Dirichlet series
| L(s) = 1 | + 4·3-s + 4·5-s + 8·9-s + 16·15-s − 3·16-s − 28·17-s + 24·23-s + 18·25-s + 20·27-s − 8·29-s − 8·31-s − 20·37-s + 8·43-s + 32·45-s − 16·47-s − 12·48-s − 112·51-s + 24·53-s − 32·59-s + 96·69-s − 24·73-s + 72·75-s − 12·80-s + 39·81-s + 24·83-s − 112·85-s − 32·87-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.78·5-s + 8/3·9-s + 4.13·15-s − 3/4·16-s − 6.79·17-s + 5.00·23-s + 18/5·25-s + 3.84·27-s − 1.48·29-s − 1.43·31-s − 3.28·37-s + 1.21·43-s + 4.77·45-s − 2.33·47-s − 1.73·48-s − 15.6·51-s + 3.29·53-s − 4.16·59-s + 11.5·69-s − 2.80·73-s + 8.31·75-s − 1.34·80-s + 13/3·81-s + 2.63·83-s − 12.1·85-s − 3.43·87-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(494.261\) |
| Root analytic conductor: | \(1.29493\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(4.556593057\) |
| \(L(\frac12)\) | \(\approx\) | \(4.556593057\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T^{4} )^{3} \) |
| 3 | \( 1 - 4 T + 8 T^{2} - 20 T^{3} + 19 p T^{4} - 104 T^{5} + 160 T^{6} - 104 p T^{7} + 19 p^{3} T^{8} - 20 p^{3} T^{9} + 8 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) | |
| 5 | \( 1 - 4 T - 2 T^{2} + 4 T^{3} + 89 T^{4} - 88 T^{5} - 288 T^{6} - 88 p T^{7} + 89 p^{2} T^{8} + 4 p^{3} T^{9} - 2 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) | |
| 7 | \( ( 1 + T^{4} )^{3} \) | |
| good | 11 | \( 1 - 56 T^{2} + 1358 T^{4} - 19128 T^{6} + 195023 T^{8} - 1950544 T^{10} + 21368868 T^{12} - 1950544 p^{2} T^{14} + 195023 p^{4} T^{16} - 19128 p^{6} T^{18} + 1358 p^{8} T^{20} - 56 p^{10} T^{22} + p^{12} T^{24} \) |
| 13 | \( 1 - 32 T^{3} - 14 T^{4} + 272 T^{5} + 512 T^{6} - 5520 T^{7} - 4301 T^{8} + 109488 T^{9} + 220800 T^{10} + 406064 T^{11} - 6852572 T^{12} + 406064 p T^{13} + 220800 p^{2} T^{14} + 109488 p^{3} T^{15} - 4301 p^{4} T^{16} - 5520 p^{5} T^{17} + 512 p^{6} T^{18} + 272 p^{7} T^{19} - 14 p^{8} T^{20} - 32 p^{9} T^{21} + p^{12} T^{24} \) | |
| 17 | \( 1 + 28 T + 392 T^{2} + 3716 T^{3} + 26626 T^{4} + 151924 T^{5} + 720808 T^{6} + 3009708 T^{7} + 12126655 T^{8} + 52482936 T^{9} + 50512 p^{3} T^{10} + 4066888 p^{2} T^{11} + 5125539580 T^{12} + 4066888 p^{3} T^{13} + 50512 p^{5} T^{14} + 52482936 p^{3} T^{15} + 12126655 p^{4} T^{16} + 3009708 p^{5} T^{17} + 720808 p^{6} T^{18} + 151924 p^{7} T^{19} + 26626 p^{8} T^{20} + 3716 p^{9} T^{21} + 392 p^{10} T^{22} + 28 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 104 T^{2} + 6018 T^{4} - 249384 T^{6} + 7954675 T^{8} - 203686352 T^{10} + 4270353988 T^{12} - 203686352 p^{2} T^{14} + 7954675 p^{4} T^{16} - 249384 p^{6} T^{18} + 6018 p^{8} T^{20} - 104 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( 1 - 24 T + 288 T^{2} - 2648 T^{3} + 22078 T^{4} - 163240 T^{5} + 1065248 T^{6} - 6483944 T^{7} + 37525599 T^{8} - 203370768 T^{9} + 1048113088 T^{10} - 5277774096 T^{11} + 25830292740 T^{12} - 5277774096 p T^{13} + 1048113088 p^{2} T^{14} - 203370768 p^{3} T^{15} + 37525599 p^{4} T^{16} - 6483944 p^{5} T^{17} + 1065248 p^{6} T^{18} - 163240 p^{7} T^{19} + 22078 p^{8} T^{20} - 2648 p^{9} T^{21} + 288 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( ( 1 + 4 T + 114 T^{2} + 548 T^{3} + 6631 T^{4} + 29256 T^{5} + 241692 T^{6} + 29256 p T^{7} + 6631 p^{2} T^{8} + 548 p^{3} T^{9} + 114 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 31 | \( ( 1 + 4 T + 28 T^{2} + 76 T^{3} + 807 T^{4} + 1288 T^{5} + 32184 T^{6} + 1288 p