Dirichlet series
| L(s) = 1 | − 3·2-s − 2·3-s + 3·4-s − 3·5-s + 6·6-s + 3·7-s − 8-s + 3·9-s + 9·10-s + 7·11-s − 6·12-s + 8·13-s − 9·14-s + 6·15-s + 8·17-s − 9·18-s + 11·19-s − 9·20-s − 6·21-s − 21·22-s − 20·23-s + 2·24-s + 3·25-s − 24·26-s − 4·27-s + 9·28-s + 20·29-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s + 3/2·4-s − 1.34·5-s + 2.44·6-s + 1.13·7-s − 0.353·8-s + 9-s + 2.84·10-s + 2.11·11-s − 1.73·12-s + 2.21·13-s − 2.40·14-s + 1.54·15-s + 1.94·17-s − 2.12·18-s + 2.52·19-s − 2.01·20-s − 1.30·21-s − 4.47·22-s − 4.17·23-s + 0.408·24-s + 3/5·25-s − 4.70·26-s − 0.769·27-s + 1.70·28-s + 3.71·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 7^{12} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 7^{12} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 5^{12} \cdot 7^{12} \cdot 11^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.91886\times 10^{9}\) |
| Root analytic conductor: | \(2.47961\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 5^{12} \cdot 7^{12} \cdot 11^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.01858961732\) |
| \(L(\frac12)\) | \(\approx\) | \(0.01858961732\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{3} \) |
| 5 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{3} \) | |
| 7 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{3} \) | |
| 11 | \( 1 - 7 T - 4 T^{2} + 20 p T^{3} - 695 T^{4} - 142 p T^{5} + 1249 p T^{6} - 142 p^{2} T^{7} - 695 p^{2} T^{8} + 20 p^{4} T^{9} - 4 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 3 | \( 1 + 2 T + T^{2} + 2 p T^{4} + 14 T^{5} + 20 T^{6} + 56 T^{7} + 107 T^{8} + 46 p T^{9} + 212 T^{10} + 326 T^{11} + 313 T^{12} + 326 p T^{13} + 212 p^{2} T^{14} + 46 p^{4} T^{15} + 107 p^{4} T^{16} + 56 p^{5} T^{17} + 20 p^{6} T^{18} + 14 p^{7} T^{19} + 2 p^{9} T^{20} + p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) |
| 13 | \( 1 - 8 T + 21 T^{2} + 42 T^{3} - 486 T^{4} + 1866 T^{5} - 2498 T^{6} - 10934 T^{7} + 6615 p T^{8} - 327082 T^{9} + 29754 p T^{10} + 3579460 T^{11} - 22542989 T^{12} + 3579460 p T^{13} + 29754 p^{3} T^{14} - 327082 p^{3} T^{15} + 6615 p^{5} T^{16} - 10934 p^{5} T^{17} - 2498 p^{6} T^{18} + 1866 p^{7} T^{19} - 486 p^{8} T^{20} + 42 p^{9} T^{21} + 21 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 8 T + 41 T^{2} - 208 T^{3} + 812 T^{4} - 6652 T^{5} + 41444 T^{6} - 189394 T^{7} + 864509 T^{8} - 3069944 T^{9} + 15199408 T^{10} - 81515376 T^{11} + 336294327 T^{12} - 81515376 p T^{13} + 15199408 p^{2} T^{14} - 3069944 p^{3} T^{15} + 864509 p^{4} T^{16} - 189394 p^{5} T^{17} + 41444 p^{6} T^{18} - 6652 p^{7} T^{19} + 812 p^{8} T^{20} - 208 p^{9} T^{21} + 41 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 11 T + 10 T^{2} + 210 T^{3} - 408 T^{4} + 93 p T^{5} - 14057 T^{6} - 93088 T^{7} + 521625 T^{8} + 156095 T^{9} + 3588567 T^{10} + 2829109 T^{11} - 221700260 T^{12} + 2829109 p T^{13} + 3588567 