Dirichlet series
| L(s) = 1 | − 2·2-s − 20·3-s − 27·4-s + 40·6-s − 172·7-s + 124·8-s − 12·9-s + 800·11-s + 540·12-s + 1.01e3·13-s + 344·14-s − 763·16-s − 4.39e3·17-s + 24·18-s + 5.30e3·19-s + 3.44e3·21-s − 1.60e3·22-s + 1.14e3·23-s − 2.48e3·24-s − 2.02e3·26-s + 4.26e3·27-s + 4.64e3·28-s − 3.66e3·29-s + 1.66e4·31-s + 458·32-s − 1.60e4·33-s + 8.79e3·34-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 1.28·3-s − 0.843·4-s + 0.453·6-s − 1.32·7-s + 0.685·8-s − 0.0493·9-s + 1.99·11-s + 1.08·12-s + 1.66·13-s + 0.469·14-s − 0.745·16-s − 3.68·17-s + 0.0174·18-s + 3.37·19-s + 1.70·21-s − 0.704·22-s + 0.449·23-s − 0.878·24-s − 0.588·26-s + 1.12·27-s + 1.11·28-s − 0.809·29-s + 3.11·31-s + 0.0790·32-s − 2.55·33-s + 1.30·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(5^{12} \cdot 13^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.00568\times 10^{10}\) |
| Root analytic conductor: | \(7.21974\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 5^{12} \cdot 13^{6} ,\ ( \ : [5/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(5.633660258\) |
| \(L(\frac12)\) | \(\approx\) | \(5.633660258\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 13 | \( ( 1 - p^{2} T )^{6} \) | |
| good | 2 | \( 1 + p T + 31 T^{2} - p^{3} T^{3} + 167 p^{3} T^{4} - 5 p^{6} T^{5} + 2507 p^{4} T^{6} - 5 p^{11} T^{7} + 167 p^{13} T^{8} - p^{18} T^{9} + 31 p^{20} T^{10} + p^{26} T^{11} + p^{30} T^{12} \) |
| 3 | \( 1 + 20 T + 412 T^{2} + 4220 T^{3} + 6553 p T^{4} - 94360 p^{2} T^{5} - 596312 p^{3} T^{6} - 94360 p^{7} T^{7} + 6553 p^{11} T^{8} + 4220 p^{15} T^{9} + 412 p^{20} T^{10} + 20 p^{25} T^{11} + p^{30} T^{12} \) | |
| 7 | \( 1 + 172 T + 79134 T^{2} + 10856548 T^{3} + 409892745 p T^{4} + 317015702232 T^{5} + 61415726406116 T^{6} + 317015702232 p^{5} T^{7} + 409892745 p^{11} T^{8} + 10856548 p^{15} T^{9} + 79134 p^{20} T^{10} + 172 p^{25} T^{11} + p^{30} T^{12} \) | |
| 11 | \( 1 - 800 T + 759568 T^{2} - 406608064 T^{3} + 249396661771 T^{4} - 103203903134944 T^{5} + 48608265340441376 T^{6} - 103203903134944 p^{5} T^{7} + 249396661771 p^{10} T^{8} - 406608064 p^{15} T^{9} + 759568 p^{20} T^{10} - 800 p^{25} T^{11} + p^{30} T^{12} \) | |
| 17 | \( 1 + 4396 T + 13388314 T^{2} + 30247747580 T^{3} + 55454091910447 T^{4} + 84911134688114072 T^{5} + \)\(10\!\cdots\!96\)\( T^{6} + 84911134688114072 p^{5} T^{7} + 55454091910447 p^{10} T^{8} + 30247747580 p^{15} T^{9} + 13388314 p^{20} T^{10} + 4396 p^{25} T^{11} + p^{30} T^{12} \) | |
| 19 | \( 1 - 5304 T + 935248 p T^{2} - 41459051976 T^{3} + 81939580759323 T^{4} - 137801885409410896 T^{5} + \)\(22\!\cdots\!68\)\( T^{6} - 137801885409410896 p^{5} T^{7} + 81939580759323 p^{10} T^{8} - 41459051976 p^{15} T^{9} + 935248 p^{21} T^{10} - 5304 p^{25} T^{11} + p^{30} T^{12} \) | |
| 23 | \( 1 - 1140 T + 26990372 T^{2} - 27334511148 T^{3} + 346087829393987 T^{4} - 321185337113879016 T^{5} + \)\(27\!\cdots\!88\)\( T^{6} - 321185337113879016 p^{5} T^{7} + 346087829393987 p^{10} T^{8} - 27334511148 p^{15} T^{9} + 26990372 p^{20} T^{10} - 1140 p^{25} T^{11} + p^{30} T^{12} \) | |
| 29 | \( 1 + 3664 T + 70215246 T^{2} + 212100467280 T^{3} + 2582215649047719 T^{4} + 6374882876649641504 T^{5} + \)\(61\!\cdots\!48\)\( T^{6} + 6374882876649641504 p^{5} T^{7} + 2582215649047719 p^{10} T^{8} + 212100467280 p^{15} T^{9} + 70215246 p^{20} T^{10} + 3664 p^{25} T^{11} + p^{30} T^{12} \) | |
