Dirichlet series
| L(s) = 1 | + 2·3-s + 76·7-s + 2·9-s + 1.06e3·13-s + 152·21-s + 2.03e3·25-s + 486·27-s − 4.72e3·31-s − 612·37-s + 2.13e3·39-s − 2.40e4·43-s + 2.88e3·49-s + 5.98e4·61-s + 152·63-s − 8.08e4·67-s − 5.69e4·73-s + 4.06e3·75-s − 1.52e3·81-s + 8.11e4·91-s − 9.44e3·93-s − 1.51e5·97-s − 5.68e5·103-s − 1.22e3·111-s + 2.13e3·117-s + 1.79e6·121-s + 127-s − 4.80e4·129-s + ⋯ |
| L(s) = 1 | + 0.128·3-s + 0.586·7-s + 0.00823·9-s + 1.75·13-s + 0.0752·21-s + 0.649·25-s + 0.128·27-s − 0.882·31-s − 0.0734·37-s + 0.224·39-s − 1.98·43-s + 0.171·49-s + 2.06·61-s + 0.00482·63-s − 2.19·67-s − 1.25·73-s + 0.0833·75-s − 0.0258·81-s + 1.02·91-s − 0.113·93-s − 1.63·97-s − 5.27·103-s − 0.00942·111-s + 0.0144·117-s + 11.1·121-s + 5.50e−6·127-s − 0.254·129-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{20} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{20} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{40} \cdot 3^{20} \cdot 5^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.63698\times 10^{19}\) |
| Root analytic conductor: | \(3.10210\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{40} \cdot 3^{20} \cdot 5^{20} ,\ ( \ : [5/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(0.9594301880\) |
| \(L(\frac12)\) | \(\approx\) | \(0.9594301880\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 2 T + 2 T^{2} - 2 p^{5} T^{3} + 385 p^{2} T^{4} - 203648 p^{2} T^{5} + 419648 p^{2} T^{6} + 141056 p^{6} T^{7} - 19250 p^{10} T^{8} - 104668 p^{10} T^{9} + 27624668 p^{10} T^{10} - 104668 p^{15} T^{11} - 19250 p^{20} T^{12} + 141056 p^{21} T^{13} + 419648 p^{22} T^{14} - 203648 p^{27} T^{15} + 385 p^{32} T^{16} - 2 p^{40} T^{17} + 2 p^{40} T^{18} - 2 p^{45} T^{19} + p^{50} T^{20} \) | |
| 5 | \( 1 - 406 p T^{2} + 395409 p^{2} T^{4} - 25458336 p^{5} T^{6} + 572476398 p^{8} T^{8} - 11714848116 p^{11} T^{10} + 572476398 p^{18} T^{12} - 25458336 p^{25} T^{14} + 395409 p^{32} T^{16} - 406 p^{41} T^{18} + p^{50} T^{20} \) | |
| good | 7 | \( ( 1 - 38 T + 722 T^{2} + 1906934 T^{3} - 229399619 T^{4} - 8178273416 T^{5} + 2294599554904 T^{6} + 268865885650888 T^{7} - 13886062361869774 T^{8} - 5779254660243519204 T^{9} + \)\(13\!\cdots\!76\)\( T^{10} - 5779254660243519204 p^{5} T^{11} - 13886062361869774 p^{10} T^{12} + 268865885650888 p^{15} T^{13} + 2294599554904 p^{20} T^{14} - 8178273416 p^{25} T^{15} - 229399619 p^{30} T^{16} + 1906934 p^{35} T^{17} + 722 p^{40} T^{18} - 38 p^{45} T^{19} + p^{50} T^{20} )^{2} \) |
