Dirichlet series
| L(s) = 1 | − 44·4-s − 94·7-s + 878·16-s − 448·19-s − 1.13e3·25-s + 4.13e3·28-s − 818·31-s − 524·37-s + 752·43-s + 2.80e3·49-s − 1.78e3·61-s − 1.10e4·64-s − 2.87e3·67-s − 3.07e3·73-s + 1.97e4·76-s + 3.32e3·79-s − 4.63e3·97-s + 4.98e4·100-s − 5.82e3·103-s − 6.77e3·109-s − 8.25e4·112-s − 9.82e3·121-s + 3.59e4·124-s + 127-s + 131-s + 4.21e4·133-s + 137-s + ⋯ |
| L(s) = 1 | − 5.5·4-s − 5.07·7-s + 13.7·16-s − 5.40·19-s − 9.06·25-s + 27.9·28-s − 4.73·31-s − 2.32·37-s + 2.66·43-s + 8.17·49-s − 3.74·61-s − 21.6·64-s − 5.23·67-s − 4.93·73-s + 29.7·76-s + 4.73·79-s − 4.84·97-s + 49.8·100-s − 5.57·103-s − 5.95·109-s − 69.6·112-s − 7.38·121-s + 26.0·124-s + 0.000698·127-s + 0.000666·131-s + 27.4·133-s + 0.000623·137-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 13^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 13^{36}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(3^{36} \cdot 13^{36}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.42532\times 10^{35}\) |
| Root analytic conductor: | \(9.47322\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(18\) |
| Selberg data: | \((36,\ 3^{36} \cdot 13^{36} ,\ ( \ : [3/2]^{18} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + 11 p^{2} T^{2} + 529 p T^{4} + 18987 T^{6} + 8853 p^{5} T^{8} + 3651385 T^{10} + 41631593 T^{12} + 106463945 p^{2} T^{14} + 122893123 p^{5} T^{16} + 515070299 p^{6} T^{18} + 122893123 p^{11} T^{20} + 106463945 p^{14} T^{22} + 41631593 p^{18} T^{24} + 3651385 p^{24} T^{26} + 8853 p^{35} T^{28} + 18987 p^{36} T^{30} + 529 p^{43} T^{32} + 11 p^{50} T^{34} + p^{54} T^{36} \) |
| 5 | \( 1 + 1133 T^{2} + 125869 p T^{4} + 45404163 p T^{6} + 59400025891 T^{8} + 11960465472119 T^{10} + 1932670375994502 T^{12} + 10499020966934233 p^{2} T^{14} + 10302697688494629 p^{5} T^{16} + 251489924925832643 p^{6} T^{18} + 10302697688494629 p^{11} T^{20} + 10499020966934233 p^{14} T^{22} + 1932670375994502 p^{18} T^{24} + 11960465472119 p^{24} T^{26} + 59400025891 p^{30} T^{28} + 45404163 p^{37} T^{30} + 125869 p^{43} T^{32} + 1133 p^{48} T^{34} + p^{54} T^{36} \) | |
| 7 | \( ( 1 + 47 T + 39 p^{2} T^{2} + 60705 T^{3} + 1721985 T^{4} + 43379183 T^{5} + 1050265362 T^{6} + 22532104555 T^{7} + 462781329903 T^{8} + 8811664320047 T^{9} + 462781329903 p^{3} T^{10} + 22532104555 p^{6} T^{11} + 1050265362 p^{9} T^{12} + 43379183 p^{12} T^{13} + 1721985 p^{15} T^{14} + 60705 p^{18} T^{15} + 39 p^{23} T^{16} + 47 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 11 | \( 1 + 893 p T^{2} + 50132118 T^{4} + 179450963170 T^{6} + 507027484337219 T^{8} + 1196791091309549221 T^{10} + \)\(24\!\cdots\!84\)\( T^{12} + \)\(43\!\cdots\!29\)\( T^{14} + \)\(68\!\cdots\!80\)\( T^{16} + \)\(96\!\cdots\!53\)\( T^{18} + \)\(68\!\cdots\!80\)\( p^{6} T^{20} + \)\(43\!\cdots\!29\)\( p^{12} T^{22} + \)\(24\!\cdots\!84\)\( p^{18} T^{24} + 1196791091309549221 p^{24} T^{26} + 507027484337219 p^{30} T^{28} + 179450963170 p^{36} T^{30} + 50132118 p^{42} T^{32} + 893 p^{49} T^{34} + p^{54} T^{36} \) | |