| 31 | \( 1 - 16664 T + 222991608 T^{2} - 1841948763592 T^{3} + 13891289359405827 T^{4} - 80310268298937076688 T^{5} + \)\(47\!\cdots\!56\)\( T^{6} - 80310268298937076688 p^{5} T^{7} + 13891289359405827 p^{10} T^{8} - 1841948763592 p^{15} T^{9} + 222991608 p^{20} T^{10} - 16664 p^{25} T^{11} + p^{30} T^{12} \) | |
| 37 | \( 1 + 4488 T + 268756262 T^{2} + 1050793017672 T^{3} + 36014260500286007 T^{4} + \)\(11\!\cdots\!76\)\( T^{5} + \)\(30\!\cdots\!96\)\( T^{6} + \)\(11\!\cdots\!76\)\( p^{5} T^{7} + 36014260500286007 p^{10} T^{8} + 1050793017672 p^{15} T^{9} + 268756262 p^{20} T^{10} + 4488 p^{25} T^{11} + p^{30} T^{12} \) | |
| 41 | \( 1 - 12436 T + 438551018 T^{2} - 4844551362468 T^{3} + 106540073505916031 T^{4} - \)\(94\!\cdots\!04\)\( T^{5} + \)\(15\!\cdots\!96\)\( T^{6} - \)\(94\!\cdots\!04\)\( p^{5} T^{7} + 106540073505916031 p^{10} T^{8} - 4844551362468 p^{15} T^{9} + 438551018 p^{20} T^{10} - 12436 p^{25} T^{11} + p^{30} T^{12} \) | |
| 43 | \( 1 + 1516 T + 118385308 T^{2} + 1205424018852 T^{3} + 22780555260448539 T^{4} - 19968699827127017192 T^{5} + \)\(44\!\cdots\!56\)\( T^{6} - 19968699827127017192 p^{5} T^{7} + 22780555260448539 p^{10} T^{8} + 1205424018852 p^{15} T^{9} + 118385308 p^{20} T^{10} + 1516 p^{25} T^{11} + p^{30} T^{12} \) | |
| 47 | \( 1 + 212 T + 868921678 T^{2} - 815484002756 T^{3} + 387087648470244367 T^{4} - \)\(37\!\cdots\!72\)\( T^{5} + \)\(11\!\cdots\!80\)\( T^{6} - \)\(37\!\cdots\!72\)\( p^{5} T^{7} + 387087648470244367 p^{10} T^{8} - 815484002756 p^{15} T^{9} + 868921678 p^{20} T^{10} + 212 p^{25} T^{11} + p^{30} T^{12} \) | |
| 53 | \( 1 - 15612 T + 1302611762 T^{2} - 3861503238396 T^{3} + 464446823633000119 T^{4} + \)\(86\!\cdots\!24\)\( T^{5} + \)\(79\!\cdots\!36\)\( T^{6} + \)\(86\!\cdots\!24\)\( p^{5} T^{7} + 464446823633000119 p^{10} T^{8} - 3861503238396 p^{15} T^{9} + 1302611762 p^{20} T^{10} - 15612 p^{25} T^{11} + p^{30} T^{12} \) | |
| 59 | \( 1 + 11896 T + 2693780976 T^{2} + 10411916715592 T^{3} + 3369278181997078251 T^{4} + 78121391922971510608 T^{5} + \)\(28\!\cdots\!24\)\( T^{6} + 78121391922971510608 p^{5} T^{7} + 3369278181997078251 p^{10} T^{8} + 10411916715592 p^{15} T^{9} + 2693780976 p^{20} T^{10} + 11896 p^{25} T^{11} + p^{30} T^{12} \) | |
| 61 | \( 1 - 57232 T + 5236420718 T^{2} - 219172905169808 T^{3} + 11261991887609202663 T^{4} - \)\(35\!\cdots\!04\)\( T^{5} + \)\(12\!\cdots\!24\)\( T^{6} - \)\(35\!\cdots\!04\)\( p^{5} T^{7} + 11261991887609202663 p^{10} T^{8} - 219172905169808 p^{15} T^{9} + 5236420718 p^{20} T^{10} - 57232 p^{25} T^{11} + p^{30} T^{12} \) | |
| 67 | \( 1 - 25428 T + 5004079118 T^{2} - 103351564549004 T^{3} + 12139682587774178471 T^{4} - \)\(19\!\cdots\!60\)\( T^{5} + \)\(19\!\cdots\!44\)\( T^{6} - \)\(19\!\cdots\!60\)\( p^{5} T^{7} + 12139682587774178471 p^{10} T^{8} - 103351564549004 p^{15} T^{9} + 5004079118 p^{20} T^{10} - 25428 p^{25} T^{11} + p^{30} T^{12} \) | |
| 71 | \( 1 - 214104 T + 27159407320 T^{2} - 2418426473306760 T^{3} + \)\(16\!\cdots\!99\)\( T^{4} - \)\(94\!\cdots\!00\)\( T^{5} + \)\(44\!\cdots\!48\)\( T^{6} - \)\(94\!\cdots\!00\)\( p^{5} T^{7} + \)\(16\!\cdots\!99\)\( p^{10} T^{8} - 2418426473306760 p^{15} T^{9} + 27159407320 p^{20} T^{10} - 214104 p^{25} T^{11} + p^{30} T^{12} \) | |