| 11 | \( ( 1 - 897310 T^{2} + 396718489545 T^{4} - 117995788006701120 T^{6} + \)\(26\!\cdots\!10\)\( T^{8} - \)\(48\!\cdots\!52\)\( T^{10} + \)\(26\!\cdots\!10\)\( p^{10} T^{12} - 117995788006701120 p^{20} T^{14} + 396718489545 p^{30} T^{16} - 897310 p^{40} T^{18} + p^{50} T^{20} )^{2} \) | |
| 13 | \( ( 1 - 534 T + 142578 T^{2} + 540492738 T^{3} - 471987900919 T^{4} + 52204385334912 T^{5} + 185484149083754496 T^{6} - \)\(20\!\cdots\!84\)\( T^{7} + \)\(47\!\cdots\!26\)\( T^{8} + \)\(45\!\cdots\!28\)\( T^{9} - \)\(50\!\cdots\!76\)\( T^{10} + \)\(45\!\cdots\!28\)\( p^{5} T^{11} + \)\(47\!\cdots\!26\)\( p^{10} T^{12} - \)\(20\!\cdots\!84\)\( p^{15} T^{13} + 185484149083754496 p^{20} T^{14} + 52204385334912 p^{25} T^{15} - 471987900919 p^{30} T^{16} + 540492738 p^{35} T^{17} + 142578 p^{40} T^{18} - 534 p^{45} T^{19} + p^{50} T^{20} )^{2} \) | |
| 17 | \( 1 - 3456604260610 T^{4} - \)\(49\!\cdots\!55\)\( T^{8} + \)\(31\!\cdots\!80\)\( T^{12} + \)\(85\!\cdots\!10\)\( T^{16} - \)\(15\!\cdots\!52\)\( T^{20} + \)\(85\!\cdots\!10\)\( p^{20} T^{24} + \)\(31\!\cdots\!80\)\( p^{40} T^{28} - \)\(49\!\cdots\!55\)\( p^{60} T^{32} - 3456604260610 p^{80} T^{36} + p^{100} T^{40} \) | |
| 19 | \( ( 1 - 14326370 T^{2} + 102175395624165 T^{4} - 25805917138465231080 p T^{6} + \)\(17\!\cdots\!70\)\( T^{8} - \)\(49\!\cdots\!24\)\( T^{10} + \)\(17\!\cdots\!70\)\( p^{10} T^{12} - 25805917138465231080 p^{21} T^{14} + 102175395624165 p^{30} T^{16} - 14326370 p^{40} T^{18} + p^{50} T^{20} )^{2} \) | |
| 23 | \( 1 - 172624327595410 T^{4} + \)\(11\!\cdots\!45\)\( T^{8} - \)\(25\!\cdots\!20\)\( T^{12} - \)\(15\!\cdots\!90\)\( T^{16} + \)\(12\!\cdots\!48\)\( T^{20} - \)\(15\!\cdots\!90\)\( p^{20} T^{24} - \)\(25\!\cdots\!20\)\( p^{40} T^{28} + \)\(11\!\cdots\!45\)\( p^{60} T^{32} - 172624327595410 p^{80} T^{36} + p^{100} T^{40} \) | |
| 29 | \( ( 1 + 142841890 T^{2} + 333887829980205 p T^{4} + \)\(42\!\cdots\!80\)\( T^{6} + \)\(13\!\cdots\!10\)\( T^{8} + \)\(30\!\cdots\!48\)\( T^{10} + \)\(13\!\cdots\!10\)\( p^{10} T^{12} + \)\(42\!\cdots\!80\)\( p^{20} T^{14} + 333887829980205 p^{31} T^{16} + 142841890 p^{40} T^{18} + p^{50} T^{20} )^{2} \) | |
| 31 | \( ( 1 + 1180 T + 99102615 T^{2} + 177179496480 T^{3} + 4616852373342870 T^{4} + 7998031803847255176 T^{5} + 4616852373342870 p^{5} T^{6} + 177179496480 p^{10} T^{7} + 99102615 p^{15} T^{8} + 1180 p^{20} T^{9} + p^{25} T^{10} )^{4} \) | |
