Dirichlet series
| L(s) = 1 | − 6·4-s + 4·7-s − 4·9-s + 10·11-s − 2·13-s + 21·16-s − 16·19-s − 4·23-s − 4·27-s − 24·28-s + 20·29-s + 18·31-s + 24·36-s + 36·41-s + 12·43-s − 60·44-s − 20·47-s + 8·49-s + 12·52-s + 18·53-s − 4·61-s − 16·63-s − 56·64-s + 12·67-s + 96·76-s + 40·77-s + 30·79-s + ⋯ |
| L(s) = 1 | − 3·4-s + 1.51·7-s − 4/3·9-s + 3.01·11-s − 0.554·13-s + 21/4·16-s − 3.67·19-s − 0.834·23-s − 0.769·27-s − 4.53·28-s + 3.71·29-s + 3.23·31-s + 4·36-s + 5.62·41-s + 1.82·43-s − 9.04·44-s − 2.91·47-s + 8/7·49-s + 1.66·52-s + 2.47·53-s − 0.512·61-s − 2.01·63-s − 7·64-s + 1.46·67-s + 11.0·76-s + 4.55·77-s + 3.37·79-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{24} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{24} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 5^{24} \cdot 29^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.80418\times 10^{12}\) |
| Root analytic conductor: | \(3.40269\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 5^{24} \cdot 29^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(21.30301949\) |
| \(L(\frac12)\) | \(\approx\) | \(21.30301949\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T^{2} )^{6} \) |
| 5 | \( 1 \) | |
| 29 | \( 1 - 20 T + 190 T^{2} - 916 T^{3} - 21 T^{4} + 34760 T^{5} - 272580 T^{6} + 34760 p T^{7} - 21 p^{2} T^{8} - 916 p^{3} T^{9} + 190 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 3 | \( ( 1 + 2 T^{2} + 2 T^{3} + 7 T^{4} - 2 T^{5} + 49 T^{6} - 2 p T^{7} + 7 p^{2} T^{8} + 2 p^{3} T^{9} + 2 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 7 | \( 1 - 4 T + 8 T^{2} + 12 T^{3} + 41 T^{4} - 396 T^{5} + 1328 T^{6} + 264 T^{7} + 376 T^{8} - 21572 T^{9} + 110408 T^{10} - 53352 T^{11} - 125632 T^{12} - 53352 p T^{13} + 110408 p^{2} T^{14} - 21572 p^{3} T^{15} + 376 p^{4} T^{16} + 264 p^{5} T^{17} + 1328 p^{6} T^{18} - 396 p^{7} T^{19} + 41 p^{8} T^{20} + 12 p^{9} T^{21} + 8 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 10 T + 50 T^{2} - 116 T^{3} - 445 T^{4} + 4244 T^{5} - 13462 T^{6} - 2886 T^{7} + 238355 T^{8} - 962596 T^{9} + 1058164 T^{10} + 7514120 T^{11} - 44554078 T^{12} + 7514120 p T^{13} + 1058164 p^{2} T^{14} - 962596 p^{3} T^{15} + 238355 p^{4} T^{16} - 2886 p^{5} T^{17} - 13462 p^{6} T^{18} + 4244 p^{7} T^{19} - 445 p^{8} T^{20} - 116 p^{9} T^{21} + 50 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 2 T + 2 T^{2} - 48 T^{3} + 6 p T^{4} - 66 T^{5} + 864 T^{6} - 19294 T^{7} - 12725 T^{8} - 10382 T^{9} + 865584 T^{10} - 949466 T^{11} + 2448917 T^{12} - 949466 p T^{13} + 865584 p^{2} T^{14} - 10382 p^{3} T^{15} - 12725 p^{4} T^{16} - 19294 p^{5} T^{17} + 864 p^{6} T^{18} - 66 p^{7} T^{19} + 6 p^{9} T^{20} - 48 p^{9} T^{21} + 2 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 114 T^{2} + 5789 T^{4} - 10464 p T^{6} + 3926340 T^{8} - 72442562 T^{10} + 1249996680 T^{12} - 72442562 p^{2} T^{14} + 3926340 p^{4} T^{16} - 10464 p^{7} T^{18} + 5789 p^{8} T^{20} - 114 p^{10} T^{22} + p^{12} T^{24} \) | |
| 19 | \( 1 + 16 T + 128 T^{2} + 664 T^{3} + 1926 T^{4} - 440 T^{5} - 33120 T^{6} - 167152 T^{7} - 387297 T^{8} - 445504 T^{9} - 4657056 T^{10} - 53407792 T^{11} - 310229804 T^{12} - 53407792 p T^{13} - 4657056 p^{2} T^{14} - 445504 p^{3} T^{15} - 387297 p^{4} T^{16} - 167152 p^{5} T^{17} - 33120 p^{6} T^{18} - 440 p^{7} T^{19} + 1926 p^{8} T^{20} + 664 p^{9} T^{21} + 128 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 4 T + 8 T^{2} - 44 T^{3} + 557 T^{4} + 4164 T^{5} + 13168 T^{6} - 37592 T^{7} - 209060 T^{8} + 1057532 