Dirichlet series
| L(s) = 1 | − 30·3-s − 20·4-s + 495·9-s + 600·12-s − 100·13-s + 240·16-s + 142·17-s + 136·23-s + 600·25-s − 5.94e3·27-s − 80·29-s − 9.90e3·36-s + 3.00e3·39-s − 304·43-s − 7.20e3·48-s − 245·49-s − 4.26e3·51-s + 2.00e3·52-s − 1.55e3·53-s − 678·61-s − 2.24e3·64-s − 2.84e3·68-s − 4.08e3·69-s − 1.80e4·75-s − 4.11e3·79-s + 5.79e4·81-s + 2.40e3·87-s + ⋯ |
| L(s) = 1 | − 5.77·3-s − 5/2·4-s + 55/3·9-s + 14.4·12-s − 2.13·13-s + 15/4·16-s + 2.02·17-s + 1.23·23-s + 24/5·25-s − 42.3·27-s − 0.512·29-s − 45.8·36-s + 12.3·39-s − 1.07·43-s − 21.6·48-s − 5/7·49-s − 11.6·51-s + 5.33·52-s − 4.01·53-s − 1.42·61-s − 4.37·64-s − 5.06·68-s − 7.11·69-s − 27.7·75-s − 5.86·79-s + 79.4·81-s + 2.95·87-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 7^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 7^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 3^{10} \cdot 7^{10} \cdot 13^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.20389\times 10^{15}\) |
| Root analytic conductor: | \(5.67582\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 3^{10} \cdot 7^{10} \cdot 13^{10} ,\ ( \ : [3/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.06137463856\) |
| \(L(\frac12)\) | \(\approx\) | \(0.06137463856\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{2} T^{2} )^{5} \) |
| 3 | \( ( 1 + p T )^{10} \) | |
| 7 | \( ( 1 + p^{2} T^{2} )^{5} \) | |
| 13 | \( 1 + 100 T + 7977 T^{2} + 38112 p T^{3} + 164218 p^{2} T^{4} + 643096 p^{3} T^{5} + 164218 p^{5} T^{6} + 38112 p^{7} T^{7} + 7977 p^{9} T^{8} + 100 p^{12} T^{9} + p^{15} T^{10} \) | |
| good | 5 | \( 1 - 24 p^{2} T^{2} + 42182 p T^{4} - 51358526 T^{6} + 9354427057 T^{8} - 1325056868356 T^{10} + 9354427057 p^{6} T^{12} - 51358526 p^{12} T^{14} + 42182 p^{19} T^{16} - 24 p^{26} T^{18} + p^{30} T^{20} \) |
| 11 | \( 1 - 7925 T^{2} + 29289221 T^{4} - 70095821276 T^{6} + 126760852572914 T^{8} - 185814613141722846 T^{10} + 126760852572914 p^{6} T^{12} - 70095821276 p^{12} T^{14} + 29289221 p^{18} T^{16} - 7925 p^{24} T^{18} + p^{30} T^{20} \) | |
| 17 | \( ( 1 - 71 T + 9391 T^{2} - 15012 p T^{3} + 41219256 T^{4} - 1135479802 T^{5} + 41219256 p^{3} T^{6} - 15012 p^{7} T^{7} + 9391 p^{9} T^{8} - 71 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 19 | \( 1 - 31220 T^{2} + 431838782 T^{4} - 3151316628466 T^{6} + 11321234506766289 T^{8} - 29487820527598978740 T^{10} + 11321234506766289 p^{6} T^{12} - 3151316628466 p^{12} T^{14} + 431838782 p^{18} T^{16} - 31220 p^{24} T^{18} + p^{30} T^{20} \) | |
| 23 | \( ( 1 - 68 T + 31692 T^{2} - 2514738 T^{3} + 497123211 T^{4} - 43409763020 T^{5} + 497123211 p^{3} T^{6} - 2514738 p^{6} T^{7} + 31692 p^{9} T^{8} - 68 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 29 | \( ( 1 + 40 T + 88962 T^{2} + 5131322 T^{3} + 3538237437 T^{4} + 202652238828 T^{5} + 3538237437 p^{3} T^{6} + 5131322 p^{6} T^{7} + 88962 p^{9} T^{8} + 40 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 31 | \( 1 - 114069 T^{2} + 6604430853 T^{4} - 281278607614604 T^{6} + 10782823520174902650 T^{8} - \)\(35\!\cdots\!26\)\( T^{10} + 10782823520174902650 p^{6} T^{12} - 281278607614604 p^{12} T^{14} + 6604430853 p^{18} T^{16} - 114069 p^{24} T^{18} + p^{30} T^{20} \) | |