| 17 | \( 1 + 36044 T^{2} + 703523242 T^{4} + 9747349468880 T^{6} + 106051807115602057 T^{8} + \)\(95\!\cdots\!00\)\( T^{10} + \)\(73\!\cdots\!92\)\( T^{12} + \)\(48\!\cdots\!10\)\( T^{14} + \)\(28\!\cdots\!36\)\( T^{16} + \)\(14\!\cdots\!64\)\( T^{18} + \)\(28\!\cdots\!36\)\( p^{6} T^{20} + \)\(48\!\cdots\!10\)\( p^{12} T^{22} + \)\(73\!\cdots\!92\)\( p^{18} T^{24} + \)\(95\!\cdots\!00\)\( p^{24} T^{26} + 106051807115602057 p^{30} T^{28} + 9747349468880 p^{36} T^{30} + 703523242 p^{42} T^{32} + 36044 p^{48} T^{34} + p^{54} T^{36} \) | |
| 19 | \( ( 1 + 224 T + 58224 T^{2} + 8427580 T^{3} + 1354020417 T^{4} + 154769962830 T^{5} + 19192061764918 T^{6} + 1821895311847128 T^{7} + 185006899341615484 T^{8} + 14817295566249336020 T^{9} + 185006899341615484 p^{3} T^{10} + 1821895311847128 p^{6} T^{11} + 19192061764918 p^{9} T^{12} + 154769962830 p^{12} T^{13} + 1354020417 p^{15} T^{14} + 8427580 p^{18} T^{15} + 58224 p^{21} T^{16} + 224 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 23 | \( 1 + 108346 T^{2} + 6011978995 T^{4} + 228766722738450 T^{6} + 6690515926966059999 T^{8} + \)\(15\!\cdots\!84\)\( T^{10} + \)\(31\!\cdots\!96\)\( T^{12} + \)\(54\!\cdots\!24\)\( T^{14} + \)\(81\!\cdots\!65\)\( T^{16} + \)\(10\!\cdots\!76\)\( T^{18} + \)\(81\!\cdots\!65\)\( p^{6} T^{20} + \)\(54\!\cdots\!24\)\( p^{12} T^{22} + \)\(31\!\cdots\!96\)\( p^{18} T^{24} + \)\(15\!\cdots\!84\)\( p^{24} T^{26} + 6690515926966059999 p^{30} T^{28} + 228766722738450 p^{36} T^{30} + 6011978995 p^{42} T^{32} + 108346 p^{48} T^{34} + p^{54} T^{36} \) | |
| 29 | \( 1 + 246225 T^{2} + 30179225973 T^{4} + 2470026074923163 T^{6} + \)\(15\!\cdots\!51\)\( T^{8} + \)\(74\!\cdots\!35\)\( T^{10} + \)\(30\!\cdots\!38\)\( T^{12} + \)\(36\!\cdots\!01\)\( p T^{14} + \)\(32\!\cdots\!81\)\( T^{16} + \)\(83\!\cdots\!63\)\( T^{18} + \)\(32\!\cdots\!81\)\( p^{6} T^{20} + \)\(36\!\cdots\!01\)\( p^{13} T^{22} + \)\(30\!\cdots\!38\)\( p^{18} T^{24} + \)\(74\!\cdots\!35\)\( p^{24} T^{26} + \)\(15\!\cdots\!51\)\( p^{30} T^{28} + 2470026074923163 p^{36} T^{30} + 30179225973 p^{42} T^{32} + 246225 p^{48} T^{34} + p^{54} T^{36} \) | |
| 31 | \( ( 1 + 409 T + 208601 T^{2} + 63072427 T^{3} + 20697968957 T^{4} + 5003661353881 T^{5} + 1263047874952500 T^{6} + 255273722919554599 T^{7} + 52937892315909650875 T^{8} + \)\(90\!\cdots\!83\)\( T^{9} + 52937892315909650875 p^{3} T^{10} + 255273722919554599 p^{6} T^{11} + 1263047874952500 p^{9} T^{12} + 5003661353881 p^{12} T^{13} + 20697968957 p^{15} T^{14} + 63072427 p^{18} T^{15} + 208601 p^{21} T^{16} + 409 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 37 | \( ( 1 + 262 T + 221727 T^{2} + 35506286 T^{3} + 21090803243 T^{4} + 2185776739492 T^{5} + 1481150989048076 T^{6} + 142715708083483612 T^{7} + 96928934366998873573 T^{8} + \)\(90\!\cdots\!16\)\( T^{9} + 96928934366998873573 p^{3} T^{10} + 142715708083483612 p^{6} T^{11} + 1481150989048076 p^{9} T^{12} + 2185776739492 p^{12} T^{13} + 21090803243 p^{15} T^{14} + 35506286 p^{18} T^{15} + 221727 p^{21} T^{16} + 262 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 41 | \( 1 + 699264 T^{2} + 245173763166 T^{4} + 57510838409770172 T^{6} + \)\(10\!