| 37 | \( ( 1 + 306 T + 46818 T^{2} - 380485615158 T^{3} - 9964642263940615 T^{4} + 41748563712201644016 T^{5} + \)\(85\!\cdots\!48\)\( T^{6} + \)\(15\!\cdots\!76\)\( p T^{7} + \)\(31\!\cdots\!90\)\( T^{8} - \)\(42\!\cdots\!44\)\( T^{9} - \)\(59\!\cdots\!32\)\( T^{10} - \)\(42\!\cdots\!44\)\( p^{5} T^{11} + \)\(31\!\cdots\!90\)\( p^{10} T^{12} + \)\(15\!\cdots\!76\)\( p^{16} T^{13} + \)\(85\!\cdots\!48\)\( p^{20} T^{14} + 41748563712201644016 p^{25} T^{15} - 9964642263940615 p^{30} T^{16} - 380485615158 p^{35} T^{17} + 46818 p^{40} T^{18} + 306 p^{45} T^{19} + p^{50} T^{20} )^{2} \) | |
| 41 | \( ( 1 - 8196410 p T^{2} + 66895688235114045 T^{4} - \)\(10\!\cdots\!20\)\( T^{6} + \)\(13\!\cdots\!10\)\( T^{8} - \)\(16\!\cdots\!52\)\( T^{10} + \)\(13\!\cdots\!10\)\( p^{10} T^{12} - \)\(10\!\cdots\!20\)\( p^{20} T^{14} + 66895688235114045 p^{30} T^{16} - 8196410 p^{41} T^{18} + p^{50} T^{20} )^{2} \) | |
| 43 | \( ( 1 + 12006 T + 72072018 T^{2} + 2395361129058 T^{3} + 18621149912102201 T^{4} - \)\(21\!\cdots\!88\)\( T^{5} - \)\(10\!\cdots\!64\)\( T^{6} - \)\(22\!\cdots\!84\)\( T^{7} - \)\(41\!\cdots\!74\)\( T^{8} - \)\(28\!\cdots\!32\)\( T^{9} - \)\(16\!\cdots\!96\)\( T^{10} - \)\(28\!\cdots\!32\)\( p^{5} T^{11} - \)\(41\!\cdots\!74\)\( p^{10} T^{12} - \)\(22\!\cdots\!84\)\( p^{15} T^{13} - \)\(10\!\cdots\!64\)\( p^{20} T^{14} - \)\(21\!\cdots\!88\)\( p^{25} T^{15} + 18621149912102201 p^{30} T^{16} + 2395361129058 p^{35} T^{17} + 72072018 p^{40} T^{18} + 12006 p^{45} T^{19} + p^{50} T^{20} )^{2} \) | |
| 47 | \( 1 - 38658592902975410 T^{4} + \)\(33\!\cdots\!45\)\( T^{8} - \)\(30\!\cdots\!20\)\( T^{12} + \)\(20\!\cdots\!10\)\( T^{16} - \)\(56\!\cdots\!52\)\( T^{20} + \)\(20\!\cdots\!10\)\( p^{20} T^{24} - \)\(30\!\cdots\!20\)\( p^{40} T^{28} + \)\(33\!\cdots\!45\)\( p^{60} T^{32} - 38658592902975410 p^{80} T^{36} + p^{100} T^{40} \) | |
| 53 | \( 1 - 696230093242751410 T^{4} + \)\(22\!\cdots\!45\)\( T^{8} - \)\(45\!\cdots\!20\)\( T^{12} + \)\(65\!\cdots\!10\)\( T^{16} - \)\(97\!\cdots\!52\)\( T^{20} + \)\(65\!\cdots\!10\)\( p^{20} T^{24} - \)\(45\!\cdots\!20\)\( p^{40} T^{28} + \)\(22\!\cdots\!45\)\( p^{60} T^{32} - 696230093242751410 p^{80} T^{36} + p^{100} T^{40} \) | |
| 59 | \( ( 1 + 3943256890 T^{2} + 7441762782679563945 T^{4} + \)\(93\!\cdots\!80\)\( T^{6} + \)\(91\!\cdots\!10\)\( T^{8} + \)\(72\!\cdots\!48\)\( T^{10} + \)\(91\!\cdots\!10\)\( p^{10} T^{12} + \)\(93\!\cdots\!80\)\( p^{20} T^{14} + 7441762782679563945 p^{30} T^{16} + 3943256890 p^{40} T^{18} + p^{50} T^{20} )^{2} \) | |
| 61 | \( ( 1 - 14970 T + 2183547065 T^{2} - 39940119936120 T^{3} + 2938859331460958770 T^{4} - \)\(44\!\cdots\!24\)\( T^{5} + 2938859331460958770 p^{5} T^{6} - 39940119936120 p^{10} T^{7} + 2183547065 p^{15} T^{8} - 14970 p^{20} T^{9} + p^{25} T^{10} )^{4} \) | |