| 37 | \( 1 - 282289 T^{2} + 42893203149 T^{4} - 4386408250119772 T^{6} + \)\(33\!\cdots\!42\)\( T^{8} - \)\(19\!\cdots\!74\)\( T^{10} + \)\(33\!\cdots\!42\)\( p^{6} T^{12} - 4386408250119772 p^{12} T^{14} + 42893203149 p^{18} T^{16} - 282289 p^{24} T^{18} + p^{30} T^{20} \) | |
| 41 | \( 1 - 465910 T^{2} + 109129071677 T^{4} - 16525327265964984 T^{6} + \)\(17\!\cdots\!54\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{10} + \)\(17\!\cdots\!54\)\( p^{6} T^{12} - 16525327265964984 p^{12} T^{14} + 109129071677 p^{18} T^{16} - 465910 p^{24} T^{18} + p^{30} T^{20} \) | |
| 43 | \( ( 1 + 152 T + 290782 T^{2} + 29608328 T^{3} + 38894019489 T^{4} + 3002226709376 T^{5} + 38894019489 p^{3} T^{6} + 29608328 p^{6} T^{7} + 290782 p^{9} T^{8} + 152 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 47 | \( 1 - 589785 T^{2} + 179452417561 T^{4} - 36768248484690244 T^{6} + \)\(55\!\cdots\!58\)\( T^{8} - \)\(65\!\cdots\!06\)\( T^{10} + \)\(55\!\cdots\!58\)\( p^{6} T^{12} - 36768248484690244 p^{12} T^{14} + 179452417561 p^{18} T^{16} - 589785 p^{24} T^{18} + p^{30} T^{20} \) | |
| 53 | \( ( 1 + 775 T + 599989 T^{2} + 242294972 T^{3} + 116381137510 T^{4} + 37552863861546 T^{5} + 116381137510 p^{3} T^{6} + 242294972 p^{6} T^{7} + 599989 p^{9} T^{8} + 775 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 59 | \( 1 - 1369030 T^{2} + 892500164245 T^{4} - 371949305394195512 T^{6} + \)\(11\!\cdots\!66\)\( T^{8} - \)\(25\!\cdots\!92\)\( T^{10} + \)\(11\!\cdots\!66\)\( p^{6} T^{12} - 371949305394195512 p^{12} T^{14} + 892500164245 p^{18} T^{16} - 1369030 p^{24} T^{18} + p^{30} T^{20} \) | |
| 61 | \( ( 1 + 339 T + 1092015 T^{2} + 304236512 T^{3} + 486285548892 T^{4} + 103177217976042 T^{5} + 486285548892 p^{3} T^{6} + 304236512 p^{6} T^{7} + 1092015 p^{9} T^{8} + 339 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 67 | \( 1 - 1739674 T^{2} + 1548290935941 T^{4} - 933522928809916600 T^{6} + \)\(41\!\cdots\!14\)\( T^{8} - \)\(14\!\cdots\!72\)\( T^{10} + \)\(41\!\cdots\!14\)\( p^{6} T^{12} - 933522928809916600 p^{12} T^{14} + 1548290935941 p^{18} T^{16} - 1739674 p^{24} T^{18} + p^{30} T^{20} \) | |
| 71 | \( 1 - 1384874 T^{2} + 1144330957613 T^{4} - 701187300759590408 T^{6} + \)\(34\!\cdots\!34\)\( T^{8} - \)\(13\!\cdots\!76\)\( T^{10} + \)\(34\!\cdots\!34\)\( p^{6} T^{12} - 701187300759590408 p^{12} T^{14} + 1144330957613 p^{18} T^{16} - 1384874 p^{24} T^{18} + p^{30} T^{20} \) | |
| 73 | \( 1 - 1466232 T^{2} + 1268132031198 T^{4} - 729841903348409894 T^{6} + \)\(33\!\cdots\!17\)\( T^{8} - \)\(13\!\cdots\!60\)\( T^{10} + \)\(33\!\cdots\!17\)\( p^{6} T^{12} - 729841903348409894 p^{12} T^{14} + 1268132031198 p^{18} T^{16} - 1466232 p^{24} T^{18} + p^{30} T^{20} \) | |