p^{2} T^{14} + 156095 p^{3} T^{15} + 521625 p^{4} T^{16} - 93088 p^{5} T^{17} - 14057 p^{6} T^{18} + 93 p^{8} T^{19} - 408 p^{8} T^{20} + 210 p^{9} T^{21} + 10 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( ( 1 + 10 T + 130 T^{2} + 838 T^{3} + 6495 T^{4} + 31660 T^{5} + 186140 T^{6} + 31660 p T^{7} + 6495 p^{2} T^{8} + 838 p^{3} T^{9} + 130 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 29 | \( 1 - 20 T + 189 T^{2} - 1128 T^{3} + 3582 T^{4} - 2484 T^{5} + 47892 T^{6} - 1153090 T^{7} + 11650101 T^{8} - 71869918 T^{9} + 277829974 T^{10} - 648945544 T^{11} + 1490096163 T^{12} - 648945544 p T^{13} + 277829974 p^{2} T^{14} - 71869918 p^{3} T^{15} + 11650101 p^{4} T^{16} - 1153090 p^{5} T^{17} + 47892 p^{6} T^{18} - 2484 p^{7} T^{19} + 3582 p^{8} T^{20} - 1128 p^{9} T^{21} + 189 p^{10} T^{22} - 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 2 T - 79 T^{2} + 192 T^{3} + 3444 T^{4} - 15650 T^{5} - 93466 T^{6} + 742338 T^{7} + 1523975 T^{8} - 21358522 T^{9} + 472460 p T^{10} + 265788304 T^{11} - 1401776209 T^{12} + 265788304 p T^{13} + 472460 p^{3} T^{14} - 21358522 p^{3} T^{15} + 1523975 p^{4} T^{16} + 742338 p^{5} T^{17} - 93466 p^{6} T^{18} - 15650 p^{7} T^{19} + 3444 p^{8} T^{20} + 192 p^{9} T^{21} - 79 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 4 T - 113 T^{2} - 536 T^{3} + 5428 T^{4} + 50540 T^{5} - 46384 T^{6} - 3118282 T^{7} - 10205503 T^{8} + 3110882 p T^{9} + 823282836 T^{10} - 1820298216 T^{11} - 38006571801 T^{12} - 1820298216 p T^{13} + 823282836 p^{2} T^{14} + 3110882 p^{4} T^{15} - 10205503 p^{4} T^{16} - 3118282 p^{5} T^{17} - 46384 p^{6} T^{18} + 50540 p^{7} T^{19} + 5428 p^{8} T^{20} - 536 p^{9} T^{21} - 113 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 14 T + 203 T^{2} + 2022 T^{3} + 22624 T^{4} + 184464 T^{5} + 1615224 T^{6} + 11600134 T^{7} + 93464665 T^{8} + 608433988 T^{9} + 4439354948 T^{10} + 27553746668 T^{11} + 195101215931 T^{12} + 27553746668 p T^{13} + 4439354948 p^{2} T^{14} + 608433988 p^{3} T^{15} + 93464665 p^{4} T^{16} + 11600134 p^{5} T^{17} + 1615224 p^{6} T^{18} + 184464 p^{7} T^{19} + 22624 p^{8} T^{20} + 2022 p^{9} T^{21} + 203 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( ( 1 + 19 T + 336 T^{2} + 3874 T^{3} + 39637 T^{4} + 323308 T^{5} + 2328247 T^{6} + 323308 p T^{7} + 39637 p^{2} T^{8} + 3874 p^{3} T^{9} + 336 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 + 10 T - 9 T^{2} - 6 p T^{3} + 3858 T^{4} + 24576 T^{5} - 241614 T^{6} - 1088824 T^{7} + 16561167 T^{8} + 45852872 T^{9} - 786550898 T^{10} + 974387732 T^{11} + 61239406419 T^{12} + 974387732 p T^{13} - 786550898 p^{2} T^{14} + 45852872 p^{3} T^{15} + 16561167 p^{4} T^{16} - 1088824 p^{5} T^{17} - 241614 p^{6} T^{18} + 24576 p^{7} T^{19} + 3858 p^{8} T^{20} - 6 p^{10} T^{21} - 9 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 8 T - 147 T^{2} + 1524 T^{3} + 1782 T^{4} - 