\cdots\!37\)\( T^{8} + \)\(14\!\cdots\!48\)\( T^{10} + \)\(16\!\cdots\!24\)\( T^{12} + \)\(16\!\cdots\!46\)\( T^{14} + \)\(14\!\cdots\!04\)\( T^{16} + \)\(62\!\cdots\!44\)\( p^{2} T^{18} + \)\(14\!\cdots\!04\)\( p^{6} T^{20} + \)\(16\!\cdots\!46\)\( p^{12} T^{22} + \)\(16\!\cdots\!24\)\( p^{18} T^{24} + \)\(14\!\cdots\!48\)\( p^{24} T^{26} + \)\(10\!\cdots\!37\)\( p^{30} T^{28} + 57510838409770172 p^{36} T^{30} + 245173763166 p^{42} T^{32} + 699264 p^{48} T^{34} + p^{54} T^{36} \) | |
| 43 | \( ( 1 - 376 T + 451802 T^{2} - 166399510 T^{3} + 97595183706 T^{4} - 33681220462432 T^{5} + 13715058001818141 T^{6} - 4257560050438139424 T^{7} + \)\(14\!\cdots\!84\)\( T^{8} - \)\(38\!\cdots\!08\)\( T^{9} + \)\(14\!\cdots\!84\)\( p^{3} T^{10} - 4257560050438139424 p^{6} T^{11} + 13715058001818141 p^{9} T^{12} - 33681220462432 p^{12} T^{13} + 97595183706 p^{15} T^{14} - 166399510 p^{18} T^{15} + 451802 p^{21} T^{16} - 376 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 47 | \( 1 + 680056 T^{2} + 258124199294 T^{4} + 70295332776885440 T^{6} + \)\(15\!\cdots\!45\)\( T^{8} + \)\(27\!\cdots\!76\)\( T^{10} + \)\(43\!\cdots\!00\)\( T^{12} + \)\(59\!\cdots\!86\)\( T^{14} + \)\(73\!\cdots\!40\)\( T^{16} + \)\(80\!\cdots\!32\)\( T^{18} + \)\(73\!\cdots\!40\)\( p^{6} T^{20} + \)\(59\!\cdots\!86\)\( p^{12} T^{22} + \)\(43\!\cdots\!00\)\( p^{18} T^{24} + \)\(27\!\cdots\!76\)\( p^{24} T^{26} + \)\(15\!\cdots\!45\)\( p^{30} T^{28} + 70295332776885440 p^{36} T^{30} + 258124199294 p^{42} T^{32} + 680056 p^{48} T^{34} + p^{54} T^{36} \) | |
| 53 | \( 1 + 1297447 T^{2} + 828428025690 T^{4} + 345974974750232482 T^{6} + \)\(10\!\cdots\!91\)\( T^{8} + \)\(25\!\cdots\!33\)\( T^{10} + \)\(53\!\cdots\!88\)\( T^{12} + \)\(95\!\cdots\!17\)\( T^{14} + \)\(15\!\cdots\!48\)\( T^{16} + \)\(23\!\cdots\!29\)\( T^{18} + \)\(15\!\cdots\!48\)\( p^{6} T^{20} + \)\(95\!\cdots\!17\)\( p^{12} T^{22} + \)\(53\!\cdots\!88\)\( p^{18} T^{24} + \)\(25\!\cdots\!33\)\( p^{24} T^{26} + \)\(10\!\cdots\!91\)\( p^{30} T^{28} + 345974974750232482 p^{36} T^{30} + 828428025690 p^{42} T^{32} + 1297447 p^{48} T^{34} + p^{54} T^{36} \) | |
| 59 | \( 1 + 676173 T^{2} + 488224764967 T^{4} + 212066123608567475 T^{6} + \)\(90\!\cdots\!22\)\( T^{8} + \)\(29\!\cdots\!30\)\( T^{10} + \)\(91\!\cdots\!02\)\( T^{12} + \)\(23\!\cdots\!13\)\( T^{14} + \)\(57\!\cdots\!04\)\( T^{16} + \)\(12\!\cdots\!06\)\( T^{18} + \)\(57\!\cdots\!04\)\( p^{6} T^{20} + \)\(23\!\cdots\!13\)\( p^{12} T^{22} + \)\(91\!\cdots\!02\)\( p^{18} T^{24} + \)\(29\!\cdots\!30\)\( p^{24} T^{26} + \)\(90\!\cdots\!22\)\( p^{30} T^{28} + 212066123608567475 p^{36} T^{30} + 488224764967 p^{42} T^{32} + 676173 p^{48} T^{34} + p^{54} T^{36} \) | |
