Dirichlet series
| L(s) = 1 | − 6·3-s − 28·4-s − 4·7-s + 18·9-s + 168·12-s − 12·13-s + 352·16-s + 24·21-s − 18·27-s + 112·28-s + 48·31-s − 504·36-s + 32·37-s + 72·39-s − 2.11e3·48-s + 8·49-s + 336·52-s − 336·61-s − 72·63-s − 2.57e3·64-s − 408·67-s − 88·73-s − 172·79-s + 43·81-s − 672·84-s + 48·91-s − 288·93-s + ⋯ |
| L(s) = 1 | − 2·3-s − 7·4-s − 4/7·7-s + 2·9-s + 14·12-s − 0.923·13-s + 22·16-s + 8/7·21-s − 2/3·27-s + 4·28-s + 1.54·31-s − 14·36-s + 0.864·37-s + 1.84·39-s − 44·48-s + 8/49·49-s + 6.46·52-s − 5.50·61-s − 8/7·63-s − 40.2·64-s − 6.08·67-s − 1.20·73-s − 2.17·79-s + 0.530·81-s − 8·84-s + 0.527·91-s − 3.09·93-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 17^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 17^{20}\right)^{s/2} \, \Gamma_{\C}(s+1)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(3^{20} \cdot 17^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(721.281\) |
| Root analytic conductor: | \(1.17883\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 3^{20} \cdot 17^{20} ,\ ( \ : [1]^{20} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.003158799453\) |
| \(L(\frac12)\) | \(\approx\) | \(0.003158799453\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 + 2 p T + 2 p^{2} T^{2} + 2 p^{2} T^{3} - 151 T^{4} - 284 p T^{5} - 248 p^{2} T^{6} - 148 p^{3} T^{7} + 7606 T^{8} + 25592 p T^{9} + 30860 p^{2} T^{10} + 25592 p^{3} T^{11} + 7606 p^{4} T^{12} - 148 p^{9} T^{13} - 248 p^{10} T^{14} - 284 p^{11} T^{15} - 151 p^{12} T^{16} + 2 p^{16} T^{17} + 2 p^{18} T^{18} + 2 p^{19} T^{19} + p^{20} T^{20} \) |
| 17 | \( 1 - 556 T^{2} + 376277 T^{4} - 9618048 p T^{6} + 12047114 p^{3} T^{8} - 13848360 p^{5} T^{10} + 12047114 p^{7} T^{12} - 9618048 p^{9} T^{14} + 376277 p^{12} T^{16} - 556 p^{16} T^{18} + p^{20} T^{20} \) | |
| good | 2 | \( ( 1 + 7 p T^{2} + 59 p T^{4} + 189 p^{2} T^{6} + 3857 T^{8} + 8307 p T^{10} + 3857 p^{4} T^{12} + 189 p^{10} T^{14} + 59 p^{13} T^{16} + 7 p^{17} T^{18} + p^{20} T^{20} )^{2} \) |
| 5 | \( 1 - 229 T^{4} - 298919 T^{8} - 84354884 T^{12} - 20265406266 T^{16} + 154881797815634 T^{20} - 20265406266 p^{8} T^{24} - 84354884 p^{16} T^{28} - 298919 p^{24} T^{32} - 229 p^{32} T^{36} + p^{40} T^{40} \) | |
| 7 | \( ( 1 + 2 T + 2 T^{2} - 30 p T^{3} - 201 p T^{4} + 7692 T^{5} + 40248 T^{6} + 384756 T^{7} - 1434018 T^{8} - 40666576 T^{9} - 73051508 T^{10} - 40666576 p^{2} T^{11} - 1434018 p^{4} T^{12} + 384756 p^{6} T^{13} + 40248 p^{8} T^{14} + 7692 p^{10} T^{15} - 201 p^{13} T^{16} - 30 p^{15} T^{17} + 2 p^{16} T^{18} + 2 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 11 | \( 1 - 49397 T^{4} + 1139542697 T^{8} - 18525008626532 T^{12} + 284816397145861366 T^{16} - \)\(43\!\cdots\!94\)\( T^{20} + 284816397145861366 p^{8} T^{24} - 18525008626532 p^{16} T^{28} + 1139542697 p^{24} T^{32} - 49397 p^{32} T^{36} + p^{40} T^{40} \) | |
| 13 | \( ( 1 + 3 T + 731 T^{2} + 1496 T^{3} + 228868 T^{4} + 339162 T^{5} + 228868 p^{2} T^{6} + 1496 p^{4} T^{7} + 731 p^{6} T^{8} + 3 p^{8} T^{9} + p^{10} T^{10} )^{4} \) | |
| 19 | \( ( 1 - 1553 T^{2} + 1561469 T^{4} - 1061918172 T^{6} + 558831105746 T^{8} - 225700114197222 T^{10} + 558831105746 p^{4} T^{12} - 1061918172 p^{8} T^{14} + 1561469 p^{12} T^{16} - 1553 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 23 | \( 1 + 999919 T^{4} + 666146959677 T^{8} + 319199544490856132 T^{12} + \)\(12\!\cdots\!38\)\( T^{16} + \)\(37\!\cdots\!50\)\( T^{20} + \)\(12\!\cdots\!38\)\( p^{8} T^{24} + 319199544490856132 p^{16} T^{28} + 666146959677 p^{24} T^{32} + 999919 p^{32} T^{36} + p^{40} T^{40} \) | |
