Dirichlet series
| L(s) = 1 | + 8·9-s − 4·11-s + 8·19-s + 4·25-s − 10·29-s − 18·31-s − 2·41-s + 23·49-s − 22·59-s − 8·61-s + 34·71-s + 20·79-s + 24·81-s + 48·89-s − 32·99-s + 10·101-s + 8·109-s − 56·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 8/3·9-s − 1.20·11-s + 1.83·19-s + 4/5·25-s − 1.85·29-s − 3.23·31-s − 0.312·41-s + 23/7·49-s − 2.86·59-s − 1.02·61-s + 4.03·71-s + 2.25·79-s + 8/3·81-s + 5.08·89-s − 3.21·99-s + 0.995·101-s + 0.766·109-s − 5.09·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{12} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{12} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 5^{12} \cdot 23^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.01190\times 10^{14}\) |
| Root analytic conductor: | \(3.83307\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{48} \cdot 5^{12} \cdot 23^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(30.66805406\) |
| \(L(\frac12)\) | \(\approx\) | \(30.66805406\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 4 T^{2} + 12 T^{3} + 43 T^{4} + 12 T^{5} - 48 T^{6} + 12 p T^{7} + 43 p^{2} T^{8} + 12 p^{3} T^{9} - 4 p^{4} T^{10} + p^{6} T^{12} \) | |
| 23 | \( ( 1 + T^{2} )^{6} \) | |
| good | 3 | \( 1 - 8 T^{2} + 40 T^{4} - 56 p T^{6} + 616 T^{8} - 2008 T^{10} + 6238 T^{12} - 2008 p^{2} T^{14} + 616 p^{4} T^{16} - 56 p^{7} T^{18} + 40 p^{8} T^{20} - 8 p^{10} T^{22} + p^{12} T^{24} \) |
| 7 | \( 1 - 23 T^{2} + 344 T^{4} - 3923 T^{6} + 5569 p T^{8} - 334294 T^{10} + 2519840 T^{12} - 334294 p^{2} T^{14} + 5569 p^{5} T^{16} - 3923 p^{6} T^{18} + 344 p^{8} T^{20} - 23 p^{10} T^{22} + p^{12} T^{24} \) | |
| 11 | \( ( 1 + 2 T + 34 T^{2} + 82 T^{3} + 619 T^{4} + 1640 T^{5} + 7796 T^{6} + 1640 p T^{7} + 619 p^{2} T^{8} + 82 p^{3} T^{9} + 34 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 13 | \( 1 - 48 T^{2} + 1240 T^{4} - 20544 T^{6} + 236232 T^{8} - 1991712 T^{10} + 18769518 T^{12} - 1991712 p^{2} T^{14} + 236232 p^{4} T^{16} - 20544 p^{6} T^{18} + 1240 p^{8} T^{20} - 48 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( 1 - 147 T^{2} + 10012 T^{4} - 422363 T^{6} + 12556895 T^{8} - 16852866 p T^{10} + 5329948648 T^{12} - 16852866 p^{3} T^{14} + 12556895 p^{4} T^{16} - 422363 p^{6} T^{18} + 10012 p^{8} T^{20} - 147 p^{10} T^{22} + p^{12} T^{24} \) | |
| 19 | \( ( 1 - 4 T + 72 T^{2} - 240 T^{3} + 2459 T^{4} - 7308 T^{5} + 55432 T^{6} - 7308 p T^{7} + 2459 p^{2} T^{8} - 240 p^{3} T^{9} + 72 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 29 | \( ( 1 + 5 T + 90 T^{2} + 275 T^{3} + 4100 T^{4} + 12353 T^{5} + 147060 T^{6} + 12353 p T^{7} + 4100 p^{2} T^{8} + 275 p^{3} T^{9} + 90 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 31 | \( ( 1 + 9 T + 128 T^{2} + 557 T^{3} + 4572 T^{4} + 6569 T^{5} + 97946 T^{6} + 6569 p T^{7} + 4572 p^{2} T^{8} + 557 p^{3} T^{9} + 128 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 37 | \( 1 - 87 T^{2} + 6484 T^{4} - 364523 T^{6} + 17299847 T^{8} - 702666966 T^{10} + 28185279944 T^{12} - 702666966 p^{2} T^{14} + 17299847 p^{4} T^{16} - 364523 p^{6} T^{18} + 6484 p^{8} T^{20} - 87 p^{10} T^{22} + p^{12} T^{24} \) | |
| 41 | \( ( 1 + T + 182 T^{2} + 47 T^{3} + 14976 T^{4} - 2147 T^{5} + 753988 T^{6} - 2147 p T^{7} + 14976 p^{2} T^{8} + 47 p^{3} T^{9} + 182 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 43 | \( 1 - 176 T^{2} + 20822 T^{4} - 1758416 T^{6} + 120611391 T^{8} - 6717447872 T^{10} + 316423407796 T^{12} - 6717447872 p^{2} T^{14} + 120611391 p^{4} T^{16} - 1758416 p^{6} T^{18} + 20822 p^{8} T^{20} - 176 p^{10} T^{22} + p^{12} T^{24} \) | |
| 47 | \( 1 - 336 T^{2} + 56608 T^{4} - 6349784 T^{6} + 528335600 T^{8} - 34386609360 T^{10} + 1796765512750 T^{12} - 34386609360 p^{2} T^{14} + 528335600 p^{4} T^{16} - 6349784 p^{6} T^{18} + 56608 p^{8} T^{20} - 336 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( 1 - 363 T^{2} + 65936 T^{4} - 7923639 T^{6} + 706621863 T^{8} - 49902431622 T^{10} + 2898048453808 T^{12} - 49902431622 p^{2} T^{14} + 706621863 p^{4} T^{16} - 7923639 p^{6} T^{18} + 65936 p^{8} T^{20} - 363 p^{10} T^{22} + p^{12} T^{24} \) | |