| 73 | \( 1 + 60808 T + 7683503646 T^{2} + 466685836081256 T^{3} + 32200043594087922015 T^{4} + \)\(16\!\cdots\!20\)\( T^{5} + \)\(83\!\cdots\!88\)\( T^{6} + \)\(16\!\cdots\!20\)\( p^{5} T^{7} + 32200043594087922015 p^{10} T^{8} + 466685836081256 p^{15} T^{9} + 7683503646 p^{20} T^{10} + 60808 p^{25} T^{11} + p^{30} T^{12} \) | |
| 79 | \( 1 - 456 p T + 11615543106 T^{2} - 230917448345448 T^{3} + 62062444497721483599 T^{4} - \)\(74\!\cdots\!44\)\( T^{5} + \)\(22\!\cdots\!68\)\( T^{6} - \)\(74\!\cdots\!44\)\( p^{5} T^{7} + 62062444497721483599 p^{10} T^{8} - 230917448345448 p^{15} T^{9} + 11615543106 p^{20} T^{10} - 456 p^{26} T^{11} + p^{30} T^{12} \) | |
| 83 | \( 1 + 34684 T + 7848433574 T^{2} + 97618429712676 T^{3} + 32967984989601535079 T^{4} - \)\(25\!\cdots\!48\)\( T^{5} + \)\(13\!\cdots\!44\)\( T^{6} - \)\(25\!\cdots\!48\)\( p^{5} T^{7} + 32967984989601535079 p^{10} T^{8} + 97618429712676 p^{15} T^{9} + 7848433574 p^{20} T^{10} + 34684 p^{25} T^{11} + p^{30} T^{12} \) | |
| 89 | \( 1 + 52028 T + 7379546418 T^{2} + 426261312231404 T^{3} + 56427851113243582527 T^{4} + \)\(36\!\cdots\!68\)\( T^{5} + \)\(30\!\cdots\!08\)\( T^{6} + \)\(36\!\cdots\!68\)\( p^{5} T^{7} + 56427851113243582527 p^{10} T^{8} + 426261312231404 p^{15} T^{9} + 7379546418 p^{20} T^{10} + 52028 p^{25} T^{11} + p^{30} T^{12} \) | |
| 97 | \( 1 + 129316 T + 26692221250 T^{2} + 3616210172433684 T^{3} + \)\(39\!\cdots\!07\)\( T^{4} + \)\(43\!\cdots\!40\)\( T^{5} + \)\(41\!\cdots\!04\)\( T^{6} + \)\(43\!\cdots\!40\)\( p^{5} T^{7} + \)\(39\!\cdots\!07\)\( p^{10} T^{8} + 3616210172433684 p^{15} T^{9} + 26692221250 p^{20} T^{10} + 129316 p^{25} T^{11} + p^{30} T^{12} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.37494558120357542643514262308, −5.07534695243356204283293911182, −4.99031563794642622927811443719, −4.91482120844438773169771813983, −4.81043956800408024057518885081, −4.53421273073423733768006889466, −4.21394190890656730350530220217, −4.03169695043063648149652713051, −3.87562120534043960472924085283, −3.86860504976223299157370362557, −3.47802380528109059262488724646, −3.26609505600707075345612751489, −3.14400769858179058408582426565, −2.82466216104059592068167792476, −2.64595666002383962832282486320, −2.36928963599832644546840189012, −2.12146003094111640263027226297, −1.70407370363802893987987347740, −1.49785837399324914407825110159, −1.45201455924205559157608851914, −0.915255879415186512158254405173, −0.73793836101671324056521906019, −0.52474212714070350780990581765, −0.44491041399355830094882298927, −0.43909575249550549659992038078, 0.43909575249550549659992038078, 0.44491041399355830094882298927, 0.52474212714070350780990581765, 0.73793836101671324056521906019, 0.915255879415186512158254405173, 1.45201455924205559157608851914, 1.49785837399324914407825110159, 1.70407370363802893987987347740, 2.12146003094111640263027226297, 2.36928963599832644546840189012, 2.64595666002383962832282486320, 2.82466216104059592068167792476, 3.14400769858179058408582426565, 3.26609505600707075345612751489, 3.47802380528109059262488724646, 3.86860504976223299157370362557, 3.87562120534043960472924085283, 4.03169695043063648149652713051, 4.21394190890656730350530220217, 4.53421273073423733768006889466, 4.81043956800408024057518885081, 4.91482120844438773169771813983, 4.99031563794642622927811443719, 5.07534695243356204283293911182, 5.37494558120357542643514262308
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.