| 61 | \( ( 1 + 892 T + 1501998 T^{2} + 1008672712 T^{3} + 1068842282873 T^{4} + 600348580160568 T^{5} + 482548128170874108 T^{6} + \)\(23\!\cdots\!42\)\( T^{7} + \)\(15\!\cdots\!52\)\( T^{8} + \)\(61\!\cdots\!80\)\( T^{9} + \)\(15\!\cdots\!52\)\( p^{3} T^{10} + \)\(23\!\cdots\!42\)\( p^{6} T^{11} + 482548128170874108 p^{9} T^{12} + 600348580160568 p^{12} T^{13} + 1068842282873 p^{15} T^{14} + 1008672712 p^{18} T^{15} + 1501998 p^{21} T^{16} + 892 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 67 | \( ( 1 + 1436 T + 2450646 T^{2} + 2434789058 T^{3} + 2537415076829 T^{4} + 1994319608602590 T^{5} + 1584335217830138190 T^{6} + \)\(10\!\cdots\!64\)\( T^{7} + \)\(10\!\cdots\!70\)\( p T^{8} + \)\(36\!\cdots\!04\)\( T^{9} + \)\(10\!\cdots\!70\)\( p^{4} T^{10} + \)\(10\!\cdots\!64\)\( p^{6} T^{11} + 1584335217830138190 p^{9} T^{12} + 1994319608602590 p^{12} T^{13} + 2537415076829 p^{15} T^{14} + 2434789058 p^{18} T^{15} + 2450646 p^{21} T^{16} + 1436 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 71 | \( 1 + 3279710 T^{2} + 5363602688599 T^{4} + 5805661625463535634 T^{6} + \)\(46\!\cdots\!79\)\( T^{8} + \)\(30\!\cdots\!16\)\( T^{10} + \)\(16\!\cdots\!76\)\( T^{12} + \)\(79\!\cdots\!60\)\( T^{14} + \)\(33\!\cdots\!29\)\( T^{16} + \)\(12\!\cdots\!48\)\( T^{18} + \)\(33\!\cdots\!29\)\( p^{6} T^{20} + \)\(79\!\cdots\!60\)\( p^{12} T^{22} + \)\(16\!\cdots\!76\)\( p^{18} T^{24} + \)\(30\!\cdots\!16\)\( p^{24} T^{26} + \)\(46\!\cdots\!79\)\( p^{30} T^{28} + 5805661625463535634 p^{36} T^{30} + 5363602688599 p^{42} T^{32} + 3279710 p^{48} T^{34} + p^{54} T^{36} \) | |
| 73 | \( ( 1 + 1539 T + 2876345 T^{2} + 2997652541 T^{3} + 3535908537239 T^{4} + 2970226451357985 T^{5} + 37390831756719434 p T^{6} + \)\(19\!\cdots\!31\)\( T^{7} + \)\(14\!\cdots\!33\)\( T^{8} + \)\(88\!\cdots\!77\)\( T^{9} + \)\(14\!\cdots\!33\)\( p^{3} T^{10} + \)\(19\!\cdots\!31\)\( p^{6} T^{11} + 37390831756719434 p^{10} T^{12} + 2970226451357985 p^{12} T^{13} + 3535908537239 p^{15} T^{14} + 2997652541 p^{18} T^{15} + 2876345 p^{21} T^{16} + 1539 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 79 | \( ( 1 - 1663 T + 3663344 T^{2} - 4125874372 T^{3} + 5474869374465 T^{4} - 4854453092580035 T^{5} + 5041232713703067784 T^{6} - \)\(38\!\cdots\!35\)\( T^{7} + \)\(33\!\cdots\!24\)\( T^{8} - \)\(21\!\cdots\!67\)\( T^{9} + \)\(33\!\cdots\!24\)\( p^{3} T^{10} - \)\(38\!\cdots\!35\)\( p^{6} T^{11} + 5041232713703067784 p^{9} T^{12} - 4854453092580035 p^{12} T^{13} + 5474869374465 p^{15} T^{14} - 4125874372 p^{18} T^{15} + 3663344 p^{21} T^{16} - 1663 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 83 | \( 1 + 1615377 T^{2} + 1463872281853 T^{4} + 992266729016036539 T^{6} + \)\(74\!\cdots\!23\)\( T^{8} + \)\(54\!\cdots\!39\)\( T^{10} + \)\(40\!\cdots\!70\)\( T^{12} + \)\(25\!\cdots\!73\)\( T^{14} + \)\(14\!\cdots\!77\)\( T^{16} + \)\(81\!\cdots\!47\)\( T^{18} + \)\(14\!\cdots\!77\)\( p^{6} T^{20} + \)\(25\!\cdots\!73\)\( p^{12} T^{22} + \)\(40\!\cdots\!70\)\( p^{18} T^{24} + \)\(54\!\cdots\!39\)\( p^{24} T^{26} + \)\(74\!\cdots\!23\)\( p^{30} T^{28} + 992266729016036539 p^{36} T^{30} + 1463872281853 p^{42} T^{32} + 1615377 p^{48} T^{34} + p^{54} T^{36} \) | |
