Dirichlet series
| L(s) = 1 | − 64·4-s + 584·13-s + 1.72e3·16-s − 4.69e3·25-s − 1.01e4·43-s + 1.66e4·49-s − 3.73e4·52-s − 1.54e3·61-s − 2.22e4·64-s − 2.00e4·79-s + 3.00e5·100-s + 8.29e4·103-s − 1.44e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.67e5·169-s + 6.50e5·172-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 4·4-s + 3.45·13-s + 6.72·16-s − 7.51·25-s − 5.49·43-s + 6.92·49-s − 13.8·52-s − 0.414·61-s − 5.42·64-s − 3.21·79-s + 30.0·100-s + 7.81·103-s − 9.90·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 5.87·169-s + 21.9·172-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s+2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(3^{40} \cdot 13^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.48336\times 10^{21}\) |
| Root analytic conductor: | \(3.47768\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 3^{40} \cdot 13^{20} ,\ ( \ : [2]^{20} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(2.270622690\) |
| \(L(\frac12)\) | \(\approx\) | \(2.270622690\) |
| \(L(3)\) | 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 | \( ( 1 - 292 T + 44061 T^{2} + 74752 p T^{3} - 8354014 p^{2} T^{4} + 13616328 p^{4} T^{5} - 8354014 p^{6} T^{6} + 74752 p^{9} T^{7} + 44061 p^{12} T^{8} - 292 p^{16} T^{9} + p^{20} T^{10} )^{2} \) | |
| good | 2 | \( ( 1 + p^{5} T^{2} + 675 T^{4} + 5183 p T^{6} + 22901 p^{2} T^{8} + 130725 p^{3} T^{10} + 22901 p^{10} T^{12} + 5183 p^{17} T^{14} + 675 p^{24} T^{16} + p^{37} T^{18} + p^{40} T^{20} )^{2} \) |
| 5 | \( ( 1 + 2348 T^{2} + 2579841 T^{4} + 1850034544 T^{6} + 962334635198 T^{8} + 491073539586696 T^{10} + 962334635198 p^{8} T^{12} + 1850034544 p^{16} T^{14} + 2579841 p^{24} T^{16} + 2348 p^{32} T^{18} + p^{40} T^{20} )^{2} \) | |
| 7 | \( ( 1 - 8314 T^{2} + 50633577 T^{4} - 211270901808 T^{6} + 710705247661398 T^{8} - 38355679690431084 p^{2} T^{10} + 710705247661398 p^{8} T^{12} - 211270901808 p^{16} T^{14} + 50633577 p^{24} T^{16} - 8314 p^{32} T^{18} + p^{40} T^{20} )^{2} \) | |
| 11 | \( ( 1 + 72482 T^{2} + 2815289133 T^{4} + 76211737412248 T^{6} + 1584544883923058018 T^{8} + \)\(26\!\cdots\!40\)\( T^{10} + 1584544883923058018 p^{8} T^{12} + 76211737412248 p^{16} T^{14} + 2815289133 p^{24} T^{16} + 72482 p^{32} T^{18} + p^{40} T^{20} )^{2} \) | |
| 17 | \( ( 1 - 164008 T^{2} + 31916636469 T^{4} - 3223343711257248 T^{6} + \)\(38\!\cdots\!26\)\( T^{8} - \)\(29\!\cdots\!12\)\( T^{10} + \)\(38\!\cdots\!26\)\( p^{8} T^{12} - 3223343711257248 p^{16} T^{14} + 31916636469 p^{24} T^{16} - 164008 p^{32} T^{18} + p^{40} T^{20} )^{2} \) | |
| 19 | \( ( 1 - 486442 T^{2} + 6835769523 p T^{4} - 26028073563144240 T^{6} + \)\(45\!\cdots\!22\)\( T^{8} - \)\(65\!\cdots\!84\)\( T^{10} + \)\(45\!\cdots\!22\)\( p^{8} T^{12} - 26028073563144240 p^{16} T^{14} + 6835769523 p^{25} T^{16} - 486442 p^{32} T^{18} + p^{40} T^{20} )^{2} \) | |
| 23 | \( ( 1 - 1643170 T^{2} + 1444079303613 T^{4} - 842531965424497272 T^{6} + \)\(35\!\cdots\!14\)\( T^{8} - \)\(11\!\cdots\!24\)\( T^{10} + \)\(35\!\cdots\!14\)\( p^{8} T^{12} - 842531965424497272 p^{16} T^{14} + 1444079303613 p^{24} T^{16} - 1643170 p^{32} T^{18} + p^{40} T^{20} )^{2} \) | |
