Dirichlet series
| L(s) = 1 | + 2·2-s − 6·4-s − 12·7-s − 14·8-s − 14·9-s − 8·11-s + 2·13-s − 24·14-s + 13·16-s − 8·17-s − 28·18-s − 26·19-s − 16·22-s + 12·23-s + 4·26-s + 4·27-s + 72·28-s − 12·29-s − 50·31-s + 40·32-s − 16·34-s + 84·36-s + 8·37-s − 52·38-s − 4·41-s + 26·43-s + 48·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 3·4-s − 4.53·7-s − 4.94·8-s − 4.66·9-s − 2.41·11-s + 0.554·13-s − 6.41·14-s + 13/4·16-s − 1.94·17-s − 6.59·18-s − 5.96·19-s − 3.41·22-s + 2.50·23-s + 0.784·26-s + 0.769·27-s + 13.6·28-s − 2.22·29-s − 8.98·31-s + 7.07·32-s − 2.74·34-s + 14·36-s + 1.31·37-s − 8.43·38-s − 0.624·41-s + 3.96·43-s + 7.23·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 7^{12} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 7^{12} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(5^{24} \cdot 7^{12} \cdot 23^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.21483\times 10^{18}\) |
| Root analytic conductor: | \(5.66919\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 5^{24} \cdot 7^{12} \cdot 23^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 7 | \( ( 1 + T )^{12} \) | |
| 23 | \( ( 1 - T )^{12} \) | |
| good | 2 | \( 1 - p T + 5 p T^{2} - 9 p T^{3} + 55 T^{4} - 23 p^{2} T^{5} + 107 p T^{6} - 83 p^{2} T^{7} + 329 p T^{8} - 469 p T^{9} + 1671 T^{10} - 1097 p T^{11} + 3613 T^{12} - 1097 p^{2} T^{13} + 1671 p^{2} T^{14} - 469 p^{4} T^{15} + 329 p^{5} T^{16} - 83 p^{7} T^{17} + 107 p^{7} T^{18} - 23 p^{9} T^{19} + 55 p^{8} T^{20} - 9 p^{10} T^{21} + 5 p^{11} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) |
| 3 | \( 1 + 14 T^{2} - 4 T^{3} + 4 p^{3} T^{4} - 40 T^{5} + 625 T^{6} - 80 p T^{7} + 958 p T^{8} - 376 p T^{9} + 3664 p T^{10} - 1352 p T^{11} + 35812 T^{12} - 1352 p^{2} T^{13} + 3664 p^{3} T^{14} - 376 p^{4} T^{15} + 958 p^{5} T^{16} - 80 p^{6} T^{17} + 625 p^{6} T^{18} - 40 p^{7} T^{19} + 4 p^{11} T^{20} - 4 p^{9} T^{21} + 14 p^{10} T^{22} + p^{12} T^{24} \) | |
| 11 | \( 1 + 8 T + 107 T^{2} + 694 T^{3} + 5424 T^{4} + 29570 T^{5} + 173242 T^{6} + 810066 T^{7} + 3885411 T^{8} + 15777182 T^{9} + 64461251 T^{10} + 228757088 T^{11} + 73739736 p T^{12} + 228757088 p T^{13} + 64461251 p^{2} T^{14} + 15777182 p^{3} T^{15} + 3885411 p^{4} T^{16} + 810066 p^{5} T^{17} + 173242 p^{6} T^{18} + 29570 p^{7} T^{19} + 5424 p^{8} T^{20} + 694 p^{9} T^{21} + 107 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 2 T + 81 T^{2} - 92 T^{3} + 3013 T^{4} - 982 T^{5} + 70653 T^{6} + 33340 T^{7} + 1225551 T^{8} + 1478546 T^{9} + 17756861 T^{10} + 29894704 T^{11} + 236277458 T^{12} + 29894704 p T^{13} + 17756861 p^{2} T^{14} + 1478546 p^{3} T^{15} + 1225551 p^{4} T^{16} + 33340 p^{5} T^{17} + 70653 p^{6} T^{18} - 982 p^{7} T^{19} + 3013 p^{8} T^{20} - 92 p^{9} T^{21} + 81 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 8 T + 143 T^{2} + 1074 T^{3} + 10390 T^{4} + 69858 T^{5} + 494790 T^{6} + 2909608 T^{7} + 16897571 T^{8} + 86342196 T^{9} + 431097207 T^{10} + 1919135912 T^{11} + 8374637380 T^{12} + 1919135912 p T^{13} + 431097207 p^{2} T^{14} + 86342196 p^{3} T^{15} + 16897571 p^{4} T^{16} + 2909608 p^{5} T^{17} + 494790 p^{6} T^{18} + 69858 p^{7} T^{19} + 10390 p^{8} T^{20} + 1074 p^{9} T^{21} + 143 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 26 T + 426 T^{2} + 5010 T^{3} + 47727 T^{4} + 380838 T^{5} + 2657599 