80880 T^{5} + 992028 T^{6} - 4590856 T^{7} - 61015695 T^{8} + 736489784 T^{9} - 1583635154 T^{10} - 22887937510 T^{11} + 268675624335 T^{12} - 22887937510 p T^{13} - 1583635154 p^{2} T^{14} + 736489784 p^{3} T^{15} - 61015695 p^{4} T^{16} - 4590856 p^{5} T^{17} + 992028 p^{6} T^{18} - 80880 p^{7} T^{19} + 1782 p^{8} T^{20} + 1524 p^{9} T^{21} - 147 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 11 T - 2 T^{2} - 934 T^{3} - 6584 T^{4} - 17191 T^{5} + 184103 T^{6} + 1724484 T^{7} + 4252081 T^{8} - 46091339 T^{9} - 4038141 T^{10} + 1064076711 T^{11} + 17281503340 T^{12} + 1064076711 p T^{13} - 4038141 p^{2} T^{14} - 46091339 p^{3} T^{15} + 4252081 p^{4} T^{16} + 1724484 p^{5} T^{17} + 184103 p^{6} T^{18} - 17191 p^{7} T^{19} - 6584 p^{8} T^{20} - 934 p^{9} T^{21} - 2 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 34 T + 623 T^{2} + 8472 T^{3} + 92344 T^{4} + 866074 T^{5} + 7467144 T^{6} + 60252694 T^{7} + 465927965 T^{8} + 3515853378 T^{9} + 25950424088 T^{10} + 193450691428 T^{11} + 1489856071591 T^{12} + 193450691428 p T^{13} + 25950424088 p^{2} T^{14} + 3515853378 p^{3} T^{15} + 465927965 p^{4} T^{16} + 60252694 p^{5} T^{17} + 7467144 p^{6} T^{18} + 866074 p^{7} T^{19} + 92344 p^{8} T^{20} + 8472 p^{9} T^{21} + 623 p^{10} T^{22} + 34 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( ( 1 + 27 T + 574 T^{2} + 7848 T^{3} + 94627 T^{4} + 894694 T^{5} + 8025073 T^{6} + 894694 p T^{7} + 94627 p^{2} T^{8} + 7848 p^{3} T^{9} + 574 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( 1 - 18 T + 291 T^{2} - 3004 T^{3} + 31406 T^{4} - 320202 T^{5} + 3428968 T^{6} - 36151926 T^{7} + 361259877 T^{8} - 3434626866 T^{9} + 29435587546 T^{10} - 249445830364 T^{11} + 1997231375803 T^{12} - 249445830364 p T^{13} + 29435587546 p^{2} T^{14} - 3434626866 p^{3} T^{15} + 361259877 p^{4} T^{16} - 36151926 p^{5} T^{17} + 3428968 p^{6} T^{18} - 320202 p^{7} T^{19} + 31406 p^{8} T^{20} - 3004 p^{9} T^{21} + 291 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 24 T + T^{2} + 4450 T^{3} - 28776 T^{4} - 331874 T^{5} + 4749108 T^{6} - 2202364 T^{7} - 328312807 T^{8} + 2042899906 T^{9} + 5666654576 T^{10} - 91393396742 T^{11} + 415318170835 T^{12} - 91393396742 p T^{13} + 5666654576 p^{2} T^{14} + 2042899906 p^{3} T^{15} - 328312807 p^{4} T^{16} - 2202364 p^{5} T^{17} + 4749108 p^{6} T^{18} - 331874 p^{7} T^{19} - 28776 p^{8} T^{20} + 4450 p^{9} T^{21} + p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 2 T + 37 T^{2} + 358 T^{3} - 924 T^{4} + 30264 T^{5} - 147962 T^{6} + 2045204 T^{7} - 15791209 T^{8} - 313560672 T^{9} - 188508940 T^{10} - 16891597742 T^{11} - 78184020885 T^{12} - 16891597742 p T^{13} - 188508940 p^{2} T^{14} - 313560672 p^{3} T^{15} - 15791209 p^{4} T^{16} + 2045204 p^{5} T^{17} - 147962 p^{6} T^{18} + 30264 p^{7} T^{19} - 924 p^{8} T^{20} + 358 p^{9} T^{21} + 37 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 