Dirichlet series
| L(s) = 1 | − 8·4-s − 6·7-s − 2·13-s + 30·16-s − 92·19-s + 192·25-s + 48·28-s − 156·31-s + 148·37-s + 186·43-s + 267·49-s + 16·52-s + 210·61-s − 192·64-s − 158·67-s + 156·73-s + 736·76-s − 132·79-s + 12·91-s − 14·97-s − 1.53e3·100-s − 964·103-s − 364·109-s − 180·112-s − 460·121-s + 1.24e3·124-s + 127-s + ⋯ |
| L(s) = 1 | − 2·4-s − 6/7·7-s − 0.153·13-s + 15/8·16-s − 4.84·19-s + 7.67·25-s + 12/7·28-s − 5.03·31-s + 4·37-s + 4.32·43-s + 5.44·49-s + 4/13·52-s + 3.44·61-s − 3·64-s − 2.35·67-s + 2.13·73-s + 9.68·76-s − 1.67·79-s + 0.131·91-s − 0.144·97-s − 15.3·100-s − 9.35·103-s − 3.33·109-s − 1.60·112-s − 3.80·121-s + 10.0·124-s + 0.00787·127-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s+1)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(3^{40} \cdot 13^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.17602\times 10^{10}\) |
| Root analytic conductor: | \(1.78550\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 3^{40} \cdot 13^{20} ,\ ( \ : [1]^{20} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(1.181001084\) |
| \(L(\frac12)\) | \(\approx\) | \(1.181001084\) |
| \(L(2)\) | 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 | \( ( 1 + T - 207 T^{2} - 3976 T^{3} + 2447 p T^{4} + 3951 p^{2} T^{5} + 2447 p^{3} T^{6} - 3976 p^{4} T^{7} - 207 p^{6} T^{8} + p^{8} T^{9} + p^{10} T^{10} )^{2} \) | |
| good | 2 | \( 1 + p^{3} T^{2} + 17 p T^{4} + 7 p^{5} T^{6} + 1091 T^{8} + 333 p^{3} T^{10} + 6401 p T^{12} + 8341 p^{3} T^{14} + 110561 T^{16} + 60433 p^{3} T^{18} + 238467 p^{4} T^{20} + 60433 p^{7} T^{22} + 110561 p^{8} T^{24} + 8341 p^{15} T^{26} + 6401 p^{17} T^{28} + 333 p^{23} T^{30} + 1091 p^{24} T^{32} + 7 p^{33} T^{34} + 17 p^{33} T^{36} + p^{39} T^{38} + p^{40} T^{40} \) |
| 5 | \( ( 1 - 96 T^{2} + 4983 T^{4} - 37176 p T^{6} + 229241 p^{2} T^{8} - 153490752 T^{10} + 229241 p^{6} T^{12} - 37176 p^{9} T^{14} + 4983 p^{12} T^{16} - 96 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 7 | \( ( 1 + 3 T - 120 T^{2} - 745 T^{3} + 4343 T^{4} + 42810 T^{5} - 108432 T^{6} + 300598 T^{7} + 19432339 T^{8} - 47715229 T^{9} - 1578286380 T^{10} - 47715229 p^{2} T^{11} + 19432339 p^{4} T^{12} + 300598 p^{6} T^{13} - 108432 p^{8} T^{14} + 42810 p^{10} T^{15} + 4343 p^{12} T^{16} - 745 p^{14} T^{17} - 120 p^{16} T^{18} + 3 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 11 | \( 1 + 460 T^{2} + 94459 T^{4} + 10688764 T^{6} + 534655963 T^{8} - 38977643720 T^{10} - 13881430037658 T^{12} - 2344613964651912 T^{14} - 246653396678846391 T^{16} - 11698723426973347084 T^{18} - \)\(12\!\cdots\!43\)\( T^{20} - 11698723426973347084 p^{4} T^{22} - 246653396678846391 p^{8} T^{24} - 2344613964651912 p^{12} T^{26} - 13881430037658 p^{16} T^{28} - 38977643720 p^{20} T^{30} + 534655963 p^{24} T^{32} + 10688764 p^{28} T^{34} + 94459 p^{32} T^{36} + 460 p^{36} T^{38} + p^{40} T^{40} \) | |
| 17 | \( 1 + 130 p T^{2} + 2540113 T^{4} + 2050544846 T^{6} + 1309410717548 T^{8} + 701379615304470 T^{10} + 325682097995297839 T^{12} + \)\(13\!\cdots\!10\)\( T^{14} + \)\(49\!\cdots\!55\)\( T^{16} + \)\(56\!\cdots\!64\)\( p^{2} T^{18} + \)\(59\!\cdots\!48\)\( p^{4} T^{20} + \)\(56\!\cdots\!64\)\( p^{6} T^{22} + \)\(49\!\cdots\!55\)\( p^{8} T^{24} + \)\(13\!\cdots\!10\)\( p^{12} T^{26} + 325682097995297839 p^{16} T^{28} + 701379615304470 p^{20} T^{30} + 1309410717548 p^{24} T^{32} + 2050544846 p^{28} T^{34} + 2540113 p^{32} T^{36} + 130 p^{37} T^{38} + p^{40} T^{40} \) | |
