Dirichlet series
| L(s) = 1 | + 9·3-s + 13·5-s − 3·7-s − 24·9-s − 45·13-s + 117·15-s + 17·17-s + 147·19-s − 27·21-s + 164·23-s − 321·25-s − 400·27-s − 177·29-s + 275·31-s − 39·35-s + 745·37-s − 405·39-s − 967·41-s + 380·43-s − 312·45-s + 769·47-s − 1.45e3·49-s + 153·51-s + 701·53-s + 1.32e3·57-s + 1.29e3·59-s + 1.35e3·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.16·5-s − 0.161·7-s − 8/9·9-s − 0.960·13-s + 2.01·15-s + 0.242·17-s + 1.77·19-s − 0.280·21-s + 1.48·23-s − 2.56·25-s − 2.85·27-s − 1.13·29-s + 1.59·31-s − 0.188·35-s + 3.31·37-s − 1.66·39-s − 3.68·41-s + 1.34·43-s − 1.03·45-s + 2.38·47-s − 4.25·49-s + 0.420·51-s + 1.81·53-s + 3.07·57-s + 2.84·59-s + 2.85·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{30} \cdot 11^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.69329\times 10^{17}\) |
| Root analytic conductor: | \(7.55737\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{30} \cdot 11^{20} ,\ ( \ : [3/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(237.3796845\) |
| \(L(\frac12)\) | \(\approx\) | \(237.3796845\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) | |
| good | 3 | \( 1 - p^{2} T + 35 p T^{2} - 761 T^{3} + 4730 T^{4} - 28721 T^{5} + 50405 p T^{6} - 860033 T^{7} + 5046880 T^{8} - 340937 p^{4} T^{9} + 155781023 T^{10} - 340937 p^{7} T^{11} + 5046880 p^{6} T^{12} - 860033 p^{9} T^{13} + 50405 p^{13} T^{14} - 28721 p^{15} T^{15} + 4730 p^{18} T^{16} - 761 p^{21} T^{17} + 35 p^{25} T^{18} - p^{29} T^{19} + p^{30} T^{20} \) |
| 5 | \( 1 - 13 T + 98 p T^{2} - 8742 T^{3} + 31789 p T^{4} - 2617808 T^{5} + 7763527 p T^{6} - 531081282 T^{7} + 1413051422 p T^{8} - 83552674603 T^{9} + 990768241926 T^{10} - 83552674603 p^{3} T^{11} + 1413051422 p^{7} T^{12} - 531081282 p^{9} T^{13} + 7763527 p^{13} T^{14} - 2617808 p^{15} T^{15} + 31789 p^{19} T^{16} - 8742 p^{21} T^{17} + 98 p^{25} T^{18} - 13 p^{27} T^{19} + p^{30} T^{20} \) | |
| 7 | \( 1 + 3 T + 1468 T^{2} - 3076 T^{3} + 1231035 T^{4} - 4425906 T^{5} + 761339781 T^{6} - 3246267488 T^{7} + 360136558344 T^{8} - 1566471860117 T^{9} + 137530603531910 T^{10} - 1566471860117 p^{3} T^{11} + 360136558344 p^{6} T^{12} - 3246267488 p^{9} T^{13} + 761339781 p^{12} T^{14} - 4425906 p^{15} T^{15} + 1231035 p^{18} T^{16} - 3076 p^{21} T^{17} + 1468 p^{24} T^{18} + 3 p^{27} T^{19} + p^{30} T^{20} \) | |
| 13 | \( 1 + 45 T + 6334 T^{2} + 229550 T^{3} + 24463805 T^{4} + 835008360 T^{5} + 71875410319 T^{6} + 2433322753490 T^{7} + 198367987987690 T^{8} + 500822144993895 p T^{9} + 467725587283127334 T^{10} + 500822144993895 p^{4} T^{11} + 198367987987690 p^{6} T^{12} + 2433322753490 p^{9} T^{13} + 71875410319 p^{12} T^{14} + 835008360 p^{15} T^{15} + 24463805 p^{18} T^{16} + 229550 p^{21} T^{17} + 6334 p^{24} T^{18} + 45 p^{27} T^{19} + p^{30} T^{20} \) | |