| 29 | \( 1 + 1262926 T^{4} + 193664844029 T^{8} - 356721082445205432 T^{12} - \)\(25\!\cdots\!26\)\( T^{16} - \)\(16\!\cdots\!20\)\( T^{20} - \)\(25\!\cdots\!26\)\( p^{8} T^{24} - 356721082445205432 p^{16} T^{28} + 193664844029 p^{24} T^{32} + 1262926 p^{32} T^{36} + p^{40} T^{40} \) | |
| 31 | \( ( 1 - 24 T + 288 T^{2} - 6684 T^{3} + 640713 T^{4} - 45212 T^{5} - 161102328 T^{6} + 13789790220 T^{7} + 100237235142 T^{8} + 6772996031652 T^{9} - 180369808368688 T^{10} + 6772996031652 p^{2} T^{11} + 100237235142 p^{4} T^{12} + 13789790220 p^{6} T^{13} - 161102328 p^{8} T^{14} - 45212 p^{10} T^{15} + 640713 p^{12} T^{16} - 6684 p^{14} T^{17} + 288 p^{16} T^{18} - 24 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 37 | \( ( 1 - 16 T + 128 T^{2} - 75660 T^{3} + 2915421 T^{4} + 48899692 T^{5} + 1706648840 T^{6} - 144018725788 T^{7} - 7455409034286 T^{8} + 158238600425996 T^{9} + 5538921613778544 T^{10} + 158238600425996 p^{2} T^{11} - 7455409034286 p^{4} T^{12} - 144018725788 p^{6} T^{13} + 1706648840 p^{8} T^{14} + 48899692 p^{10} T^{15} + 2915421 p^{12} T^{16} - 75660 p^{14} T^{17} + 128 p^{16} T^{18} - 16 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 41 | \( 1 + 921319 T^{4} + 10381881146885 T^{8} - 9513125067202480860 T^{12} + \)\(20\!\cdots\!70\)\( T^{16} - \)\(15\!\cdots\!26\)\( T^{20} + \)\(20\!\cdots\!70\)\( p^{8} T^{24} - 9513125067202480860 p^{16} T^{28} + 10381881146885 p^{24} T^{32} + 921319 p^{32} T^{36} + p^{40} T^{40} \) | |
| 43 | \( ( 1 - 11269 T^{2} + 55646041 T^{4} - 160030985380 T^{6} + 319193769913046 T^{8} - 569461068957094126 T^{10} + 319193769913046 p^{4} T^{12} - 160030985380 p^{8} T^{14} + 55646041 p^{12} T^{16} - 11269 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 47 | \( ( 1 - 7354 T^{2} + 32799613 T^{4} - 114224391896 T^{6} + 319120409364162 T^{8} - 755615485965456604 T^{10} + 319120409364162 p^{4} T^{12} - 114224391896 p^{8} T^{14} + 32799613 p^{12} T^{16} - 7354 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 53 | \( ( 1 + 19334 T^{2} + 176953405 T^{4} + 19524963048 p T^{6} + 4361360075139314 T^{8} + 13958042733776994660 T^{10} + 4361360075139314 p^{4} T^{12} + 19524963048 p^{9} T^{14} + 176953405 p^{12} T^{16} + 19334 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 59 | \( ( 1 + 13380 T^{2} + 103345829 T^{4} + 622958312000 T^{6} + 2941525785385162 T^{8} + 11188316528321583864 T^{10} + 2941525785385162 p^{4} T^{12} + 622958312000 p^{8} T^{14} + 103345829 p^{12} T^{16} + 13380 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 61 | \( ( 1 + 168 T + 14112 T^{2} + 1121644 T^{3} + 77725293 T^{4} + 3808445948 T^{5} + 2819708456 p T^{6} + 7464539240852 T^{7} + 111534704581602 T^{8} - 6792060489288484 T^{9} - 459280612518856784 T^{10} - 6792060489288484 p^{2} T^{11} + 111534704581602 p^{4} T^{12} + 7464539240852 p^{6} T^{13} + 2819708456 p^{9} T^{14} + 3808445948 p^{10} T^{15} + 77725293 p^{12} T^{16} + 1121644 p^{14} T^{17} + 14112 p^{16} T^{18} + 168 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 67 | \( ( 1 + 102 T + 10893 T^{2} + 1116120 T^{3} + 89995978 T^{4} + 5402407076 T^{5} + 89995978 p^{2} T^{6} + 1116120 p^{4} T^{7} + 10893 p^{6} T^{8} + 102 p^{8} T^{9} + p^{10} T^{10} )^{4} \) | |
| 71 | \( 1 + 46045090 T^{4} + 841681893718269 T^{8} + \)\(82\!\cdots\!60\)\( T^{12} - \)\(27\!\cdots\!54\)\( T^{16} - \)\(67\!\cdots\!48\)\( p^{4} T^{20} - \)\(27\!\cdots\!54\)\( p^{8} T^{24} + \)\(82\!\cdots\!60\)\( p^{16} T^{28} + 841681893718269 p^{24} T^{32} + 46045090 p^{32} T^{36} + p^{40} T^{40} \) | |