| 19 | \( ( 1 + 46 T + 561 T^{2} + 1374 p T^{3} + 1115955 T^{4} + 11871444 T^{5} + 291022210 T^{6} + 12369430276 T^{7} + 115747824583 T^{8} + 2244304781886 T^{9} + 94372864194305 T^{10} + 2244304781886 p^{2} T^{11} + 115747824583 p^{4} T^{12} + 12369430276 p^{6} T^{13} + 291022210 p^{8} T^{14} + 11871444 p^{10} T^{15} + 1115955 p^{12} T^{16} + 1374 p^{15} T^{17} + 561 p^{16} T^{18} + 46 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 23 | \( 1 + 3108 T^{2} + 4601395 T^{4} + 4716126068 T^{6} + 4107391919811 T^{8} + 3269132309517944 T^{10} + 2371897016139069886 T^{12} + \)\(15\!\cdots\!72\)\( T^{14} + \)\(98\!\cdots\!33\)\( T^{16} + \)\(57\!\cdots\!68\)\( T^{18} + \)\(31\!\cdots\!37\)\( T^{20} + \)\(57\!\cdots\!68\)\( p^{4} T^{22} + \)\(98\!\cdots\!33\)\( p^{8} T^{24} + \)\(15\!\cdots\!72\)\( p^{12} T^{26} + 2371897016139069886 p^{16} T^{28} + 3269132309517944 p^{20} T^{30} + 4107391919811 p^{24} T^{32} + 4716126068 p^{28} T^{34} + 4601395 p^{32} T^{36} + 3108 p^{36} T^{38} + p^{40} T^{40} \) | |
| 29 | \( 1 + 4264 T^{2} + 10819369 T^{4} + 20134943320 T^{6} + 29982732784336 T^{8} + 37314335405332048 T^{10} + 40021311641957782347 T^{12} + \)\(13\!\cdots\!24\)\( p T^{14} + \)\(33\!\cdots\!71\)\( T^{16} + \)\(32\!\cdots\!24\)\( p^{2} T^{18} + \)\(32\!\cdots\!12\)\( p^{4} T^{20} + \)\(32\!\cdots\!24\)\( p^{6} T^{22} + \)\(33\!\cdots\!71\)\( p^{8} T^{24} + \)\(13\!\cdots\!24\)\( p^{13} T^{26} + 40021311641957782347 p^{16} T^{28} + 37314335405332048 p^{20} T^{30} + 29982732784336 p^{24} T^{32} + 20134943320 p^{28} T^{34} + 10819369 p^{32} T^{36} + 4264 p^{36} T^{38} + p^{40} T^{40} \) | |
| 31 | \( ( 1 + 39 T + 3833 T^{2} + 129428 T^{3} + 6846726 T^{4} + 175251050 T^{5} + 6846726 p^{2} T^{6} + 129428 p^{4} T^{7} + 3833 p^{6} T^{8} + 39 p^{8} T^{9} + p^{10} T^{10} )^{4} \) | |
| 37 | \( ( 1 - 2 p T + 417 T^{2} + 64238 T^{3} - 1317790 T^{4} + 93707482 T^{5} - 4667503575 T^{6} - 12280668934 T^{7} + 5108628144413 T^{8} - 142661148175128 T^{9} + 3737926799435984 T^{10} - 142661148175128 p^{2} T^{11} + 5108628144413 p^{4} T^{12} - 12280668934 p^{6} T^{13} - 4667503575 p^{8} T^{14} + 93707482 p^{10} T^{15} - 1317790 p^{12} T^{16} + 64238 p^{14} T^{17} + 417 p^{16} T^{18} - 2 p^{19} T^{19} + p^{20} T^{20} )^{2} \) | |
| 41 | \( 1 + 9154 T^{2} + 42032145 T^{4} + 124734269614 T^{6} + 262870384020316 T^{8} + 407833703424403094 T^{10} + \)\(48\!\cdots\!79\)\( T^{12} + \)\(50\!\cdots\!62\)\( T^{14} + \)\(73\!\cdots\!79\)\( T^{16} + \)\(15\!\cdots\!76\)\( T^{18} + \)\(28\!\cdots\!56\)\( T^{20} + \)\(15\!\cdots\!76\)\( p^{4} T^{22} + \)\(73\!\cdots\!79\)\( p^{8} T^{24} + \)\(50\!\cdots\!62\)\( p^{12} T^{26} + \)\(48\!\cdots\!79\)\( p^{16} T^{28} + 407833703424403094 p^{20} T^{30} + 262870384020316 p^{24} T^{32} + 124734269614 p^{28} T^{34} + 42032145 p^{32} T^{36} + 9154 p^{36} T^{38} + p^{40} T^{40} \) | |