T^{6} + 16489928 T^{7} + 93329855 T^{8} + 25651846 p T^{9} + 2393508283 T^{10} + 585836240 p T^{11} + 49603973866 T^{12} + 585836240 p^{2} T^{13} + 2393508283 p^{2} T^{14} + 25651846 p^{4} T^{15} + 93329855 p^{4} T^{16} + 16489928 p^{5} T^{17} + 2657599 p^{6} T^{18} + 380838 p^{7} T^{19} + 47727 p^{8} T^{20} + 5010 p^{9} T^{21} + 426 p^{10} T^{22} + 26 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 12 T + 258 T^{2} + 88 p T^{3} + 31367 T^{4} + 262496 T^{5} + 2401642 T^{6} + 17323380 T^{7} + 130559371 T^{8} + 825033200 T^{9} + 5385163876 T^{10} + 30202298184 T^{11} + 174902665114 T^{12} + 30202298184 p T^{13} + 5385163876 p^{2} T^{14} + 825033200 p^{3} T^{15} + 130559371 p^{4} T^{16} + 17323380 p^{5} T^{17} + 2401642 p^{6} T^{18} + 262496 p^{7} T^{19} + 31367 p^{8} T^{20} + 88 p^{10} T^{21} + 258 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 50 T + 1356 T^{2} + 25672 T^{3} + 377722 T^{4} + 147692 p T^{5} + 47514831 T^{6} + 433678672 T^{7} + 3549751230 T^{8} + 26406623386 T^{9} + 180056634294 T^{10} + 1130557827586 T^{11} + 6550831526410 T^{12} + 1130557827586 p T^{13} + 180056634294 p^{2} T^{14} + 26406623386 p^{3} T^{15} + 3549751230 p^{4} T^{16} + 433678672 p^{5} T^{17} + 47514831 p^{6} T^{18} + 147692 p^{8} T^{19} + 377722 p^{8} T^{20} + 25672 p^{9} T^{21} + 1356 p^{10} T^{22} + 50 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 8 T + 271 T^{2} - 1588 T^{3} + 33050 T^{4} - 143046 T^{5} + 2536026 T^{6} - 8188974 T^{7} + 145490287 T^{8} - 366377890 T^{9} + 6874214803 T^{10} - 14648570878 T^{11} + 275823336652 T^{12} - 14648570878 p T^{13} + 6874214803 p^{2} T^{14} - 366377890 p^{3} T^{15} + 145490287 p^{4} T^{16} - 8188974 p^{5} T^{17} + 2536026 p^{6} T^{18} - 143046 p^{7} T^{19} + 33050 p^{8} T^{20} - 1588 p^{9} T^{21} + 271 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 4 T + 251 T^{2} + 780 T^{3} + 29258 T^{4} + 68680 T^{5} + 2208081 T^{6} + 4031628 T^{7} + 129639772 T^{8} + 206766552 T^{9} + 6549282855 T^{10} + 9944814512 T^{11} + 288865734980 T^{12} + 9944814512 p T^{13} + 6549282855 p^{2} T^{14} + 206766552 p^{3} T^{15} + 129639772 p^{4} T^{16} + 4031628 p^{5} T^{17} + 2208081 p^{6} T^{18} + 68680 p^{7} T^{19} + 29258 p^{8} T^{20} + 780 p^{9} T^{21} + 251 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 26 T + 578 T^{2} - 8682 T^{3} + 117563 T^{4} - 1307620 T^{5} + 13584507 T^{6} - 124442736 T^{7} + 1088354331 T^{8} - 8668947386 T^{9} + 66529730527 T^{10} - 470551819326 T^{11} + 3210480833674 T^{12} - 470551819326 p T^{13} + 66529730527 p^{2} T^{14} - 8668947386 p^{3} T^{15} + 1088354331 p^{4} T^{16} - 124442736 p^{5} T^{17} + 13584507 p^{6} T^{18} - 1307620 p^{7} T^{19} + 117563 p^{8} T^{20} - 8682 p^{9} T^{21} + 578 p^{10} T^{22} - 26 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 16 T + 368 T^{2} - 4392 T^{3} + 62248 T^{4} - 618200 T^{5} + 6857049 T^{6} - 27108 p^{2} T^{7} + 11983898 p T^{8} - 4408165880 T^{9} + 36381823766 T^{10} - 256816993148 T^{11} + 1895036675956 T^{12} - 256816993148 p T^{13} + 36381823766 p^{2} T^{14} - 4408165880 p^{3} T^{15} + 11983898 p^{5} T^{16} - 27108 p^{7} T^{17} + 6857049 p^{6} T^{18} - 618200 p^{7} T^{19} + 62248 p^{8} T^{20} - 4392 p^{9} T^{21} + 368 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 18 T + 501 T^{2} + 6670 T^{3} + 111442 T^{4} + 1217844 