T^{9} + 8891528 T^{10} - 12752472 T^{11} - 300659188 T^{12} - 12752472 p T^{13} + 8891528 p^{2} T^{14} + 1057532 p^{3} T^{15} - 209060 p^{4} T^{16} - 37592 p^{5} T^{17} + 13168 p^{6} T^{18} + 4164 p^{7} T^{19} + 557 p^{8} T^{20} - 44 p^{9} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 18 T + 162 T^{2} - 1108 T^{3} + 7022 T^{4} - 38970 T^{5} + 177728 T^{6} - 562610 T^{7} - 1224125 T^{8} + 36206394 T^{9} - 318710528 T^{10} + 2092544538 T^{11} - 12204011947 T^{12} + 2092544538 p T^{13} - 318710528 p^{2} T^{14} + 36206394 p^{3} T^{15} - 1224125 p^{4} T^{16} - 562610 p^{5} T^{17} + 177728 p^{6} T^{18} - 38970 p^{7} T^{19} + 7022 p^{8} T^{20} - 1108 p^{9} T^{21} + 162 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( ( 1 + 78 T^{2} + 4087 T^{4} + 2048 T^{5} + 181988 T^{6} + 2048 p T^{7} + 4087 p^{2} T^{8} + 78 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 - 36 T + 648 T^{2} - 8500 T^{3} + 96614 T^{4} - 991204 T^{5} + 9202472 T^{6} - 79071956 T^{7} + 639402735 T^{8} - 4861976872 T^{9} + 34964876112 T^{10} - 240539411016 T^{11} + 1580340953044 T^{12} - 240539411016 p T^{13} + 34964876112 p^{2} T^{14} - 4861976872 p^{3} T^{15} + 639402735 p^{4} T^{16} - 79071956 p^{5} T^{17} + 9202472 p^{6} T^{18} - 991204 p^{7} T^{19} + 96614 p^{8} T^{20} - 8500 p^{9} T^{21} + 648 p^{10} T^{22} - 36 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( ( 1 - 6 T + 142 T^{2} - 826 T^{3} + 11203 T^{4} - 54910 T^{5} + 593889 T^{6} - 54910 p T^{7} + 11203 p^{2} T^{8} - 826 p^{3} T^{9} + 142 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( ( 1 + 10 T + 257 T^{2} + 2098 T^{3} + 28217 T^{4} + 185824 T^{5} + 1725070 T^{6} + 185824 p T^{7} + 28217 p^{2} T^{8} + 2098 p^{3} T^{9} + 257 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 53 | \( 1 - 18 T + 162 T^{2} - 1504 T^{3} + 15294 T^{4} - 118302 T^{5} + 782816 T^{6} - 5454882 T^{7} + 24437819 T^{8} - 32627602 T^{9} - 142193616 T^{10} + 4238370506 T^{11} - 50841481499 T^{12} + 4238370506 p T^{13} - 142193616 p^{2} T^{14} - 32627602 p^{3} T^{15} + 24437819 p^{4} T^{16} - 5454882 p^{5} T^{17} + 782816 p^{6} T^{18} - 118302 p^{7} T^{19} + 15294 p^{8} T^{20} - 1504 p^{9} T^{21} + 162 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 474 T^{2} + 110549 T^{4} - 16828120 T^{6} + 1862046236 T^{8} - 157638931370 T^{10} + 10464380942136 T^{12} - 157638931370 p^{2} T^{14} + 1862046236 p^{4} T^{16} - 16828120 p^{6} T^{18} + 110549 p^{8} T^{20} - 474 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( 1 + 4 T + 8 T^{2} + 108 T^{3} + 12565 T^{4} + 54932 T^{5} + 125040 T^{6} + 2076272 T^{7} + 74633764 T^{8} + 317718676 T^{9} + 835676680 T^{10} + 16485548232 T^{11} + 308656466956 T^{12} + 16485548232 p T^{13} + 835676680 p^{2} T^{14} + 317718676 p^{3} T^{15} + 74633764 p^{4} T^{16} + 2076272 p^{5} T^{17} + 125040 p^{6} T^{18} + 54932 p^{7} T^{19} + 12565 p^{8} T^{20} + 108 p^{9} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 12 T + 72 T^{2} + 76 T^{3} - 210 p T^{4} + 160980 T^{5} - 915832 T^{6} - 2992980 T^{7} + 106963135 T^{8} - 876795192 T^{9} + 2664254064 T^{10} + 38247489176 T^{11} - 569794686772 T^{12} + 38247489176 p T^{13} + 2664254064 p^{2} T^{14} - 876795192 p^{3} T^{15} + 106963135 p^{4} T^{16} - 2992980 p^{5} T^{17} - 915832 p^{6} T^{18} + 160980 p^{7} T^{19} - 210 p^{9} T^{20} + 76 p^{9} T^{21} + 72 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 464 T^{2} + 117258 T^{4} - 20147056 T^{6} + 2584809519 T^{8} - 258124212992 T^{10} + 20495817922828 T^{12} - 