| 89 | \( 1 + 7047944 T^{2} + 23690315489038 T^{4} + 51170048384084054224 T^{6} + \)\(81\!\cdots\!53\)\( T^{8} + \)\(10\!\cdots\!28\)\( T^{10} + \)\(10\!\cdots\!20\)\( T^{12} + \)\(11\!\cdots\!22\)\( p T^{14} + \)\(81\!\cdots\!52\)\( T^{16} + \)\(60\!\cdots\!04\)\( T^{18} + \)\(81\!\cdots\!52\)\( p^{6} T^{20} + \)\(11\!\cdots\!22\)\( p^{13} T^{22} + \)\(10\!\cdots\!20\)\( p^{18} T^{24} + \)\(10\!\cdots\!28\)\( p^{24} T^{26} + \)\(81\!\cdots\!53\)\( p^{30} T^{28} + 51170048384084054224 p^{36} T^{30} + 23690315489038 p^{42} T^{32} + 7047944 p^{48} T^{34} + p^{54} T^{36} \) | |
| 97 | \( ( 1 + 2315 T + 8817092 T^{2} + 16409603038 T^{3} + 33732033347328 T^{4} + 51335459657503414 T^{5} + 73919792083107514691 T^{6} + \)\(92\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!48\)\( T^{8} + \)\(10\!\cdots\!85\)\( T^{9} + \)\(10\!\cdots\!48\)\( p^{3} T^{10} + \)\(92\!\cdots\!60\)\( p^{6} T^{11} + 73919792083107514691 p^{9} T^{12} + 51335459657503414 p^{12} T^{13} + 33732033347328 p^{15} T^{14} + 16409603038 p^{18} T^{15} + 8817092 p^{21} T^{16} + 2315 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{36} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.36505758918027776979352908941, −2.25659511194147252937559694572, −2.15803436231745390960378165167, −2.13882477340985930419855421065, −2.09385266107344490345306421717, −1.97466027217988873012678551928, −1.93081365299388562573594371294, −1.93070940922155039215162174084, −1.91420515897444940085359527251, −1.84723476049794759591875406933, −1.81906796135614986218457452066, −1.55264553787112602384256484725, −1.38136089240566019292117426859, −1.37996261788351131013411805294, −1.36391205226987486293852414099, −1.33334161540118623206424438797, −1.31812466557117650968130118127, −1.26778905899924705866542381601, −1.22374253243533967593818165086, −1.07537457060475889183637101100, −1.05395959182500777650808121561, −0.925500302341701506420165629701, −0.809994875465153795619840224963, −0.808851099311222215629989615432, −0.64359857222449548705797521950, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.64359857222449548705797521950, 0.808851099311222215629989615432, 0.809994875465153795619840224963, 0.925500302341701506420165629701, 1.05395959182500777650808121561, 1.07537457060475889183637101100, 1.22374253243533967593818165086, 1.26778905899924705866542381601, 1.31812466557117650968130118127, 1.33334161540118623206424438797, 1.36391205226987486293852414099, 1.37996261788351131013411805294, 1.38136089240566019292117426859, 1.55264553787112602384256484725, 1.81906796135614986218457452066, 1.84723476049794759591875406933, 1.91420515897444940085359527251, 1.93070940922155039215162174084, 1.93081365299388562573594371294, 1.97466027217988873012678551928, 2.09385266107344490345306421717, 2.13882477340985930419855421065, 2.15803436231745390960378165167, 2.25659511194147252937559694572, 2.36505758918027776979352908941
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.