| 29 | \( ( 1 - 4292080 T^{2} + 9193203281061 T^{4} - 13019247705467840448 T^{6} + \)\(13\!\cdots\!86\)\( T^{8} - \)\(10\!\cdots\!52\)\( T^{10} + \)\(13\!\cdots\!86\)\( p^{8} T^{12} - 13019247705467840448 p^{16} T^{14} + 9193203281061 p^{24} T^{16} - 4292080 p^{32} T^{18} + p^{40} T^{20} )^{2} \) | |
| 31 | \( ( 1 - 1371010 T^{2} + 3537862797609 T^{4} - 3678119009923785744 T^{6} + \)\(55\!\cdots\!50\)\( T^{8} - \)\(42\!\cdots\!24\)\( T^{10} + \)\(55\!\cdots\!50\)\( p^{8} T^{12} - 3678119009923785744 p^{16} T^{14} + 3537862797609 p^{24} T^{16} - 1371010 p^{32} T^{18} + p^{40} T^{20} )^{2} \) | |
| 37 | \( ( 1 - 6084082 T^{2} + 28666922436477 T^{4} - 91247970197646753528 T^{6} + \)\(23\!\cdots\!98\)\( T^{8} - \)\(49\!\cdots\!64\)\( T^{10} + \)\(23\!\cdots\!98\)\( p^{8} T^{12} - 91247970197646753528 p^{16} T^{14} + 28666922436477 p^{24} T^{16} - 6084082 p^{32} T^{18} + p^{40} T^{20} )^{2} \) | |
| 41 | \( ( 1 + 8997308 T^{2} + 60490600748385 T^{4} + \)\(28\!\cdots\!12\)\( T^{6} + \)\(11\!\cdots\!14\)\( T^{8} + \)\(34\!\cdots\!64\)\( T^{10} + \)\(11\!\cdots\!14\)\( p^{8} T^{12} + \)\(28\!\cdots\!12\)\( p^{16} T^{14} + 60490600748385 p^{24} T^{16} + 8997308 p^{32} T^{18} + p^{40} T^{20} )^{2} \) | |
| 43 | \( ( 1 + 2540 T + 11438097 T^{2} + 26553953136 T^{3} + 64690427472030 T^{4} + 124530964474227720 T^{5} + 64690427472030 p^{4} T^{6} + 26553953136 p^{8} T^{7} + 11438097 p^{12} T^{8} + 2540 p^{16} T^{9} + p^{20} T^{10} )^{4} \) | |
| 47 | \( ( 1 + 588382 p T^{2} + 413683563717261 T^{4} + \)\(41\!\cdots\!84\)\( T^{6} + \)\(30\!\cdots\!02\)\( T^{8} + \)\(16\!\cdots\!08\)\( T^{10} + \)\(30\!\cdots\!02\)\( p^{8} T^{12} + \)\(41\!\cdots\!84\)\( p^{16} T^{14} + 413683563717261 p^{24} T^{16} + 588382 p^{33} T^{18} + p^{40} T^{20} )^{2} \) | |
| 53 | \( ( 1 - 40680280 T^{2} + 688243041050517 T^{4} - \)\(64\!\cdots\!32\)\( T^{6} + \)\(40\!\cdots\!54\)\( T^{8} - \)\(25\!\cdots\!92\)\( T^{10} + \)\(40\!\cdots\!54\)\( p^{8} T^{12} - \)\(64\!\cdots\!32\)\( p^{16} T^{14} + 688243041050517 p^{24} T^{16} - 40680280 p^{32} T^{18} + p^{40} T^{20} )^{2} \) | |
| 59 | \( ( 1 + 86280794 T^{2} + 3601131722137773 T^{4} + \)\(96\!\cdots\!72\)\( T^{6} + \)\(18\!\cdots\!90\)\( T^{8} + \)\(25\!\cdots\!12\)\( T^{10} + \)\(18\!\cdots\!90\)\( p^{8} T^{12} + \)\(96\!\cdots\!72\)\( p^{16} T^{14} + 3601131722137773 p^{24} T^{16} + 86280794 p^{32} T^{18} + p^{40} T^{20} )^{2} \) | |
| 61 | \( ( 1 + 386 T + 38054817 T^{2} + 35242018144 T^{3} + 861176001460910 T^{4} + 515883636595432764 T^{5} + 861176001460910 p^{4} T^{6} + 35242018144 p^{8} T^{7} + 38054817 p^{12} T^{8} + 386 p^{16} T^{9} + p^{20} T^{10} )^{4} \) | |
| 67 | \( ( 1 - 167596114 T^{2} + 13066690832261577 T^{4} - \)\(62\!\cdots\!36\)\( T^{6} + \)\(20\!\cdots\!06\)\( T^{8} - \)\(48\!\cdots\!20\)\( T^{10} + \)\(20\!\cdots\!06\)\( p^{8} T^{12} - \)\(62\!\cdots\!36\)\( p^{16} T^{14} + 13066690832261577 p^{24} T^{16} - 167596114 p^{32} T^{18} + p^{40} T^{20} )^{2} \) | |
| 71 | \( ( 1 + 216066458 T^{2} + 21829185725118669 T^{4} + \)\(13\!\cdots\!92\)\( T^{6} + \)\(57\!\cdots\!22\)\( T^{8} + \)\(17\!\cdots\!88\)\( T^{10} + \)\(57\!\cdots\!22\)\( p^{8} T^{12} + \)\(13\!\cdots\!92\)\( p^{16} T^{14} + 21829185725118669 p^{24} T^{16} + 216066458 p^{32} T^{18} + p^{40} T^{20} )^{2} \) | |
