Dirichlet series
| L(s) = 1 | + 14·4-s + 25·9-s + 20·11-s + 91·16-s + 20·19-s + 25-s + 50·29-s − 50·31-s + 350·36-s − 34·41-s + 280·44-s + 67·49-s + 30·59-s − 14·61-s + 364·64-s − 40·71-s + 280·76-s + 106·79-s + 305·81-s + 36·89-s + 500·99-s + 14·100-s − 56·101-s + 100·109-s + 700·116-s + 210·121-s − 700·124-s + ⋯ |
| L(s) = 1 | + 7·4-s + 25/3·9-s + 6.03·11-s + 91/4·16-s + 4.58·19-s + 1/5·25-s + 9.28·29-s − 8.98·31-s + 58.3·36-s − 5.30·41-s + 42.2·44-s + 67/7·49-s + 3.90·59-s − 1.79·61-s + 91/2·64-s − 4.74·71-s + 32.1·76-s + 11.9·79-s + 33.8·81-s + 3.81·89-s + 50.2·99-s + 7/5·100-s − 5.57·101-s + 9.57·109-s + 64.9·116-s + 19.0·121-s − 62.8·124-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 11^{20} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 11^{20} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(5^{20} \cdot 11^{20} \cdot 19^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.67835\times 10^{18}\) |
| Root analytic conductor: | \(2.88866\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 5^{20} \cdot 11^{20} \cdot 19^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(21412.46646\) |
| \(L(\frac12)\) | \(\approx\) | \(21412.46646\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 - T^{2} + 6 T^{3} - 3 p T^{4} - 78 T^{5} - 4 p T^{6} + 46 T^{7} - 186 T^{8} + 194 p T^{9} + 3658 T^{10} + 194 p^{2} T^{11} - 186 p^{2} T^{12} + 46 p^{3} T^{13} - 4 p^{5} T^{14} - 78 p^{5} T^{15} - 3 p^{7} T^{16} + 6 p^{7} T^{17} - p^{8} T^{18} + p^{10} T^{20} \) |
| 11 | \( ( 1 - T )^{20} \) | |
| 19 | \( ( 1 - T )^{20} \) | |
| good | 2 | \( 1 - 7 p T^{2} + 105 T^{4} - 35 p^{4} T^{6} + 2375 T^{8} - 8489 T^{10} + 13225 p T^{12} - 73389 T^{14} + 183901 T^{16} - 26239 p^{4} T^{18} + 219275 p^{2} T^{20} - 26239 p^{6} T^{22} + 183901 p^{4} T^{24} - 73389 p^{6} T^{26} + 13225 p^{9} T^{28} - 8489 p^{10} T^{30} + 2375 p^{12} T^{32} - 35 p^{18} T^{34} + 105 p^{16} T^{36} - 7 p^{19} T^{38} + p^{20} T^{40} \) |
| 3 | \( 1 - 25 T^{2} + 320 T^{4} - 2752 T^{6} + 17681 T^{8} - 89780 T^{10} + 124607 p T^{12} - 1317965 T^{14} + 4096720 T^{16} - 11927182 T^{18} + 34991938 T^{20} - 11927182 p^{2} T^{22} + 4096720 p^{4} T^{24} - 1317965 p^{6} T^{26} + 124607 p^{9} T^{28} - 89780 p^{10} T^{30} + 17681 p^{12} T^{32} - 2752 p^{14} T^{34} + 320 p^{16} T^{36} - 25 p^{18} T^{38} + p^{20} T^{40} \) | |
| 7 | \( 1 - 67 T^{2} + 2165 T^{4} - 45651 T^{6} + 102902 p T^{8} - 9247759 T^{10} + 101885544 T^{12} - 992843119 T^{14} + 8695670261 T^{16} - 69204436940 T^{18} + 504960126182 T^{20} - 69204436940 p^{2} T^{22} + 8695670261 p^{4} T^{24} - 992843119 p^{6} T^{26} + 101885544 p^{8} T^{28} - 9247759 p^{10} T^{30} + 102902 p^{13} T^{32} - 45651 p^{14} T^{34} + 2165 p^{16} T^{36} - 67 p^{18} T^{38} + p^{20} T^{40} \) | |
| 13 | \( 1 - 123 T^{2} + 7523 T^{4} - 301384 T^{6} + 8797079 T^{8} - 197351666 T^{10} + 3506817162 T^{12} - 50424510744 T^{14} + 46570390824 p T^{16} - 6593015171059 T^{18} + 77546268776518 T^{20} - 6593015171059 p^{2} T^{22} + 46570390824 p^{5} T^{24} - 50424510744 p^{6} T^{26} + 3506817162 p^{8} T^{28} - 197351666 p^{10} T^{30} + 8797079 p^{12} T^{32} - 301384 p^{14} T^{34} + 7523 p^{16} T^{36} - 123 p^{18} T^{38} + p^{20} T^{40} \) | |