| 59 | \( ( 1 + 11 T + 166 T^{2} + 1553 T^{3} + 20679 T^{4} + 146450 T^{5} + 1334612 T^{6} + 146450 p T^{7} + 20679 p^{2} T^{8} + 1553 p^{3} T^{9} + 166 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 61 | \( ( 1 + 4 T + 168 T^{2} + 688 T^{3} + 18499 T^{4} + 62460 T^{5} + 1289496 T^{6} + 62460 p T^{7} + 18499 p^{2} T^{8} + 688 p^{3} T^{9} + 168 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( 1 - 463 T^{2} + 109624 T^{4} - 17536211 T^{6} + 2094468599 T^{8} - 195622111750 T^{10} + 14608971974176 T^{12} - 195622111750 p^{2} T^{14} + 2094468599 p^{4} T^{16} - 17536211 p^{6} T^{18} + 109624 p^{8} T^{20} - 463 p^{10} T^{22} + p^{12} T^{24} \) | |
| 71 | \( ( 1 - 17 T + 364 T^{2} - 4465 T^{3} + 58576 T^{4} - 543329 T^{5} + 5347658 T^{6} - 543329 p T^{7} + 58576 p^{2} T^{8} - 4465 p^{3} T^{9} + 364 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 73 | \( 1 - 500 T^{2} + 129232 T^{4} - 22622492 T^{6} + 2953369992 T^{8} - 301175413652 T^{10} + 24526171431798 T^{12} - 301175413652 p^{2} T^{14} + 2953369992 p^{4} T^{16} - 22622492 p^{6} T^{18} + 129232 p^{8} T^{20} - 500 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( ( 1 - 10 T + 464 T^{2} - 3742 T^{3} + 90539 T^{4} - 575388 T^{5} + 9499464 T^{6} - 575388 p T^{7} + 90539 p^{2} T^{8} - 3742 p^{3} T^{9} + 464 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 - 219 T^{2} + 25624 T^{4} - 1742555 T^{6} + 157441487 T^{8} - 19627426770 T^{10} + 2122064270128 T^{12} - 19627426770 p^{2} T^{14} + 157441487 p^{4} T^{16} - 1742555 p^{6} T^{18} + 25624 p^{8} T^{20} - 219 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( ( 1 - 24 T + 392 T^{2} - 5508 T^{3} + 65051 T^{4} - 687684 T^{5} + 6959384 T^{6} - 687684 p T^{7} + 65051 p^{2} T^{8} - 5508 p^{3} T^{9} + 392 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 - 460 T^{2} + 84202 T^{4} - 7964300 T^{6} + 639516863 T^{8} - 93340242856 T^{10} + 11938996972972 T^{12} - 93340242856 p^{2} T^{14} + 639516863 p^{4} T^{16} - 7964300 p^{6} T^{18} + 84202 p^{8} T^{20} - 460 p^{10} T^{22} + p^{12} T^{24} \) | |
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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
−2.86176984065097590887610397820, −2.85153720424181199580208034890, −2.75791025198023463508048543064, −2.61969864924976145340974369367, −2.32860705803556020730667019216, −2.29828139215640907898175892403, −2.23187766836765641796634372713, −2.20848990426242274659050106936, −2.11450080795921441732554050191, −2.02203744558350765694063104444, −1.92488127742330185084175562371, −1.87895918312123885299030802349, −1.81847951678729403683005820594, −1.68320684064780484707927325645, −1.41072135574155377755105233294, −1.36841272747922754528522968270, −1.28099793020103856137944449013, −1.20207467614542138522259820524, −1.06658553904558180934472198866, −1.01610558145078036519581777756, −0.67270569309599845651983317144, −0.61633565659423295991926050197, −0.39842541223044207724829568557, −0.38066344387041226806939585246, −0.34816800345945935418330573742, 0.34816800345945935418330573742, 0.38066344387041226806939585246, 0.39842541223044207724829568557, 0.61633565659423295991926050197, 0.67270569309599845651983317144, 1.01610558145078036519581777756, 1.06658553904558180934472198866, 1.20207467614542138522259820524, 1.28099793020103856137944449013, 1.36841272747922754528522968270, 1.41072135574155377755105233294, 1.68320684064780484707927325645, 1.81847951678729403683005820594, 1.87895918312123885299030802349, 1.92488127742330185084175562371, 2.02203744558350765694063104444, 2.11450080795921441732554050191, 2.20848990426242274659050106936, 2.23187766836765641796634372713, 2.29828139215640907898175892403, 2.32860705803556020730667019216, 2.61969864924976145340974369367, 2.75791025198023463508048543064, 2.85153720424181199580208034890, 2.86176984065097590887610397820
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.