 | \( ( 1 - 163738 T + 21434820786 T^{2} - 26054361956976404 T^{3} + \)\(42\!\cdots\!73\)\( T^{4} + \)\(19\!\cdots\!16\)\( T^{5} + \)\(30\!\cdots\!58\)\( T^{6} - \)\(95\!\cdots\!42\)\( p T^{7} - \)\(92\!\cdots\!56\)\( p^{2} T^{8} - \)\(95\!\cdots\!42\)\( p^{7} T^{9} + \)\(30\!\cdots\!58\)\( p^{12} T^{10} + \)\(19\!\cdots\!16\)\( p^{18} T^{11} + \)\(42\!\cdots\!73\)\( p^{24} T^{12} - 26054361956976404 p^{30} T^{13} + 21434820786 p^{36} T^{14} - 163738 p^{42} T^{15} + p^{48} T^{16} )^{2} \) | |
| 67 | \( ( 1 + 856646 T + 237680599050 T^{2} + 44660147790203176 T^{3} + \)\(19\!\cdots\!54\)\( T^{4} + \)\(17\!\cdots\!26\)\( T^{5} - \)\(22\!\cdots\!20\)\( T^{6} - \)\(77\!\cdots\!70\)\( T^{7} - \)\(16\!\cdots\!65\)\( T^{8} - \)\(77\!\cdots\!70\)\( p^{6} T^{9} - \)\(22\!\cdots\!20\)\( p^{12} T^{10} + \)\(17\!\cdots\!26\)\( p^{18} T^{11} + \)\(19\!\cdots\!54\)\( p^{24} T^{12} + 44660147790203176 p^{30} T^{13} + 237680599050 p^{36} T^{14} + 856646 p^{42} T^{15} + p^{48} T^{16} )^{2} \) | |
| 71 | \( ( 1 - 212021007668 T^{2} + \)\(13\!\cdots\!04\)\( T^{4} - \)\(57\!\cdots\!84\)\( T^{6} + \)\(84\!\cdots\!38\)\( T^{8} - \)\(57\!\cdots\!84\)\( p^{12} T^{10} + \)\(13\!\cdots\!04\)\( p^{24} T^{12} - 212021007668 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
| 73 | \( ( 1 + 1094608 T + 746233393222 T^{2} + 336788359861632640 T^{3} + \)\(13\!\cdots\!03\)\( T^{4} + 336788359861632640 p^{6} T^{5} + 746233393222 p^{12} T^{6} + 1094608 p^{18} T^{7} + p^{24} T^{8} )^{4} \) | |
| 79 | \( ( 1 - 8398 p T - 418625624382 T^{2} + 357327569862004408 T^{3} + \)\(12\!\cdots\!22\)\( T^{4} - \)\(11\!\cdots\!54\)\( T^{5} - \)\(17\!\cdots\!52\)\( T^{6} + \)\(10\!\cdots\!86\)\( T^{7} + \)\(44\!\cdots\!87\)\( T^{8} + \)\(10\!\cdots\!86\)\( p^{6} T^{9} - \)\(17\!\cdots\!52\)\( p^{12} T^{10} - \)\(11\!\cdots\!54\)\( p^{18} T^{11} + \)\(12\!\cdots\!22\)\( p^{24} T^{12} + 357327569862004408 p^{30} T^{13} - 418625624382 p^{36} T^{14} - 8398 p^{43} T^{15} + p^{48} T^{16} )^{2} \) | |
| 83 | \( 1 + 1910425105592 T^{2} + \)\(19\!\cdots\!20\)\( T^{4} + \)\(13\!\cdots\!88\)\( T^{6} + \)\(71\!\cdots\!14\)\( T^{8} + \)\(30\!\cdots\!92\)\( T^{10} + \)\(10\!\cdots\!60\)\( T^{12} + \)\(35\!\cdots\!80\)\( T^{14} + \)\(11\!\cdots\!15\)\( T^{16} + \)\(35\!\cdots\!80\)\( p^{12} T^{18} + \)\(10\!\cdots\!60\)\( p^{24} T^{20} + \)\(30\!\cdots\!92\)\( p^{36} T^{22} + \)\(71\!\cdots\!14\)\( p^{48} T^{24} + \)\(13\!\cdots\!88\)\( p^{60} T^{26} + \)\(19\!\cdots\!20\)\( p^{72} T^{28} + 1910425105592 p^{84} T^{30} + p^{96} T^{32} \) | |
| 89 | \( ( 1 - 1054207551824 T^{2} + \)\(82\!\cdots\!70\)\( T^{4} - \)\(51\!\cdots\!84\)\( T^{6} + \)\(24\!\cdots\!75\)\( T^{8} - \)\(51\!\cdots\!84\)\( p^{12} T^{10} + \)\(82\!\cdots\!70\)\( p^{24} T^{12} - 1054207551824 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 1100032 T - 795205493880 T^{2} + 76552783641695104 T^{3} + \)\(10\!\cdots\!86\)\( T^{4} + \)\(55\!\cdots\!24\)\( T^{5} - \)\(41\!\cdots\!36\)\( T^{6} - \)\(32\!\cdots\!72\)\( T^{7} + \)\(17\!\cdots\!35\)\( T^{8} - \)\(32\!\cdots\!72\)\( p^{6} T^{9} - \)\(41\!\cdots\!36\)\( p^{12} T^{10} + \)\(55\!\cdots\!24\)\( p^{18} T^{11} + \)\(10\!\cdots\!86\)\( p^{24} T^{12} + 76552783641695104 p^{30} T^{13} - 795205493880 p^{36} T^{14} - 1100032 p^{42} T^{15} + p^{48} T^{16} )^{2} \) | |
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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
−2.55467492832781708549310436732, −2.48761028161523344722164781598, −2.20229639773814485372237465074, −2.14071407719404987935548485310, −2.10423559074695765176991569975, −1.97459679587121924297873192027, −1.92593367040919988815429426238, −1.88166588210095038349436286948, −1.81361539811072551388989294444, −1.75422484896071429779228433891, −1.74380972623232787267677881507, −1.71228143720571799504071691712, −1.59478939391931861009815509811, −1.16472361840111412673354432846, −1.14412784467035063000242543253, −1.08298386799578489664757642043, −0.858829791280845570190690621125, −0.56595895653891365981176091996, −0.55722113871086762789354934469, −0.54847233948898516086311336804, −0.39541696214694971454449522988, −0.27722908133271016241331928977, −0.23304344721031213084842847812, −0.18706353721051567005751132866, −0.04888604433456453144672460628, 0.04888604433456453144672460628, 0.18706353721051567005751132866, 0.23304344721031213084842847812, 0.27722908133271016241331928977, 0.39541696214694971454449522988, 0.54847233948898516086311336804, 0.55722113871086762789354934469, 0.56595895653891365981176091996, 0.858829791280845570190690621125, 1.08298386799578489664757642043, 1.14412784467035063000242543253, 1.16472361840111412673354432846, 1.59478939391931861009815509811, 1.71228143720571799504071691712, 1.74380972623232787267677881507, 1.75422484896071429779228433891, 1.81361539811072551388989294444, 1.88166588210095038349436286948, 1.92593367040919988815429426238, 1.97459679587121924297873192027, 2.10423559074695765176991569975, 2.14071407719404987935548485310, 2.20229639773814485372237465074, 2.48761028161523344722164781598, 2.55467492832781708549310436732
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.