T^{7} + 807 p^{2} T^{8} + 76 p^{3} T^{9} + 28 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 37 | \( 1 + 20 T + 200 T^{2} + 1052 T^{3} + 3634 T^{4} + 30620 T^{5} + 438952 T^{6} + 98788 p T^{7} + 16552975 T^{8} + 43779560 T^{9} + 283860944 T^{10} + 4234271992 T^{11} + 34611552220 T^{12} + 4234271992 p T^{13} + 283860944 p^{2} T^{14} + 43779560 p^{3} T^{15} + 16552975 p^{4} T^{16} + 98788 p^{6} T^{17} + 438952 p^{6} T^{18} + 30620 p^{7} T^{19} + 3634 p^{8} T^{20} + 1052 p^{9} T^{21} + 200 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 212 T^{2} + 22434 T^{4} - 1643716 T^{6} + 97594575 T^{8} - 5014736296 T^{10} + 222514140252 T^{12} - 5014736296 p^{2} T^{14} + 97594575 p^{4} T^{16} - 1643716 p^{6} T^{18} + 22434 p^{8} T^{20} - 212 p^{10} T^{22} + p^{12} T^{24} \) | |
| 43 | \( 1 - 8 T + 32 T^{2} - 184 T^{3} + 5014 T^{4} - 25096 T^{5} + 57248 T^{6} + 105352 T^{7} + 1869663 T^{8} + 20409392 T^{9} - 214625984 T^{10} + 2505871696 T^{11} - 14493786700 T^{12} + 2505871696 p T^{13} - 214625984 p^{2} T^{14} + 20409392 p^{3} T^{15} + 1869663 p^{4} T^{16} + 105352 p^{5} T^{17} + 57248 p^{6} T^{18} - 25096 p^{7} T^{19} + 5014 p^{8} T^{20} - 184 p^{9} T^{21} + 32 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 16 T + 128 T^{2} + 48 T^{3} - 3930 T^{4} + 2832 T^{5} + 549504 T^{6} + 4835248 T^{7} + 9727055 T^{8} - 102211168 T^{9} - 484788992 T^{10} + 7463204576 T^{11} + 87433873108 T^{12} + 7463204576 p T^{13} - 484788992 p^{2} T^{14} - 102211168 p^{3} T^{15} + 9727055 p^{4} T^{16} + 4835248 p^{5} T^{17} + 549504 p^{6} T^{18} + 2832 p^{7} T^{19} - 3930 p^{8} T^{20} + 48 p^{9} T^{21} + 128 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 24 T + 288 T^{2} - 56 p T^{3} + 27318 T^{4} - 189144 T^{5} + 1076384 T^{6} - 4855704 T^{7} - 9792929 T^{8} + 439179408 T^{9} - 4825144512 T^{10} + 45292675216 T^{11} - 372958615948 T^{12} + 45292675216 p T^{13} - 4825144512 p^{2} T^{14} + 439179408 p^{3} T^{15} - 9792929 p^{4} T^{16} - 4855704 p^{5} T^{17} + 1076384 p^{6} T^{18} - 189144 p^{7} T^{19} + 27318 p^{8} T^{20} - 56 p^{10} T^{21} + 288 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( ( 1 + 16 T + 270 T^{2} + 3160 T^{3} + 33193 T^{4} + 284568 T^{5} + 2445376 T^{6} + 284568 p T^{7} + 33193 p^{2} T^{8} + 3160 p^{3} T^{9} + 270 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 61 | \( ( 1 + 178 T^{2} + 664 T^{3} + 12681 T^{4} + 121240 T^{5} + 672800 T^{6} + 121240 p T^{7} + 12681 p^{2} T^{8} + 664 p^{3} T^{9} + 178 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 67 | \( 1 + 64 T^{3} + 2390 T^{4} - 9024 T^{5} + 2048 T^{6} - 508672 T^{7} + 20286015 T^{8} + 4116224 T^{9} + 3266560 T^{10} - 440818304 T^{11} + 142523500468 T^{12} - 440818304 p T^{13} + 3266560 p^{2} T^{14} + 4116224 p^{3} T^{15} + 20286015 p^{4} T^{16} - 508672 p^{5} T^{17} + 2048 p^{6} T^{18} - 9024 p^{7} T^{19} + 2390 p^{8} T^{20} + 64 p^{9} T^{21} + p^{12} T^{24} \) | |