| 43 | \( ( 1 - 93 T - 952 T^{2} + 226711 T^{3} + 5748643 T^{4} - 419206602 T^{5} - 23159166968 T^{6} + 533004860034 T^{7} + 61850466847695 T^{8} - 90608675096885 T^{9} - 147295639924257956 T^{10} - 90608675096885 p^{2} T^{11} + 61850466847695 p^{4} T^{12} + 533004860034 p^{6} T^{13} - 23159166968 p^{8} T^{14} - 419206602 p^{10} T^{15} + 5748643 p^{12} T^{16} + 226711 p^{14} T^{17} - 952 p^{16} T^{18} - 93 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 47 | \( ( 1 - 14180 T^{2} + 96753117 T^{4} - 421963480960 T^{6} + 1336108016778086 T^{8} - 3299575856060509944 T^{10} + 1336108016778086 p^{4} T^{12} - 421963480960 p^{8} T^{14} + 96753117 p^{12} T^{16} - 14180 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 53 | \( ( 1 - 21714 T^{2} + 225341783 T^{4} - 1468217966468 T^{6} + 6641835292788625 T^{8} - 21778701048290226166 T^{10} + 6641835292788625 p^{4} T^{12} - 1468217966468 p^{8} T^{14} + 225341783 p^{12} T^{16} - 21714 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 59 | \( 1 + 21640 T^{2} + 235669587 T^{4} + 1751085041512 T^{6} + 10139027156091751 T^{8} + 49339496999248094096 T^{10} + \)\(21\!\cdots\!78\)\( T^{12} + \)\(81\!\cdots\!52\)\( T^{14} + \)\(28\!\cdots\!01\)\( T^{16} + \)\(98\!\cdots\!00\)\( T^{18} + \)\(33\!\cdots\!93\)\( T^{20} + \)\(98\!\cdots\!00\)\( p^{4} T^{22} + \)\(28\!\cdots\!01\)\( p^{8} T^{24} + \)\(81\!\cdots\!52\)\( p^{12} T^{26} + \)\(21\!\cdots\!78\)\( p^{16} T^{28} + 49339496999248094096 p^{20} T^{30} + 10139027156091751 p^{24} T^{32} + 1751085041512 p^{28} T^{34} + 235669587 p^{32} T^{36} + 21640 p^{36} T^{38} + p^{40} T^{40} \) | |
| 61 | \( ( 1 - 105 T - 1280 T^{2} + 431167 T^{3} - 17516174 T^{4} + 1319692215 T^{5} - 117518630456 T^{6} + 1919069209395 T^{7} + 510116202103401 T^{8} - 27421053848356492 T^{9} + 339892031386839712 T^{10} - 27421053848356492 p^{2} T^{11} + 510116202103401 p^{4} T^{12} + 1919069209395 p^{6} T^{13} - 117518630456 p^{8} T^{14} + 1319692215 p^{10} T^{15} - 17516174 p^{12} T^{16} + 431167 p^{14} T^{17} - 1280 p^{16} T^{18} - 105 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 67 | \( ( 1 + 79 T - 4632 T^{2} - 454045 T^{3} + 6461495 T^{4} + 793475986 T^{5} + 71098233624 T^{6} + 5968188795758 T^{7} - 276814574905417 T^{8} - 131906849599671 p T^{9} + 1969244692432387700 T^{10} - 131906849599671 p^{3} T^{11} - 276814574905417 p^{4} T^{12} + 5968188795758 p^{6} T^{13} + 71098233624 p^{8} T^{14} + 793475986 p^{10} T^{15} + 6461495 p^{12} T^{16} - 454045 p^{14} T^{17} - 4632 p^{16} T^{18} + 79 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 71 | \( 1 + 15588 T^{2} + 137992723 T^{4} + 1000324189652 T^{6} + 6387142450999395 T^{8} + 37241761881481754120 T^{10} + \)\(21\!\cdots\!14\)\( T^{12} + \)\(12\!\cdots\!72\)\( T^{14} + \)\(76\!\cdots\!97\)\( T^{16} + \)\(42\!\cdots\!36\)\( T^{18} + \)\(22\!\cdots\!45\)\( T^{20} + \)\(42\!\cdots\!36\)\( p^{4} T^{22} + \)\(76\!\cdots\!97\)\( p^{8} T^{24} + \)\(12\!\cdots\!72\)\( p^{12} T^{26} + \)\(21\!\cdots\!14\)\( p^{16} T^{28} + 37241761881481754120 p^{20} T^{30} + 6387142450999395 p^{24} T^{32} + 1000324189652 p^{28} T^{34} + 137992723 p^{32} T^{36} + 15588 p^{36} T^{38} + p^{40} T^{40} \) | |
| 73 | \( ( 1 - 39 T + 22635 T^{2} - 713246 T^{3} + 219748813 T^{4} - 5392773837 T^{5} + 219748813 p^{2} T^{6} - 713246 p^{4} T^{7} + 22635 p^{6} T^{8} - 39 p^{8} T^{9} + p^{10} T^{10} )^{4} \) | |
| 79 | \( ( 1 + 33 T + 9177 T^{2} + 89884 T^{3} + 54632230 T^{4} + 27398166 T^{5} + 54632230 p^{2} T^{6} + 89884 p^{4} T^{7} + 9177 p^{6} T^{8} + 33 p^{8} T^{9} + p^{10} T^{10} )^{4} \) | |