| 17 | \( 1 - 183 T^{2} + 16635 T^{4} - 1004196 T^{6} + 45406263 T^{8} - 1644440170 T^{10} + 49778074406 T^{12} - 1295726469120 T^{14} + 29545514664408 T^{16} - 596783791814523 T^{18} + 10739275426967198 T^{20} - 596783791814523 p^{2} T^{22} + 29545514664408 p^{4} T^{24} - 1295726469120 p^{6} T^{26} + 49778074406 p^{8} T^{28} - 1644440170 p^{10} T^{30} + 45406263 p^{12} T^{32} - 1004196 p^{14} T^{34} + 16635 p^{16} T^{36} - 183 p^{18} T^{38} + p^{20} T^{40} \) | |
| 23 | \( 1 - 240 T^{2} + 28949 T^{4} - 2346267 T^{6} + 143885759 T^{8} - 7117424097 T^{10} + 295212903458 T^{12} - 10525652792422 T^{14} + 327849860942226 T^{16} - 9011830157016750 T^{18} + 219829684373170062 T^{20} - 9011830157016750 p^{2} T^{22} + 327849860942226 p^{4} T^{24} - 10525652792422 p^{6} T^{26} + 295212903458 p^{8} T^{28} - 7117424097 p^{10} T^{30} + 143885759 p^{12} T^{32} - 2346267 p^{14} T^{34} + 28949 p^{16} T^{36} - 240 p^{18} T^{38} + p^{20} T^{40} \) | |
| 29 | \( ( 1 - 25 T + 443 T^{2} - 5752 T^{3} + 63525 T^{4} - 596634 T^{5} + 5000656 T^{6} - 37273906 T^{7} + 253040478 T^{8} - 1555785771 T^{9} + 8782432306 T^{10} - 1555785771 p T^{11} + 253040478 p^{2} T^{12} - 37273906 p^{3} T^{13} + 5000656 p^{4} T^{14} - 596634 p^{5} T^{15} + 63525 p^{6} T^{16} - 5752 p^{7} T^{17} + 443 p^{8} T^{18} - 25 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 31 | \( ( 1 + 25 T + 14 p T^{2} + 5625 T^{3} + 60422 T^{4} + 553994 T^{5} + 4490800 T^{6} + 32695185 T^{7} + 218000025 T^{8} + 1343827931 T^{9} + 7740382172 T^{10} + 1343827931 p T^{11} + 218000025 p^{2} T^{12} + 32695185 p^{3} T^{13} + 4490800 p^{4} T^{14} + 553994 p^{5} T^{15} + 60422 p^{6} T^{16} + 5625 p^{7} T^{17} + 14 p^{9} T^{18} + 25 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 37 | \( 1 - 329 T^{2} + 52701 T^{4} - 5609766 T^{6} + 457347378 T^{8} - 30966520918 T^{10} + 1820591198507 T^{12} - 95054679727935 T^{14} + 4461784088601287 T^{16} - 189798744628884884 T^{18} + 7352781899668586460 T^{20} - 189798744628884884 p^{2} T^{22} + 4461784088601287 p^{4} T^{24} - 95054679727935 p^{6} T^{26} + 1820591198507 p^{8} T^{28} - 30966520918 p^{10} T^{30} + 457347378 p^{12} T^{32} - 5609766 p^{14} T^{34} + 52701 p^{16} T^{36} - 329 p^{18} T^{38} + p^{20} T^{40} \) | |
| 41 | \( ( 1 + 17 T + 366 T^{2} + 3758 T^{3} + 45886 T^{4} + 322052 T^{5} + 2916180 T^{6} + 14681882 T^{7} + 120179089 T^{8} + 481580003 T^{9} + 4538928348 T^{10} + 481580003 p T^{11} + 120179089 p^{2} T^{12} + 14681882 p^{3} T^{13} + 2916180 p^{4} T^{14} + 322052 p^{5} T^{15} + 45886 p^{6} T^{16} + 3758 p^{7} T^{17} + 366 p^{8} T^{18} + 17 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 43 | \( 1 - 481 T^{2} + 115301 T^{4} - 18363443 T^{6} + 2186593192 T^{8} - 207704040477 T^{10} + 16393821827720 T^{12} - 1104768554083781 T^{14} + 64735131753253903 T^{16} - 3337090781070703082 T^{18} + \)\(15\!\cdots\!02\)\( T^{20} - 3337090781070703082 p^{2} T^{22} + 64735131753253903 p^{4} T^{24} - 1104768554083781 p^{6} T^{26} + 16393821827720 p^{8} T^{28} - 207704040477 p^{10} T^{30} + 2186593192 p^{12} T^{32} - 18363443 p^{14} T^{34} + 115301 p^{16} T^{36} - 481 p^{18} T^{38} + p^{20} T^{40} \) | |
| 47 | \( 1 - 334 T^{2} + 65254 T^{4} - 9168129 T^{6} + 1024507075 T^{8} - 95387971644 T^{10} + 7630892678738 T^{12} - 534024003910266 T^{14} + 33110521109736100 T^{16} - 1832159063514982107 T^{18} + 90894566316942001472 T^{20} - 1832159063514982107 p^{2} T^{22} + 33110521109736100 p^{4} T^{24} - 534024003910266 p^{6} T^{26} + 7630892678738 p^{8} T^{28} - 95387971644 p^{10} T^{30} + 1024507075 p^{12} T^{32} - 9168129 p^{14} T^{34} + 65254 p^{16} T^{36} - 334 p^{18} T^{38} + p^{20} T^{40} \) | |