33 T + 458 T^{2} - 3422 T^{3} + 10378 T^{4} + 37195 T^{5} - 385151 T^{6} + 6287456 T^{7} - 150583523 T^{8} + 1988086837 T^{9} - 11566884811 T^{10} - 35185377927 T^{11} + 973893245584 T^{12} - 35185377927 p T^{13} - 11566884811 p^{2} T^{14} + 1988086837 p^{3} T^{15} - 150583523 p^{4} T^{16} + 6287456 p^{5} T^{17} - 385151 p^{6} T^{18} + 37195 p^{7} T^{19} + 10378 p^{8} T^{20} - 3422 p^{9} T^{21} + 458 p^{10} T^{22} - 33 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( ( 1 - T + 162 T^{2} + 464 T^{3} + 9129 T^{4} + 190140 T^{5} + 379175 T^{6} + 190140 p T^{7} + 9129 p^{2} T^{8} + 464 p^{3} T^{9} + 162 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 + T - 426 T^{2} - 270 T^{3} + 73556 T^{4} - 173473 T^{5} - 4987855 T^{6} + 77280082 T^{7} - 246776083 T^{8} - 12010332991 T^{9} + 88528979833 T^{10} + 606830088507 T^{11} - 10705540358772 T^{12} + 606830088507 p T^{13} + 88528979833 p^{2} T^{14} - 12010332991 p^{3} T^{15} - 246776083 p^{4} T^{16} + 77280082 p^{5} T^{17} - 4987855 p^{6} T^{18} - 173473 p^{7} T^{19} + 73556 p^{8} T^{20} - 270 p^{9} T^{21} - 426 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.37726449251953195815398243368, −3.30226889617428739925101696131, −3.17473834677764141356395509406, −3.13492660159982194267883639129, −3.07345953164706073192284151449, −2.94160995371044314345368647414, −2.70897720906908180577188674283, −2.65851828071486907745386926496, −2.65321986619237678445020600225, −2.13024099910473945987015559345, −2.12195321212066970955334864090, −2.04404147906481375950566914892, −1.88741011747581955470639903396, −1.73279635046169675756055331060, −1.67803649767918143725619313730, −1.63578667690677343910393696846, −1.45152467645235386494808124926, −1.27457411787885438378233693195, −1.24685430139700864818950460586, −1.18699149398509815358286796310, −1.03059547035610991195038525619, −0.76981619727739260895534205753, −0.56896224799729290766681829142, −0.34700528466490223138013440953, −0.03473545924074228053625561194, 0.03473545924074228053625561194, 0.34700528466490223138013440953, 0.56896224799729290766681829142, 0.76981619727739260895534205753, 1.03059547035610991195038525619, 1.18699149398509815358286796310, 1.24685430139700864818950460586, 1.27457411787885438378233693195, 1.45152467645235386494808124926, 1.63578667690677343910393696846, 1.67803649767918143725619313730, 1.73279635046169675756055331060, 1.88741011747581955470639903396, 2.04404147906481375950566914892, 2.12195321212066970955334864090, 2.13024099910473945987015559345, 2.65321986619237678445020600225, 2.65851828071486907745386926496, 2.70897720906908180577188674283, 2.94160995371044314345368647414, 3.07345953164706073192284151449, 3.13492660159982194267883639129, 3.17473834677764141356395509406, 3.30226889617428739925101696131, 3.37726449251953195815398243368
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.