| 17 | \( 1 - p T + 29311 T^{2} - 783635 T^{3} + 395007710 T^{4} - 919823877 p T^{5} + 3309790779629 T^{6} - 179820128293335 T^{7} + 20286233421651980 T^{8} - 1324535529344024217 T^{9} + \)\(10\!\cdots\!37\)\( T^{10} - 1324535529344024217 p^{3} T^{11} + 20286233421651980 p^{6} T^{12} - 179820128293335 p^{9} T^{13} + 3309790779629 p^{12} T^{14} - 919823877 p^{16} T^{15} + 395007710 p^{18} T^{16} - 783635 p^{21} T^{17} + 29311 p^{24} T^{18} - p^{28} T^{19} + p^{30} T^{20} \) | |
| 19 | \( 1 - 147 T + 48217 T^{2} - 6355839 T^{3} + 1150646466 T^{4} - 130839311515 T^{5} + 17482859638091 T^{6} - 90264264867949 p T^{7} + 186677081633975572 T^{8} - 15941730877678194643 T^{9} + \)\(14\!\cdots\!47\)\( T^{10} - 15941730877678194643 p^{3} T^{11} + 186677081633975572 p^{6} T^{12} - 90264264867949 p^{10} T^{13} + 17482859638091 p^{12} T^{14} - 130839311515 p^{15} T^{15} + 1150646466 p^{18} T^{16} - 6355839 p^{21} T^{17} + 48217 p^{24} T^{18} - 147 p^{27} T^{19} + p^{30} T^{20} \) | |
| 23 | \( 1 - 164 T + 80218 T^{2} - 10764332 T^{3} + 3018899789 T^{4} - 335925170336 T^{5} + 70840833168920 T^{6} - 6715597101085696 T^{7} + 1197419657487088866 T^{8} - 4351156388573710584 p T^{9} + \)\(16\!\cdots\!56\)\( T^{10} - 4351156388573710584 p^{4} T^{11} + 1197419657487088866 p^{6} T^{12} - 6715597101085696 p^{9} T^{13} + 70840833168920 p^{12} T^{14} - 335925170336 p^{15} T^{15} + 3018899789 p^{18} T^{16} - 10764332 p^{21} T^{17} + 80218 p^{24} T^{18} - 164 p^{27} T^{19} + p^{30} T^{20} \) | |
| 29 | \( 1 + 177 T + 140770 T^{2} + 19459110 T^{3} + 8429634525 T^{4} + 833939524968 T^{5} + 281544482005335 T^{6} + 15785763296073458 T^{7} + 6184866014275669550 T^{8} + 87364583861049328495 T^{9} + \)\(12\!\cdots\!34\)\( T^{10} + 87364583861049328495 p^{3} T^{11} + 6184866014275669550 p^{6} T^{12} + 15785763296073458 p^{9} T^{13} + 281544482005335 p^{12} T^{14} + 833939524968 p^{15} T^{15} + 8429634525 p^{18} T^{16} + 19459110 p^{21} T^{17} + 140770 p^{24} T^{18} + 177 p^{27} T^{19} + p^{30} T^{20} \) | |
| 31 | \( 1 - 275 T + 4546 p T^{2} - 40163592 T^{3} + 11906913757 T^{4} - 2993586441690 T^{5} + 704903601737103 T^{6} - 152843682499826572 T^{7} + 1005975238101600050 p T^{8} - \)\(58\!\cdots\!55\)\( T^{9} + \)\(10\!\cdots\!58\)\( T^{10} - \)\(58\!\cdots\!55\)\( p^{3} T^{11} + 1005975238101600050 p^{7} T^{12} - 152843682499826572 p^{9} T^{13} + 704903601737103 p^{12} T^{14} - 2993586441690 p^{15} T^{15} + 11906913757 p^{18} T^{16} - 40163592 p^{21} T^{17} + 4546 p^{25} T^{18} - 275 p^{27} T^{19} + p^{30} T^{20} \) | |
| 37 | \( 1 - 745 T + 512026 T^{2} - 229361406 T^{3} + 90903930337 T^{4} - 28707055121400 T^{5} + 8064587971377147 T^{6} - 1949378882730623834 T^{7} + \)\(43\!\cdots\!06\)\( T^{8} - \)\(92\!\cdots\!87\)\( T^{9} + \)\(20\!\cdots\!86\)\( T^{10} - \)\(92\!\cdots\!87\)\( p^{3} T^{11} + \)\(43\!\cdots\!06\)\( p^{6} T^{12} - 1949378882730623834 p^{9} T^{13} + 8064587971377147 p^{12} T^{14} - 28707055121400 p^{15} T^{15} + 90903930337 p^{18} T^{16} - 229361406 p^{21} T^{17} + 512026 p^{24} T^{18} - 745 p^{27} T^{19} + p^{30} T^{20} \) | |