T^{5} + 15555492 T^{6} + 146270876 T^{7} + 1555182919 T^{8} + 12868554898 T^{9} + 118426221347 T^{10} + 870464673070 T^{11} + 7064049896396 T^{12} + 870464673070 p T^{13} + 118426221347 p^{2} T^{14} + 12868554898 p^{3} T^{15} + 1555182919 p^{4} T^{16} + 146270876 p^{5} T^{17} + 15555492 p^{6} T^{18} + 1217844 p^{7} T^{19} + 111442 p^{8} T^{20} + 6670 p^{9} T^{21} + 501 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 18 T + 443 T^{2} + 6112 T^{3} + 93728 T^{4} + 1071802 T^{5} + 12989744 T^{6} + 128973906 T^{7} + 1333184899 T^{8} + 11791088468 T^{9} + 107695184445 T^{10} + 857739367502 T^{11} + 7038996899912 T^{12} + 857739367502 p T^{13} + 107695184445 p^{2} T^{14} + 11791088468 p^{3} T^{15} + 1333184899 p^{4} T^{16} + 128973906 p^{5} T^{17} + 12989744 p^{6} T^{18} + 1071802 p^{7} T^{19} + 93728 p^{8} T^{20} + 6112 p^{9} T^{21} + 443 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 8 T + 535 T^{2} - 4494 T^{3} + 140094 T^{4} - 1173188 T^{5} + 23726874 T^{6} - 189410822 T^{7} + 2879329515 T^{8} - 21162096094 T^{9} + 261954953003 T^{10} - 1727032895578 T^{11} + 18222879036420 T^{12} - 1727032895578 p T^{13} + 261954953003 p^{2} T^{14} - 21162096094 p^{3} T^{15} + 2879329515 p^{4} T^{16} - 189410822 p^{5} T^{17} + 23726874 p^{6} T^{18} - 1173188 p^{7} T^{19} + 140094 p^{8} T^{20} - 4494 p^{9} T^{21} + 535 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 38 T + 1149 T^{2} - 24514 T^{3} + 454030 T^{4} - 7065866 T^{5} + 99332140 T^{6} - 1242394940 T^{7} + 14332969401 T^{8} - 150908335254 T^{9} + 1482069378827 T^{10} - 13442018585536 T^{11} + 114292032364392 T^{12} - 13442018585536 p T^{13} + 1482069378827 p^{2} T^{14} - 150908335254 p^{3} T^{15} + 14332969401 p^{4} T^{16} - 1242394940 p^{5} T^{17} + 99332140 p^{6} T^{18} - 7065866 p^{7} T^{19} + 454030 p^{8} T^{20} - 24514 p^{9} T^{21} + 1149 p^{10} T^{22} - 38 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 24 T + 784 T^{2} + 14490 T^{3} + 280248 T^{4} + 4197644 T^{5} + 61318753 T^{6} + 768898944 T^{7} + 9221912504 T^{8} + 98820077666 T^{9} + 1008817898632 T^{10} + 9347436776276 T^{11} + 82499882792140 T^{12} + 9347436776276 p T^{13} + 1008817898632 p^{2} T^{14} + 98820077666 p^{3} T^{15} + 9221912504 p^{4} T^{16} + 768898944 p^{5} T^{17} + 61318753 p^{6} T^{18} + 4197644 p^{7} T^{19} + 280248 p^{8} T^{20} + 14490 p^{9} T^{21} + 784 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 14 T + 389 T^{2} + 4160 T^{3} + 66639 T^{4} + 553494 T^{5} + 7090493 T^{6} + 47822356 T^{7} + 554889109 T^{8} + 3067431698 T^{9} + 35193699921 T^{10} + 160607916420 T^{11} + 2266296169786 T^{12} + 160607916420 p T^{13} + 35193699921 p^{2} T^{14} + 3067431698 p^{3} T^{15} + 554889109 p^{4} T^{16} + 47822356 p^{5} T^{17} + 7090493 p^{6} T^{18} + 553494 p^{7} T^{19} + 66639 p^{8} T^{20} + 4160 p^{9} T^{21} + 389 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 44 T + 1394 T^{2} + 32652 T^{3} + 646501 T^{4} + 10969906 T^{5} + 165979585 T^{6} + 2258446966 T^{7} + 28109218637 T^{8} + 321722375442 T^{9} + 3414112084085 T^{10} + 33680849723590 T^{11} + 309968024368258 T^{12} + 33680849723590 p T^{13} + 3414112084085 p^{2} T^{14} + 321722375442 p^{3} T^{15} + 28109218637 p^{4} T^{16} + 2258446966 p^{5} T^{17} + 165979585 p^{6} T^{18} + 10969906 p^{7} T^{19} + 646501 p^{8} T^{20} + 32652 p^{9} T^{21} + 1394 