| 73 | \( ( 1 + 44 T + 968 T^{2} + 260212 T^{3} + 43631597 T^{4} + 569077224 T^{5} + 16659154432 T^{6} + 4132603830104 T^{7} - 508240888129358 T^{8} - 36832221269672992 T^{9} - 618228517417385872 T^{10} - 36832221269672992 p^{2} T^{11} - 508240888129358 p^{4} T^{12} + 4132603830104 p^{6} T^{13} + 16659154432 p^{8} T^{14} + 569077224 p^{10} T^{15} + 43631597 p^{12} T^{16} + 260212 p^{14} T^{17} + 968 p^{16} T^{18} + 44 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 79 | \( ( 1 + 86 T + 3698 T^{2} + 1208386 T^{3} + 112193393 T^{4} - 687277140 T^{5} + 256101361144 T^{6} + 39042892939700 T^{7} - 4748003714867570 T^{8} - 349348433521745464 T^{9} + 6196331957462056364 T^{10} - 349348433521745464 p^{2} T^{11} - 4748003714867570 p^{4} T^{12} + 39042892939700 p^{6} T^{13} + 256101361144 p^{8} T^{14} - 687277140 p^{10} T^{15} + 112193393 p^{12} T^{16} + 1208386 p^{14} T^{17} + 3698 p^{16} T^{18} + 86 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 83 | \( ( 1 + 46118 T^{2} + 1065465421 T^{4} + 15974388717224 T^{6} + 170921739224064962 T^{8} + \)\(13\!\cdots\!52\)\( T^{10} + 170921739224064962 p^{4} T^{12} + 15974388717224 p^{8} T^{14} + 1065465421 p^{12} T^{16} + 46118 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 89 | \( ( 1 - 37560 T^{2} + 741433421 T^{4} - 10303345961408 T^{6} + 110946757857010706 T^{8} - \)\(96\!\cdots\!40\)\( T^{10} + 110946757857010706 p^{4} T^{12} - 10303345961408 p^{8} T^{14} + 741433421 p^{12} T^{16} - 37560 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 97 | \( ( 1 - 344 T + 59168 T^{2} - 9228056 T^{3} + 1554742269 T^{4} - 218140284448 T^{5} + 25627776047488 T^{6} - 3079063032121504 T^{7} + 367406717608960386 T^{8} - 37675057321967293136 T^{9} + \)\(35\!\cdots\!40\)\( T^{10} - 37675057321967293136 p^{2} T^{11} + 367406717608960386 p^{4} T^{12} - 3079063032121504 p^{6} T^{13} + 25627776047488 p^{8} T^{14} - 218140284448 p^{10} T^{15} + 1554742269 p^{12} T^{16} - 9228056 p^{14} T^{17} + 59168 p^{16} T^{18} - 344 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.38549962010018980463018238165, −4.27959268187426612567497846893, −4.26908537409041858698961972536, −3.91525289456957796707245914097, −3.88225832960449190327758644504, −3.84798756616340850923630319535, −3.68034937597626458754816291276, −3.59500975303000650118074296160, −3.47155967695374933627002290862, −3.41620295348580162977362965382, −3.34900612313177160386476749513, −2.98360122805855990470169271551, −2.93773667186458141783288952507, −2.88956570576010021588512465017, −2.85397376786806699113474345143, −2.51244201856209923316347503053, −2.32917241639422969793909616785, −2.18064317268591340114730821828, −1.91531148190911712509107658719, −1.62947752055654176613140840549, −1.41639691455380562271269218035, −1.01454088253031458862148546103, −0.841413746315322516417052789639, −0.24677731281228048928395084702, −0.12648475512791649827639796495, 0.12648475512791649827639796495, 0.24677731281228048928395084702, 0.841413746315322516417052789639, 1.01454088253031458862148546103, 1.41639691455380562271269218035, 1.62947752055654176613140840549, 1.91531148190911712509107658719, 2.18064317268591340114730821828, 2.32917241639422969793909616785, 2.51244201856209923316347503053, 2.85397376786806699113474345143, 2.88956570576010021588512465017, 2.93773667186458141783288952507, 2.98360122805855990470169271551, 3.34900612313177160386476749513, 3.41620295348580162977362965382, 3.47155967695374933627002290862, 3.59500975303000650118074296160, 3.68034937597626458754816291276, 3.84798756616340850923630319535, 3.88225832960449190327758644504, 3.91525289456957796707245914097, 4.26908537409041858698961972536, 4.27959268187426612567497846893, 4.38549962010018980463018238165
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.