| 41 | \( 1 + 967 T + 801187 T^{2} + 458081453 T^{3} + 237447766902 T^{4} + 102086269534067 T^{5} + 40747805734848665 T^{6} + 14260581852998578969 T^{7} + \)\(46\!\cdots\!60\)\( T^{8} + \)\(13\!\cdots\!95\)\( T^{9} + \)\(37\!\cdots\!37\)\( T^{10} + \)\(13\!\cdots\!95\)\( p^{3} T^{11} + \)\(46\!\cdots\!60\)\( p^{6} T^{12} + 14260581852998578969 p^{9} T^{13} + 40747805734848665 p^{12} T^{14} + 102086269534067 p^{15} T^{15} + 237447766902 p^{18} T^{16} + 458081453 p^{21} T^{17} + 801187 p^{24} T^{18} + 967 p^{27} T^{19} + p^{30} T^{20} \) | |
| 43 | \( 1 - 380 T + 437091 T^{2} - 123976816 T^{3} + 94829437654 T^{4} - 22998632734388 T^{5} + 14003596961094614 T^{6} - 2935144462999915524 T^{7} + \)\(15\!\cdots\!53\)\( T^{8} - \)\(28\!\cdots\!72\)\( T^{9} + \)\(13\!\cdots\!90\)\( T^{10} - \)\(28\!\cdots\!72\)\( p^{3} T^{11} + \)\(15\!\cdots\!53\)\( p^{6} T^{12} - 2935144462999915524 p^{9} T^{13} + 14003596961094614 p^{12} T^{14} - 22998632734388 p^{15} T^{15} + 94829437654 p^{18} T^{16} - 123976816 p^{21} T^{17} + 437091 p^{24} T^{18} - 380 p^{27} T^{19} + p^{30} T^{20} \) | |
| 47 | \( 1 - 769 T + 871164 T^{2} - 438176036 T^{3} + 294199168187 T^{4} - 112716072360314 T^{5} + 59232794041803557 T^{6} - 19110477439710886184 T^{7} + \)\(87\!\cdots\!40\)\( T^{8} - \)\(25\!\cdots\!09\)\( T^{9} + \)\(10\!\cdots\!10\)\( T^{10} - \)\(25\!\cdots\!09\)\( p^{3} T^{11} + \)\(87\!\cdots\!40\)\( p^{6} T^{12} - 19110477439710886184 p^{9} T^{13} + 59232794041803557 p^{12} T^{14} - 112716072360314 p^{15} T^{15} + 294199168187 p^{18} T^{16} - 438176036 p^{21} T^{17} + 871164 p^{24} T^{18} - 769 p^{27} T^{19} + p^{30} T^{20} \) | |
| 53 | \( 1 - 701 T + 1240446 T^{2} - 754322710 T^{3} + 704028631785 T^{4} - 376851974413608 T^{5} + 244276067867561579 T^{6} - \)\(11\!\cdots\!50\)\( T^{7} + \)\(58\!\cdots\!30\)\( T^{8} - \)\(24\!\cdots\!71\)\( T^{9} + \)\(10\!\cdots\!10\)\( T^{10} - \)\(24\!\cdots\!71\)\( p^{3} T^{11} + \)\(58\!\cdots\!30\)\( p^{6} T^{12} - \)\(11\!\cdots\!50\)\( p^{9} T^{13} + 244276067867561579 p^{12} T^{14} - 376851974413608 p^{15} T^{15} + 704028631785 p^{18} T^{16} - 754322710 p^{21} T^{17} + 1240446 p^{24} T^{18} - 701 p^{27} T^{19} + p^{30} T^{20} \) | |
| 59 | \( 1 - 1291 T + 2349977 T^{2} - 2225967751 T^{3} + 2334488914178 T^{4} - 1734227961097123 T^{5} + 1328707124194402675 T^{6} - \)\(80\!\cdots\!63\)\( T^{7} + \)\(48\!\cdots\!60\)\( T^{8} - \)\(24\!\cdots\!15\)\( T^{9} + \)\(12\!\cdots\!39\)\( T^{10} - \)\(24\!\cdots\!15\)\( p^{3} T^{11} + \)\(48\!\cdots\!60\)\( p^{6} T^{12} - \)\(80\!\cdots\!63\)\( p^{9} T^{13} + 1328707124194402675 p^{12} T^{14} - 1734227961097123 p^{15} T^{15} + 2334488914178 p^{18} T^{16} - 2225967751 p^{21} T^{17} + 2349977 p^{24} T^{18} - 1291 p^{27} T^{19} + p^{30} T^{20} \) | |
| 61 | \( 1 - 1359 T + 2395054 T^{2} - 2315636946 T^{3} + 39051329037 p T^{4} - 1805915521819384 T^{5} + 1374826324388492227 T^{6} - \)\(85\!\cdots\!06\)\( T^{7} + \)\(52\!\cdots\!54\)\( T^{8} - \)\(27\!\cdots\!89\)\( T^{9} + \)\(14\!\cdots\!62\)\( T^{10} - \)\(27\!\cdots\!89\)\( p^{3} T^{11} + \)\(52\!\cdots\!54\)\( p^{6} T^{12} - \)\(85\!\cdots\!06\)\( p^{9} T^{13} + 1374826324388492227 p^{12} T^{14} - 1805915521819384 p^{15} T^{15} + 39051329037 p^{19} T^{16} - 2315636946 p^{21} T^{17} + 2395054 p^{24} T^{18} - 1359 p^{27} T^{19} + p^{30} T^{20} \) | |