| 67 | \( ( 1 + 40402 T + 816160802 T^{2} + 56750705339414 T^{3} + 1513054354031878441 T^{4} - \)\(25\!\cdots\!16\)\( T^{5} - \)\(64\!\cdots\!16\)\( T^{6} - \)\(51\!\cdots\!12\)\( T^{7} - \)\(56\!\cdots\!74\)\( T^{8} - \)\(14\!\cdots\!64\)\( T^{9} - \)\(28\!\cdots\!64\)\( T^{10} - \)\(14\!\cdots\!64\)\( p^{5} T^{11} - \)\(56\!\cdots\!74\)\( p^{10} T^{12} - \)\(51\!\cdots\!12\)\( p^{15} T^{13} - \)\(64\!\cdots\!16\)\( p^{20} T^{14} - \)\(25\!\cdots\!16\)\( p^{25} T^{15} + 1513054354031878441 p^{30} T^{16} + 56750705339414 p^{35} T^{17} + 816160802 p^{40} T^{18} + 40402 p^{45} T^{19} + p^{50} T^{20} )^{2} \) | |
| 71 | \( ( 1 + 1757836790 T^{2} + 7165648622494675245 T^{4} - \)\(37\!\cdots\!20\)\( T^{6} - \)\(36\!\cdots\!90\)\( T^{8} - \)\(65\!\cdots\!52\)\( T^{10} - \)\(36\!\cdots\!90\)\( p^{10} T^{12} - \)\(37\!\cdots\!20\)\( p^{20} T^{14} + 7165648622494675245 p^{30} T^{16} + 1757836790 p^{40} T^{18} + p^{50} T^{20} )^{2} \) | |
| 73 | \( ( 1 + 28478 T + 405498242 T^{2} - 5062508150546 T^{3} - 663294161859905075 T^{4} - \)\(15\!\cdots\!72\)\( T^{5} - \)\(41\!\cdots\!08\)\( T^{6} - \)\(44\!\cdots\!96\)\( T^{7} + \)\(10\!\cdots\!70\)\( T^{8} + \)\(61\!\cdots\!88\)\( T^{9} + \)\(24\!\cdots\!32\)\( T^{10} + \)\(61\!\cdots\!88\)\( p^{5} T^{11} + \)\(10\!\cdots\!70\)\( p^{10} T^{12} - \)\(44\!\cdots\!96\)\( p^{15} T^{13} - \)\(41\!\cdots\!08\)\( p^{20} T^{14} - \)\(15\!\cdots\!72\)\( p^{25} T^{15} - 663294161859905075 p^{30} T^{16} - 5062508150546 p^{35} T^{17} + 405498242 p^{40} T^{18} + 28478 p^{45} T^{19} + p^{50} T^{20} )^{2} \) | |
| 79 | \( ( 1 - 14332206610 T^{2} + \)\(11\!\cdots\!45\)\( T^{4} - \)\(65\!\cdots\!20\)\( T^{6} + \)\(27\!\cdots\!10\)\( T^{8} - \)\(95\!\cdots\!52\)\( T^{10} + \)\(27\!\cdots\!10\)\( p^{10} T^{12} - \)\(65\!\cdots\!20\)\( p^{20} T^{14} + \)\(11\!\cdots\!45\)\( p^{30} T^{16} - 14332206610 p^{40} T^{18} + p^{50} T^{20} )^{2} \) | |
| 83 | \( 1 - 22278402240889867010 T^{4} - \)\(50\!\cdots\!55\)\( T^{8} + \)\(16\!\cdots\!80\)\( T^{12} + \)\(73\!\cdots\!10\)\( T^{16} - \)\(55\!\cdots\!52\)\( T^{20} + \)\(73\!\cdots\!10\)\( p^{20} T^{24} + \)\(16\!\cdots\!80\)\( p^{40} T^{28} - \)\(50\!\cdots\!55\)\( p^{60} T^{32} - 22278402240889867010 p^{80} T^{36} + p^{100} T^{40} \) | |
| 89 | \( ( 1 + 31923325390 T^{2} + 5442565536758014005 p T^{4} + \)\(46\!\cdots\!80\)\( T^{6} + \)\(33\!\cdots\!10\)\( T^{8} + \)\(19\!\cdots\!48\)\( T^{10} + \)\(33\!\cdots\!10\)\( p^{10} T^{12} + \)\(46\!\cdots\!80\)\( p^{20} T^{14} + 5442565536758014005 p^{31} T^{16} + 31923325390 p^{40} T^{18} + p^{50} T^{20} )^{2} \) | |