| 71 | \( 1 - 440 T^{2} + 91446 T^{4} - 12419864 T^{6} + 1305779695 T^{8} - 116298170672 T^{10} + 8924106004404 T^{12} - 116298170672 p^{2} T^{14} + 1305779695 p^{4} T^{16} - 12419864 p^{6} T^{18} + 91446 p^{8} T^{20} - 440 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( 1 + 24 T + 288 T^{2} + 2936 T^{3} + 13126 T^{4} - 142696 T^{5} - 2894944 T^{6} - 37628616 T^{7} - 323587345 T^{8} - 985106832 T^{9} + 7253083968 T^{10} + 194887191344 T^{11} + 2401164656404 T^{12} + 194887191344 p T^{13} + 7253083968 p^{2} T^{14} - 985106832 p^{3} T^{15} - 323587345 p^{4} T^{16} - 37628616 p^{5} T^{17} - 2894944 p^{6} T^{18} - 142696 p^{7} T^{19} + 13126 p^{8} T^{20} + 2936 p^{9} T^{21} + 288 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 508 T^{2} + 135562 T^{4} - 24649964 T^{6} + 3381685343 T^{8} - 366753014104 T^{10} + 32137748903980 T^{12} - 366753014104 p^{2} T^{14} + 3381685343 p^{4} T^{16} - 24649964 p^{6} T^{18} + 135562 p^{8} T^{20} - 508 p^{10} T^{22} + p^{12} T^{24} \) | |
| 83 | \( 1 - 24 T + 288 T^{2} - 3784 T^{3} + 70066 T^{4} - 929320 T^{5} + 9284000 T^{6} - 107957496 T^{7} + 1400664563 T^{8} - 14071243520 T^{9} + 124154702400 T^{10} - 1299362373632 T^{11} + 13371946678180 T^{12} - 1299362373632 p T^{13} + 124154702400 p^{2} T^{14} - 14071243520 p^{3} T^{15} + 1400664563 p^{4} T^{16} - 107957496 p^{5} T^{17} + 9284000 p^{6} T^{18} - 929320 p^{7} T^{19} + 70066 p^{8} T^{20} - 3784 p^{9} T^{21} + 288 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( ( 1 + 24 T + 358 T^{2} + 5272 T^{3} + 62767 T^{4} + 696208 T^{5} + 7360436 T^{6} + 696208 p T^{7} + 62767 p^{2} T^{8} + 5272 p^{3} T^{9} + 358 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 - 8 T + 32 T^{2} + 1240 T^{3} + 7270 T^{4} - 77896 T^{5} + 1159328 T^{6} + 23481432 T^{7} - 27667953 T^{8} + 164181296 T^{9} + 23294478656 T^{10} + 157379895024 T^{11} - 641510857644 T^{12} + 157379895024 p T^{13} + 23294478656 p^{2} T^{14} + 164181296 p^{3} T^{15} - 27667953 p^{4} T^{16} + 23481432 p^{5} T^{17} + 1159328 p^{6} T^{18} - 77896 p^{7} T^{19} + 7270 p^{8} T^{20} + 1240 p^{9} T^{21} + 32 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−4.44846747768548499038172437545, −4.27119608773980869787695386903, −4.24154280934798370694093078539, −3.94626396319876935588304232935, −3.68395181293659637421381507639, −3.68144744977542060039123926699, −3.51598663734738544973810977710, −3.49587184510857598534807987591, −3.40119169248587176864998601308, −3.14824810878796305445025086750, −2.99063287127576143716061488326, −2.81389086568044070757467002087, −2.80495176298863783918874763525, −2.71079382891300965046633340424, −2.61550670295574961636560240702, −2.49035819914518134272934781093, −2.16968843195902093665338916735, −2.14586791538991250400438387430, −2.13312028394773834110502183153, −1.68399124395867391503759872758, −1.68352935641657835248037316607, −1.63528317652275521628122922433, −1.37242599095823206284890245803, −1.00809102719402969751972245783, −0.37338600230011037618993655286, 0.37338600230011037618993655286, 1.00809102719402969751972245783, 1.37242599095823206284890245803, 1.63528317652275521628122922433, 1.68352935641657835248037316607, 1.68399124395867391503759872758, 2.13312028394773834110502183153, 2.14586791538991250400438387430, 2.16968843195902093665338916735, 2.49035819914518134272934781093, 2.61550670295574961636560240702, 2.71079382891300965046633340424, 2.80495176298863783918874763525, 2.81389086568044070757467002087, 2.99063287127576143716061488326, 3.14824810878796305445025086750, 3.40119169248587176864998601308, 3.49587184510857598534807987591, 3.51598663734738544973810977710, 3.68144744977542060039123926699, 3.68395181293659637421381507639, 3.94626396319876935588304232935, 4.24154280934798370694093078539, 4.27119608773980869787695386903, 4.44846747768548499038172437545
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.