| 67 | \( 1 - 2260 T + 4167351 T^{2} - 5353583296 T^{3} + 6079737484342 T^{4} - 5725389317370340 T^{5} + 4916614601782280022 T^{6} - \)\(36\!\cdots\!04\)\( T^{7} + \)\(25\!\cdots\!21\)\( T^{8} - \)\(15\!\cdots\!52\)\( T^{9} + \)\(92\!\cdots\!26\)\( T^{10} - \)\(15\!\cdots\!52\)\( p^{3} T^{11} + \)\(25\!\cdots\!21\)\( p^{6} T^{12} - \)\(36\!\cdots\!04\)\( p^{9} T^{13} + 4916614601782280022 p^{12} T^{14} - 5725389317370340 p^{15} T^{15} + 6079737484342 p^{18} T^{16} - 5353583296 p^{21} T^{17} + 4167351 p^{24} T^{18} - 2260 p^{27} T^{19} + p^{30} T^{20} \) | |
| 71 | \( 1 - 465 T + 2543894 T^{2} - 681674940 T^{3} + 2787022472633 T^{4} - 239003391760150 T^{5} + 1794444347712500027 T^{6} + \)\(15\!\cdots\!20\)\( T^{7} + \)\(80\!\cdots\!46\)\( T^{8} + \)\(17\!\cdots\!15\)\( T^{9} + \)\(30\!\cdots\!38\)\( T^{10} + \)\(17\!\cdots\!15\)\( p^{3} T^{11} + \)\(80\!\cdots\!46\)\( p^{6} T^{12} + \)\(15\!\cdots\!20\)\( p^{9} T^{13} + 1794444347712500027 p^{12} T^{14} - 239003391760150 p^{15} T^{15} + 2787022472633 p^{18} T^{16} - 681674940 p^{21} T^{17} + 2543894 p^{24} T^{18} - 465 p^{27} T^{19} + p^{30} T^{20} \) | |
| 73 | \( 1 + 111 T + 1494115 T^{2} + 53938097 T^{3} + 1355531330318 T^{4} + 896864123675 T^{5} + 869775814175007205 T^{6} - 30149689173597314187 T^{7} + \)\(44\!\cdots\!08\)\( T^{8} - \)\(22\!\cdots\!77\)\( T^{9} + \)\(18\!\cdots\!73\)\( T^{10} - \)\(22\!\cdots\!77\)\( p^{3} T^{11} + \)\(44\!\cdots\!08\)\( p^{6} T^{12} - 30149689173597314187 p^{9} T^{13} + 869775814175007205 p^{12} T^{14} + 896864123675 p^{15} T^{15} + 1355531330318 p^{18} T^{16} + 53938097 p^{21} T^{17} + 1494115 p^{24} T^{18} + 111 p^{27} T^{19} + p^{30} T^{20} \) | |
| 79 | \( 1 + 1827 T + 3657782 T^{2} + 4812275544 T^{3} + 6248132212421 T^{4} + 6636441008655610 T^{5} + 6791970654852979911 T^{6} + \)\(60\!\cdots\!16\)\( T^{7} + \)\(52\!\cdots\!42\)\( T^{8} + \)\(40\!\cdots\!63\)\( T^{9} + \)\(29\!\cdots\!82\)\( T^{10} + \)\(40\!\cdots\!63\)\( p^{3} T^{11} + \)\(52\!\cdots\!42\)\( p^{6} T^{12} + \)\(60\!\cdots\!16\)\( p^{9} T^{13} + 6791970654852979911 p^{12} T^{14} + 6636441008655610 p^{15} T^{15} + 6248132212421 p^{18} T^{16} + 4812275544 p^{21} T^{17} + 3657782 p^{24} T^{18} + 1827 p^{27} T^{19} + p^{30} T^{20} \) | |
| 83 | \( 1 - 4947 T + 15164603 T^{2} - 33603759111 T^{3} + 59693149094102 T^{4} - 88459051161285707 T^{5} + \)\(11\!\cdots\!41\)\( T^{6} - \)\(12\!\cdots\!75\)\( T^{7} + \)\(12\!\cdots\!64\)\( T^{8} - \)\(11\!\cdots\!03\)\( T^{9} + \)\(88\!\cdots\!57\)\( T^{10} - \)\(11\!\cdots\!03\)\( p^{3} T^{11} + \)\(12\!\cdots\!64\)\( p^{6} T^{12} - \)\(12\!\cdots\!75\)\( p^{9} T^{13} + \)\(11\!\cdots\!41\)\( p^{12} T^{14} - 88459051161285707 p^{15} T^{15} + 59693149094102 p^{18} T^{16} - 33603759111 p^{21} T^{17} + 15164603 p^{24} T^{18} - 4947 p^{27} T^{19} + p^{30} T^{20} \) | |