| 79 | \( ( 1 + 2059 T + 3960211 T^{2} + 57159140 p T^{3} + 4691026151074 T^{4} + 3451478078584370 T^{5} + 4691026151074 p^{3} T^{6} + 57159140 p^{7} T^{7} + 3960211 p^{9} T^{8} + 2059 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 83 | \( 1 - 2982741 T^{2} + 4774660866853 T^{4} - 5317888567381746412 T^{6} + \)\(44\!\cdots\!42\)\( T^{8} - \)\(28\!\cdots\!86\)\( T^{10} + \)\(44\!\cdots\!42\)\( p^{6} T^{12} - 5317888567381746412 p^{12} T^{14} + 4774660866853 p^{18} T^{16} - 2982741 p^{24} T^{18} + p^{30} T^{20} \) | |
| 89 | \( 1 - 3723577 T^{2} + 6585439541173 T^{4} - 7660407109905867052 T^{6} + \)\(68\!\cdots\!66\)\( T^{8} - \)\(52\!\cdots\!78\)\( T^{10} + \)\(68\!\cdots\!66\)\( p^{6} T^{12} - 7660407109905867052 p^{12} T^{14} + 6585439541173 p^{18} T^{16} - 3723577 p^{24} T^{18} + p^{30} T^{20} \) | |
| 97 | \( 1 - 6833293 T^{2} + 22766485412457 T^{4} - 47986380434019581476 T^{6} + \)\(70\!\cdots\!74\)\( T^{8} - \)\(75\!\cdots\!10\)\( T^{10} + \)\(70\!\cdots\!74\)\( p^{6} T^{12} - 47986380434019581476 p^{12} T^{14} + 22766485412457 p^{18} T^{16} - 6833293 p^{24} T^{18} + p^{30} T^{20} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.55450386319106493202326670061, −3.45001556370017060985878347220, −3.44315630274112204366039166236, −3.34356630205681116607326643013, −3.01924035135741894516320071741, −2.94046472372970286548343450384, −2.85922645410248951659180597969, −2.78783876618845472815769102792, −2.75024144779491417810035150937, −2.66290778617785773875781113508, −1.90361828308135470873545553188, −1.80766598310399500792362130504, −1.79541302419144007343877979242, −1.77940832726649261751327014023, −1.73802270949776781547422160951, −1.19070149106252155823233767474, −1.17070657787251522030300578276, −1.09017707556529430093463956494, −1.02361390375687379364706574236, −0.829380480746682327196835887928, −0.792279668251204655425442204192, −0.43281971698718364790007130285, −0.34370195071892877032719243669, −0.15609391541129361796216453656, −0.13893634290670934109760386879, 0.13893634290670934109760386879, 0.15609391541129361796216453656, 0.34370195071892877032719243669, 0.43281971698718364790007130285, 0.792279668251204655425442204192, 0.829380480746682327196835887928, 1.02361390375687379364706574236, 1.09017707556529430093463956494, 1.17070657787251522030300578276, 1.19070149106252155823233767474, 1.73802270949776781547422160951, 1.77940832726649261751327014023, 1.79541302419144007343877979242, 1.80766598310399500792362130504, 1.90361828308135470873545553188, 2.66290778617785773875781113508, 2.75024144779491417810035150937, 2.78783876618845472815769102792, 2.85922645410248951659180597969, 2.94046472372970286548343450384, 3.01924035135741894516320071741, 3.34356630205681116607326643013, 3.44315630274112204366039166236, 3.45001556370017060985878347220, 3.55450386319106493202326670061
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.