| 83 | \( ( 1 - 54470 T^{2} + 1418075469 T^{4} - 23076649482344 T^{6} + 259007203049641638 T^{8} - \)\(20\!\cdots\!24\)\( T^{10} + 259007203049641638 p^{4} T^{12} - 23076649482344 p^{8} T^{14} + 1418075469 p^{12} T^{16} - 54470 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 89 | \( 1 + 1650 T^{2} + 99281439 T^{4} - 2438150910102 T^{6} + 747383589372799 T^{8} - \)\(21\!\cdots\!16\)\( T^{10} + \)\(29\!\cdots\!38\)\( T^{12} - \)\(33\!\cdots\!28\)\( T^{14} + \)\(21\!\cdots\!33\)\( T^{16} - \)\(20\!\cdots\!82\)\( T^{18} + \)\(26\!\cdots\!85\)\( T^{20} - \)\(20\!\cdots\!82\)\( p^{4} T^{22} + \)\(21\!\cdots\!33\)\( p^{8} T^{24} - \)\(33\!\cdots\!28\)\( p^{12} T^{26} + \)\(29\!\cdots\!38\)\( p^{16} T^{28} - \)\(21\!\cdots\!16\)\( p^{20} T^{30} + 747383589372799 p^{24} T^{32} - 2438150910102 p^{28} T^{34} + 99281439 p^{32} T^{36} + 1650 p^{36} T^{38} + p^{40} T^{40} \) | |
| 97 | \( ( 1 + 7 T - 18132 T^{2} + 263579 T^{3} + 116020151 T^{4} - 9902019434 T^{5} + 181173440100 T^{6} + 199020135493418 T^{7} - 6166513921596127 T^{8} - 1053040378352667651 T^{9} + 73138042269738574448 T^{10} - 1053040378352667651 p^{2} T^{11} - 6166513921596127 p^{4} T^{12} + 199020135493418 p^{6} T^{13} + 181173440100 p^{8} T^{14} - 9902019434 p^{10} T^{15} + 116020151 p^{12} T^{16} + 263579 p^{14} T^{17} - 18132 p^{16} T^{18} + 7 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
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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
−3.30799375344025962135022804633, −3.08911016925123962258385426490, −2.91056588886623714445281817491, −2.81907014345091946057937448311, −2.77878142487467973513866916156, −2.76338745102619428089748818268, −2.70463584277705232468522776073, −2.68554927596161452025015354339, −2.66297618322095321722752944704, −2.64966639719665695037307089638, −2.45275756016929775727191931521, −2.24497363968741979362905926592, −2.23679051715539027702007631034, −1.94641501161333994978496193380, −1.86929232428913382894971880532, −1.86889498227493686659447867387, −1.44719398370214373399035874561, −1.24161693095580025646141999353, −1.20450730759714167318902381179, −1.11511208778456581614928225200, −1.04137830664237195357599871860, −0.902693444415888890077987364142, −0.47847730965781159153629204742, −0.27804736078475633605834837300, −0.21646368705124327691493199509, 0.21646368705124327691493199509, 0.27804736078475633605834837300, 0.47847730965781159153629204742, 0.902693444415888890077987364142, 1.04137830664237195357599871860, 1.11511208778456581614928225200, 1.20450730759714167318902381179, 1.24161693095580025646141999353, 1.44719398370214373399035874561, 1.86889498227493686659447867387, 1.86929232428913382894971880532, 1.94641501161333994978496193380, 2.23679051715539027702007631034, 2.24497363968741979362905926592, 2.45275756016929775727191931521, 2.64966639719665695037307089638, 2.66297618322095321722752944704, 2.68554927596161452025015354339, 2.70463584277705232468522776073, 2.76338745102619428089748818268, 2.77878142487467973513866916156, 2.81907014345091946057937448311, 2.91056588886623714445281817491, 3.08911016925123962258385426490, 3.30799375344025962135022804633
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.