| 89 | \( 1 - 446 T + 2441325 T^{2} - 1252299880 T^{3} + 3905900195138 T^{4} - 1270329468900118 T^{5} + 4197653601460087566 T^{6} - \)\(98\!\cdots\!30\)\( T^{7} + \)\(34\!\cdots\!45\)\( T^{8} - \)\(47\!\cdots\!98\)\( T^{9} + \)\(26\!\cdots\!54\)\( T^{10} - \)\(47\!\cdots\!98\)\( p^{3} T^{11} + \)\(34\!\cdots\!45\)\( p^{6} T^{12} - \)\(98\!\cdots\!30\)\( p^{9} T^{13} + 4197653601460087566 p^{12} T^{14} - 1270329468900118 p^{15} T^{15} + 3905900195138 p^{18} T^{16} - 1252299880 p^{21} T^{17} + 2441325 p^{24} T^{18} - 446 p^{27} T^{19} + p^{30} T^{20} \) | |
| 97 | \( 1 - 3511 T + 11262123 T^{2} - 24329654501 T^{3} + 47913425374934 T^{4} - 77449814890716019 T^{5} + \)\(11\!\cdots\!45\)\( T^{6} - \)\(15\!\cdots\!13\)\( T^{7} + \)\(18\!\cdots\!24\)\( T^{8} - \)\(19\!\cdots\!63\)\( T^{9} + \)\(19\!\cdots\!69\)\( T^{10} - \)\(19\!\cdots\!63\)\( p^{3} T^{11} + \)\(18\!\cdots\!24\)\( p^{6} T^{12} - \)\(15\!\cdots\!13\)\( p^{9} T^{13} + \)\(11\!\cdots\!45\)\( p^{12} T^{14} - 77449814890716019 p^{15} T^{15} + 47913425374934 p^{18} T^{16} - 24329654501 p^{21} T^{17} + 11262123 p^{24} T^{18} - 3511 p^{27} T^{19} + p^{30} T^{20} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.21932057204823071720398497792, −3.18518243345914781685943733120, −3.18257423479738107920405189727, −2.91647136379407015099480733340, −2.81241039129955903295528222247, −2.59684710713887902525164333368, −2.39904947228753099657644494822, −2.29474705614730014300120085406, −2.18219265132558019988593884314, −2.16234066115038205102579835509, −2.13363924673164594045404210352, −2.10585275306934672073439911399, −2.01520003214391992933168864841, −1.89485629901943241022308278772, −1.63938082135640725655403878932, −1.43011373574650399672617812202, −1.38967697979299652672934322282, −0.900889156645437734143864796719, −0.819036995548082502409904449224, −0.76545810652271181780396915821, −0.67959518183271146473886493740, −0.67021737375667726435793070160, −0.54733064244659704329630142135, −0.39234492431250643148186760964, −0.34108306525941665111022515255, 0.34108306525941665111022515255, 0.39234492431250643148186760964, 0.54733064244659704329630142135, 0.67021737375667726435793070160, 0.67959518183271146473886493740, 0.76545810652271181780396915821, 0.819036995548082502409904449224, 0.900889156645437734143864796719, 1.38967697979299652672934322282, 1.43011373574650399672617812202, 1.63938082135640725655403878932, 1.89485629901943241022308278772, 2.01520003214391992933168864841, 2.10585275306934672073439911399, 2.13363924673164594045404210352, 2.16234066115038205102579835509, 2.18219265132558019988593884314, 2.29474705614730014300120085406, 2.39904947228753099657644494822, 2.59684710713887902525164333368, 2.81241039129955903295528222247, 2.91647136379407015099480733340, 3.18257423479738107920405189727, 3.18518243345914781685943733120, 3.21932057204823071720398497792
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.