258124212992 p^{2} T^{14} + 2584809519 p^{4} T^{16} - 20147056 p^{6} T^{18} + 117258 p^{8} T^{20} - 464 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( 1 - 522 T^{2} + 137577 T^{4} - 24108888 T^{6} + 3125089480 T^{8} - 316219490818 T^{10} + 25666534884164 T^{12} - 316219490818 p^{2} T^{14} + 3125089480 p^{4} T^{16} - 24108888 p^{6} T^{18} + 137577 p^{8} T^{20} - 522 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( 1 - 30 T + 450 T^{2} - 68 p T^{3} + 66982 T^{4} - 849766 T^{5} + 9780272 T^{6} - 106652318 T^{7} + 1117026171 T^{8} - 10888802858 T^{9} + 102887509360 T^{10} - 989318965706 T^{11} + 9202329928773 T^{12} - 989318965706 p T^{13} + 102887509360 p^{2} T^{14} - 10888802858 p^{3} T^{15} + 1117026171 p^{4} T^{16} - 106652318 p^{5} T^{17} + 9780272 p^{6} T^{18} - 849766 p^{7} T^{19} + 66982 p^{8} T^{20} - 68 p^{10} T^{21} + 450 p^{10} T^{22} - 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{6} \) | |
| 89 | \( 1 - 20 T + 200 T^{2} - 3380 T^{3} + 47770 T^{4} - 298276 T^{5} + 2123720 T^{6} - 25394660 T^{7} - 65854545 T^{8} + 3454001752 T^{9} - 27040993552 T^{10} + 409406102648 T^{11} - 5872701758868 T^{12} + 409406102648 p T^{13} - 27040993552 p^{2} T^{14} + 3454001752 p^{3} T^{15} - 65854545 p^{4} T^{16} - 25394660 p^{5} T^{17} + 2123720 p^{6} T^{18} - 298276 p^{7} T^{19} + 47770 p^{8} T^{20} - 3380 p^{9} T^{21} + 200 p^{10} T^{22} - 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( ( 1 + 38 T + 919 T^{2} + 15444 T^{3} + 209164 T^{4} + 2384218 T^{5} + 24614596 T^{6} + 2384218 p T^{7} + 209164 p^{2} T^{8} + 15444 p^{3} T^{9} + 919 p^{4} T^{10} + 38 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
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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
−2.98428480753567498592406225461, −2.95176867728438406720047130006, −2.71202288011374665826810132334, −2.69930299696596910144329756013, −2.66414033486009099493361755160, −2.54983279897889415560185378282, −2.41690665338965338281321164007, −2.17909111632791760618207453973, −2.16597025082860878519825584815, −2.13453406151206826934692685854, −2.10905627674902805149225355999, −2.08537980055689689577572416063, −1.77615933517696420499915163070, −1.70372924212548784943512446255, −1.45098462203053524672901146183, −1.43747523266060651462966255518, −1.21887932628915492304579985420, −1.11907273236633721528759515011, −1.03999093383918903166549917810, −0.936684669241876736118130492489, −0.68473916334256611110191689870, −0.61223786155932895025300196749, −0.51730294771622726410220453483, −0.44838946800267142792289940208, −0.41191981824008856019541066174, 0.41191981824008856019541066174, 0.44838946800267142792289940208, 0.51730294771622726410220453483, 0.61223786155932895025300196749, 0.68473916334256611110191689870, 0.936684669241876736118130492489, 1.03999093383918903166549917810, 1.11907273236633721528759515011, 1.21887932628915492304579985420, 1.43747523266060651462966255518, 1.45098462203053524672901146183, 1.70372924212548784943512446255, 1.77615933517696420499915163070, 2.08537980055689689577572416063, 2.10905627674902805149225355999, 2.13453406151206826934692685854, 2.16597025082860878519825584815, 2.17909111632791760618207453973, 2.41690665338965338281321164007, 2.54983279897889415560185378282, 2.66414033486009099493361755160, 2.69930299696596910144329756013, 2.71202288011374665826810132334, 2.95176867728438406720047130006, 2.98428480753567498592406225461
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.