p^{10} T^{22} + 44 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 14 T + 735 T^{2} + 9146 T^{3} + 260712 T^{4} + 2894852 T^{5} + 59256494 T^{6} + 588022944 T^{7} + 9619690971 T^{8} + 85260583062 T^{9} + 1174765790903 T^{10} + 9262183130126 T^{11} + 110625794527488 T^{12} + 9262183130126 p T^{13} + 1174765790903 p^{2} T^{14} + 85260583062 p^{3} T^{15} + 9619690971 p^{4} T^{16} + 588022944 p^{5} T^{17} + 59256494 p^{6} T^{18} + 2894852 p^{7} T^{19} + 260712 p^{8} T^{20} + 9146 p^{9} T^{21} + 735 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 10 T + 363 T^{2} + 4124 T^{3} + 82060 T^{4} + 1013390 T^{5} + 14202706 T^{6} + 169764084 T^{7} + 2034510667 T^{8} + 21889141238 T^{9} + 243098243759 T^{10} + 2301367141458 T^{11} + 23914236452424 T^{12} + 2301367141458 p T^{13} + 243098243759 p^{2} T^{14} + 21889141238 p^{3} T^{15} + 2034510667 p^{4} T^{16} + 169764084 p^{5} T^{17} + 14202706 p^{6} T^{18} + 1013390 p^{7} T^{19} + 82060 p^{8} T^{20} + 4124 p^{9} T^{21} + 363 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 4 T + 577 T^{2} + 3798 T^{3} + 179283 T^{4} + 1393682 T^{5} + 39663292 T^{6} + 316257962 T^{7} + 6717625877 T^{8} + 51651879538 T^{9} + 901518714519 T^{10} + 6415544013704 T^{11} + 97366354036182 T^{12} + 6415544013704 p T^{13} + 901518714519 p^{2} T^{14} + 51651879538 p^{3} T^{15} + 6717625877 p^{4} T^{16} + 316257962 p^{5} T^{17} + 39663292 p^{6} T^{18} + 1393682 p^{7} T^{19} + 179283 p^{8} T^{20} + 3798 p^{9} T^{21} + 577 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.92093652921664057806979572263, −2.86029060047020706940276872555, −2.80061333319384272295043748919, −2.77967278217646616713775663084, −2.74614993195288608539066136816, −2.64474381208875284999748498515, −2.50557789746688656017007210309, −2.48106502112814227266030134672, −2.44711768492112610042215512663, −2.36543103380641915906680759244, −2.36334133389747209028770255251, −2.20591509628820530197605238493, −2.18434112432551545017890100464, −2.17411435960898659446090481149, −2.15886173806328322203348334608, −1.91695142253314689799608851221, −1.56870211362172210460744374225, −1.54184326909736171645214307127, −1.44035546598443374375633250329, −1.36397137525893084271601071125, −1.19456046550348227302936001671, −1.04220595790471733620901481764, −1.02930330547847502660488634441, −0.999168882474246751988793666800, −0.888710752693488787154571605370, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.888710752693488787154571605370, 0.999168882474246751988793666800, 1.02930330547847502660488634441, 1.04220595790471733620901481764, 1.19456046550348227302936001671, 1.36397137525893084271601071125, 1.44035546598443374375633250329, 1.54184326909736171645214307127, 1.56870211362172210460744374225, 1.91695142253314689799608851221, 2.15886173806328322203348334608, 2.17411435960898659446090481149, 2.18434112432551545017890100464, 2.20591509628820530197605238493, 2.36334133389747209028770255251, 2.36543103380641915906680759244, 2.44711768492112610042215512663, 2.48106502112814227266030134672, 2.50557789746688656017007210309, 2.64474381208875284999748498515, 2.74614993195288608539066136816, 2.77967278217646616713775663084, 2.80061333319384272295043748919, 2.86029060047020706940276872555, 2.92093652921664057806979572263
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.