| 73 | \( ( 1 - 92932030 T^{2} + 5979716733183453 T^{4} - \)\(27\!\cdots\!96\)\( T^{6} + \)\(10\!\cdots\!06\)\( T^{8} - \)\(31\!\cdots\!16\)\( T^{10} + \)\(10\!\cdots\!06\)\( p^{8} T^{12} - \)\(27\!\cdots\!96\)\( p^{16} T^{14} + 5979716733183453 p^{24} T^{16} - 92932030 p^{32} T^{18} + p^{40} T^{20} )^{2} \) | |
| 79 | \( ( 1 + 5012 T + 2306355 p T^{2} + 717728477104 T^{3} + 13743445928031602 T^{4} + 40810331168554723896 T^{5} + 13743445928031602 p^{4} T^{6} + 717728477104 p^{8} T^{7} + 2306355 p^{13} T^{8} + 5012 p^{16} T^{9} + p^{20} T^{10} )^{4} \) | |
| 83 | \( ( 1 + 360200282 T^{2} + 62316893195138925 T^{4} + \)\(67\!\cdots\!16\)\( T^{6} + \)\(51\!\cdots\!94\)\( T^{8} + \)\(28\!\cdots\!80\)\( T^{10} + \)\(51\!\cdots\!94\)\( p^{8} T^{12} + \)\(67\!\cdots\!16\)\( p^{16} T^{14} + 62316893195138925 p^{24} T^{16} + 360200282 p^{32} T^{18} + p^{40} T^{20} )^{2} \) | |
| 89 | \( ( 1 + 368403356 T^{2} + 69032220402309345 T^{4} + \)\(85\!\cdots\!56\)\( T^{6} + \)\(78\!\cdots\!02\)\( T^{8} + \)\(55\!\cdots\!28\)\( T^{10} + \)\(78\!\cdots\!02\)\( p^{8} T^{12} + \)\(85\!\cdots\!56\)\( p^{16} T^{14} + 69032220402309345 p^{24} T^{16} + 368403356 p^{32} T^{18} + p^{40} T^{20} )^{2} \) | |
| 97 | \( ( 1 - 317298814 T^{2} + 53459047139159709 T^{4} - \)\(67\!\cdots\!44\)\( T^{6} + \)\(71\!\cdots\!14\)\( T^{8} - \)\(66\!\cdots\!24\)\( T^{10} + \)\(71\!\cdots\!14\)\( p^{8} T^{12} - \)\(67\!\cdots\!44\)\( p^{16} T^{14} + 53459047139159709 p^{24} T^{16} - 317298814 p^{32} T^{18} + p^{40} T^{20} )^{2} \) | |
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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
−2.62169903404389563630567792955, −2.52561549067531887371061606000, −2.34059868841797007164782414944, −2.28720218975927213100406175525, −2.28278399564054753870053755453, −2.24332780562778545216798087864, −2.13658979241452386928479974088, −1.94068646339096395500271772967, −1.83368105473030669956408234081, −1.71453176720467406508669438951, −1.68072447319286591059648515842, −1.48397392443718302808388082008, −1.47528413821366615989448548102, −1.38513823266049374006998376214, −1.36553022164196356566942381466, −0.972901756266931825211220273745, −0.956610307162768393140786817750, −0.943777786427436611385193079989, −0.75802147789428591058770030837, −0.56689032514969963536062121293, −0.38479903303600424790474307852, −0.35325468065972451491866770790, −0.24131906945436732655219855891, −0.22509141795027801414844274073, −0.20872771883842544094503603778, 0.20872771883842544094503603778, 0.22509141795027801414844274073, 0.24131906945436732655219855891, 0.35325468065972451491866770790, 0.38479903303600424790474307852, 0.56689032514969963536062121293, 0.75802147789428591058770030837, 0.943777786427436611385193079989, 0.956610307162768393140786817750, 0.972901756266931825211220273745, 1.36553022164196356566942381466, 1.38513823266049374006998376214, 1.47528413821366615989448548102, 1.48397392443718302808388082008, 1.68072447319286591059648515842, 1.71453176720467406508669438951, 1.83368105473030669956408234081, 1.94068646339096395500271772967, 2.13658979241452386928479974088, 2.24332780562778545216798087864, 2.28278399564054753870053755453, 2.28720218975927213100406175525, 2.34059868841797007164782414944, 2.52561549067531887371061606000, 2.62169903404389563630567792955
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.