| 97 | \( ( 1 + 75582 T + 2856319362 T^{2} + 4780695422142 p T^{3} + 96685304909985347261 T^{4} + \)\(56\!\cdots\!84\)\( T^{5} + \)\(25\!\cdots\!44\)\( T^{6} + \)\(48\!\cdots\!88\)\( T^{7} + \)\(62\!\cdots\!14\)\( p^{2} T^{8} + \)\(15\!\cdots\!28\)\( p T^{9} + \)\(80\!\cdots\!56\)\( T^{10} + \)\(15\!\cdots\!28\)\( p^{6} T^{11} + \)\(62\!\cdots\!14\)\( p^{12} T^{12} + \)\(48\!\cdots\!88\)\( p^{15} T^{13} + \)\(25\!\cdots\!44\)\( p^{20} T^{14} + \)\(56\!\cdots\!84\)\( p^{25} T^{15} + 96685304909985347261 p^{30} T^{16} + 4780695422142 p^{36} T^{17} + 2856319362 p^{40} T^{18} + 75582 p^{45} T^{19} + p^{50} T^{20} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.96965198017690813920923352732, −2.89142982054699192583271354109, −2.66715385687664096094585376037, −2.64494146705617343527181015966, −2.57005419486072471810458502607, −2.36609088244197290938468359100, −2.35601267726193643209674312224, −2.29388131653503489712154546357, −2.08769023154263943123922531915, −1.91589445989930741745651502686, −1.84366916428046105512288019589, −1.76597047686727288602816367277, −1.64883292379373020023754088154, −1.51273228909823541130957481848, −1.30384002838482245781207267966, −1.26652985165889089574590187416, −1.18159219625583356475516899843, −1.07941628758719541280200705579, −0.974642891041274955683075830088, −0.830441569371714267422289697272, −0.71501107966253410070580163007, −0.40113647045045340966449883285, −0.31407728485414107122916429558, −0.14062673413806551418998104965, −0.06966336966313020284734351226, 0.06966336966313020284734351226, 0.14062673413806551418998104965, 0.31407728485414107122916429558, 0.40113647045045340966449883285, 0.71501107966253410070580163007, 0.830441569371714267422289697272, 0.974642891041274955683075830088, 1.07941628758719541280200705579, 1.18159219625583356475516899843, 1.26652985165889089574590187416, 1.30384002838482245781207267966, 1.51273228909823541130957481848, 1.64883292379373020023754088154, 1.76597047686727288602816367277, 1.84366916428046105512288019589, 1.91589445989930741745651502686, 2.08769023154263943123922531915, 2.29388131653503489712154546357, 2.35601267726193643209674312224, 2.36609088244197290938468359100, 2.57005419486072471810458502607, 2.64494146705617343527181015966, 2.66715385687664096094585376037, 2.89142982054699192583271354109, 2.96965198017690813920923352732
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.