| 53 | \( 1 - 419 T^{2} + 95355 T^{4} - 15124536 T^{6} + 1864730479 T^{8} - 189599719102 T^{10} + 311839879050 p T^{12} - 1266392916553032 T^{14} + 86654652159996344 T^{16} - 5339837441413160671 T^{18} + \)\(29\!\cdots\!98\)\( T^{20} - 5339837441413160671 p^{2} T^{22} + 86654652159996344 p^{4} T^{24} - 1266392916553032 p^{6} T^{26} + 311839879050 p^{9} T^{28} - 189599719102 p^{10} T^{30} + 1864730479 p^{12} T^{32} - 15124536 p^{14} T^{34} + 95355 p^{16} T^{36} - 419 p^{18} T^{38} + p^{20} T^{40} \) | |
| 59 | \( ( 1 - 15 T + 294 T^{2} - 2761 T^{3} + 34986 T^{4} - 248162 T^{5} + 2823252 T^{6} - 18905183 T^{7} + 217270217 T^{8} - 1434976283 T^{9} + 14744448036 T^{10} - 1434976283 p T^{11} + 217270217 p^{2} T^{12} - 18905183 p^{3} T^{13} + 2823252 p^{4} T^{14} - 248162 p^{5} T^{15} + 34986 p^{6} T^{16} - 2761 p^{7} T^{17} + 294 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 61 | \( ( 1 + 7 T + 407 T^{2} + 2532 T^{3} + 82071 T^{4} + 460928 T^{5} + 10718864 T^{6} + 54316636 T^{7} + 1003454596 T^{8} + 4530773673 T^{9} + 70300766506 T^{10} + 4530773673 p T^{11} + 1003454596 p^{2} T^{12} + 54316636 p^{3} T^{13} + 10718864 p^{4} T^{14} + 460928 p^{5} T^{15} + 82071 p^{6} T^{16} + 2532 p^{7} T^{17} + 407 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 67 | \( 1 - 490 T^{2} + 111474 T^{4} - 15934786 T^{6} + 1673115646 T^{8} - 146495946649 T^{10} + 11731781483280 T^{12} - 881104165698016 T^{14} + 61372265872154479 T^{16} - 4037714525093786635 T^{18} + \)\(26\!\cdots\!88\)\( T^{20} - 4037714525093786635 p^{2} T^{22} + 61372265872154479 p^{4} T^{24} - 881104165698016 p^{6} T^{26} + 11731781483280 p^{8} T^{28} - 146495946649 p^{10} T^{30} + 1673115646 p^{12} T^{32} - 15934786 p^{14} T^{34} + 111474 p^{16} T^{36} - 490 p^{18} T^{38} + p^{20} T^{40} \) | |
| 71 | \( ( 1 + 20 T + 637 T^{2} + 9851 T^{3} + 181315 T^{4} + 2300539 T^{5} + 31258552 T^{6} + 334758236 T^{7} + 3660853388 T^{8} + 33431197900 T^{9} + 305715302838 T^{10} + 33431197900 p T^{11} + 3660853388 p^{2} T^{12} + 334758236 p^{3} T^{13} + 31258552 p^{4} T^{14} + 2300539 p^{5} T^{15} + 181315 p^{6} T^{16} + 9851 p^{7} T^{17} + 637 p^{8} T^{18} + 20 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 73 | \( 1 - 496 T^{2} + 129831 T^{4} - 24753240 T^{6} + 3859614435 T^{8} - 513390323531 T^{10} + 59683679180030 T^{12} - 6177707181102767 T^{14} + 575389697704680516 T^{16} - 48483792332913620270 T^{18} + \)\(37\!\cdots\!66\)\( T^{20} - 48483792332913620270 p^{2} T^{22} + 575389697704680516 p^{4} T^{24} - 6177707181102767 p^{6} T^{26} + 59683679180030 p^{8} T^{28} - 513390323531 p^{10} T^{30} + 3859614435 p^{12} T^{32} - 24753240 p^{14} T^{34} + 129831 p^{16} T^{36} - 496 p^{18} T^{38} + p^{20} T^{40} \) | |
