Dirichlet series
| L(s) = 1 | − 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 − 48·89-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 + 167-s + 48·169-s + 173-s + ⋯ |
| L(s) = 1 | − 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 − 5.08·89-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 + 0.0773·167-s + 3.69·169-s + 0.0760·173-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 5^{12} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 5^{12} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 3^{24} \cdot 5^{12} \cdot 23^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.70344\times 10^{18}\) |
| Root analytic conductor: | \(5.74961\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 3^{24} \cdot 5^{12} \cdot 23^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.6730985802\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6730985802\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 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 | 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
−2.57507791264583602535918907254, −2.55522435632986980530060015792, −2.39802684251788955132224781395, −2.35282851102958357186560571714, −2.15725487853367412254855041578, −2.05936558929667199786509309101, −2.04507785988943521520266937616, −2.01051046746176026339970252574, −1.80628131994884161739779975848, −1.71806150941575230178112999895, −1.63567043620041701107223041886, −1.57064466168058140356831009471, −1.52938309788626654142120747812, −1.37556843612408292970145288129, −1.29299168939455527259032758138, −1.18765287134076512776041203395, −1.09143860345320873822987730507, −0.928430532343039732031306794517, −0.76587857142452346409197521986, −0.71839001199297752460081920170, −0.60066530396457327524693845423, −0.53310790546795136910185305106, −0.49889585221283479696855196283, −0.21568505222665742889237894147, −0.03752494361453562399073407931, 0.03752494361453562399073407931, 0.21568505222665742889237894147, 0.49889585221283479696855196283, 0.53310790546795136910185305106, 0.60066530396457327524693845423, 0.71839001199297752460081920170, 0.76587857142452346409197521986, 0.928430532343039732031306794517, 1.09143860345320873822987730507, 1.18765287134076512776041203395, 1.29299168939455527259032758138, 1.37556843612408292970145288129, 1.52938309788626654142120747812, 1.57064466168058140356831009471, 1.63567043620041701107223041886, 1.71806150941575230178112999895, 1.80628131994884161739779975848, 2.01051046746176026339970252574, 2.04507785988943521520266937616, 2.05936558929667199786509309101, 2.15725487853367412254855041578, 2.35282851102958357186560571714, 2.39802684251788955132224781395, 2.55522435632986980530060015792, 2.57507791264583602535918907254
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.