| 79 | \( ( 1 - 53 T + 1780 T^{2} - 43767 T^{3} + 875434 T^{4} - 14762829 T^{5} + 216544142 T^{6} - 2800753961 T^{7} + 32354540221 T^{8} - 335525558022 T^{9} + 3139646512940 T^{10} - 335525558022 p T^{11} + 32354540221 p^{2} T^{12} - 2800753961 p^{3} T^{13} + 216544142 p^{4} T^{14} - 14762829 p^{5} T^{15} + 875434 p^{6} T^{16} - 43767 p^{7} T^{17} + 1780 p^{8} T^{18} - 53 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 83 | \( 1 - 1235 T^{2} + 746011 T^{4} - 293300244 T^{6} + 84225960775 T^{8} - 18787329334326 T^{10} + 3378575053180126 T^{12} - 501694182739204020 T^{14} + 62483935349201544760 T^{16} - \)\(65\!\cdots\!59\)\( T^{18} + \)\(59\!\cdots\!50\)\( T^{20} - \)\(65\!\cdots\!59\)\( p^{2} T^{22} + 62483935349201544760 p^{4} T^{24} - 501694182739204020 p^{6} T^{26} + 3378575053180126 p^{8} T^{28} - 18787329334326 p^{10} T^{30} + 84225960775 p^{12} T^{32} - 293300244 p^{14} T^{34} + 746011 p^{16} T^{36} - 1235 p^{18} T^{38} + p^{20} T^{40} \) | |
| 89 | \( ( 1 - 18 T + 656 T^{2} - 9698 T^{3} + 205108 T^{4} - 2589623 T^{5} + 40949602 T^{6} - 446542956 T^{7} + 5762048929 T^{8} - 54369467353 T^{9} + 594975165744 T^{10} - 54369467353 p T^{11} + 5762048929 p^{2} T^{12} - 446542956 p^{3} T^{13} + 40949602 p^{4} T^{14} - 2589623 p^{5} T^{15} + 205108 p^{6} T^{16} - 9698 p^{7} T^{17} + 656 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 97 | \( 1 - 692 T^{2} + 215034 T^{4} - 40214856 T^{6} + 5221704500 T^{8} - 533630503915 T^{10} + 48655205927178 T^{12} - 4183858815972666 T^{14} + 322007425532164665 T^{16} - 21642680653481300239 T^{18} + \)\(16\!\cdots\!80\)\( T^{20} - 21642680653481300239 p^{2} T^{22} + 322007425532164665 p^{4} T^{24} - 4183858815972666 p^{6} T^{26} + 48655205927178 p^{8} T^{28} - 533630503915 p^{10} T^{30} + 5221704500 p^{12} T^{32} - 40214856 p^{14} T^{34} + 215034 p^{16} T^{36} - 692 p^{18} T^{38} + p^{20} T^{40} \) | |
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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
−2.05912090100821806503347186981, −2.01189964434016569447052999423, −1.98452077734187674249273297612, −1.97696479251698416763925092951, −1.89967573597757831520699124628, −1.85394539043002327044788270834, −1.80599012767945582536566431470, −1.77213508630100390124912520507, −1.66881963317342119357951558857, −1.57779847245114694837507273132, −1.57107794872127617993952689606, −1.50555830173936986421068117664, −1.50404188705024698592105471656, −1.33733920486374301276467875477, −1.10365675582976700726427260139, −1.09288492504500504009547605867, −1.03174674809233467599607982822, −0.991159982373842165233647102443, −0.964290341612790503501445626874, −0.921257103962304226282214713117, −0.906532090200920137242917607659, −0.77653135058547757543838673796, −0.75459201500841754253879200373, −0.69630778444675790805824135844, −0.31753370506964171560761569750, 0.31753370506964171560761569750, 0.69630778444675790805824135844, 0.75459201500841754253879200373, 0.77653135058547757543838673796, 0.906532090200920137242917607659, 0.921257103962304226282214713117, 0.964290341612790503501445626874, 0.991159982373842165233647102443, 1.03174674809233467599607982822, 1.09288492504500504009547605867, 1.10365675582976700726427260139, 1.33733920486374301276467875477, 1.50404188705024698592105471656, 1.50555830173936986421068117664, 1.57107794872127617993952689606, 1.57779847245114694837507273132, 1.66881963317342119357951558857, 1.77213508630100390124912520507, 1.80599012767945582536566431470, 1.85394539043002327044788270834, 1.89967573597757831520699124628, 1.97696479251698416763925092951, 1.98452077734187674249273297612, 2.01189964434016569447052999423, 2.05912090100821806503347186981
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.