Dirichlet series
| L(s) = 1 | + 12·3-s + 12·5-s + 78·9-s + 144·15-s + 4·17-s + 78·25-s + 364·27-s − 12·31-s + 936·45-s + 28·49-s + 48·51-s + 52·59-s − 56·61-s − 32·67-s + 8·71-s + 32·73-s + 936·75-s − 28·79-s + 1.36e3·81-s + 48·85-s − 144·93-s + 24·101-s + 32·103-s + 56·107-s + 64·121-s + 364·125-s + 127-s + ⋯ |
| L(s) = 1 | + 6.92·3-s + 5.36·5-s + 26·9-s + 37.1·15-s + 0.970·17-s + 78/5·25-s + 70.0·27-s − 2.15·31-s + 139.·45-s + 4·49-s + 6.72·51-s + 6.76·59-s − 7.17·61-s − 3.90·67-s + 0.949·71-s + 3.74·73-s + 108.·75-s − 3.15·79-s + 151.·81-s + 5.20·85-s − 14.9·93-s + 2.38·101-s + 3.15·103-s + 5.41·107-s + 5.81·121-s + 32.5·125-s + 0.0887·127-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 5^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 5^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 3^{12} \cdot 5^{12} \cdot 19^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.43129\times 10^{18}\) |
| Root analytic conductor: | \(6.03421\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{48} \cdot 3^{12} \cdot 5^{12} \cdot 19^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(19535.37784\) |
| \(L(\frac12)\) | \(\approx\) | \(19535.37784\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 - T )^{12} \) | |
| 5 | \( ( 1 - T )^{12} \) | |
| 19 | \( 1 + 26 T^{2} - 64 T^{3} + 471 T^{4} - 2368 T^{5} + 4652 T^{6} - 2368 p T^{7} + 471 p^{2} T^{8} - 64 p^{3} T^{9} + 26 p^{4} T^{10} + p^{6} T^{12} \) | |
| good | 7 | \( 1 - 4 p T^{2} + 66 p T^{4} - 5788 T^{6} + 58659 T^{8} - 506888 T^{10} + 3808092 T^{12} - 506888 p^{2} T^{14} + 58659 p^{4} T^{16} - 5788 p^{6} T^{18} + 66 p^{9} T^{20} - 4 p^{11} T^{22} + p^{12} T^{24} \) |
| 11 | \( 1 - 64 T^{2} + 194 p T^{4} - 4464 p T^{6} + 876627 T^{8} - 12769216 T^{10} + 154124140 T^{12} - 12769216 p^{2} T^{14} + 876627 p^{4} T^{16} - 4464 p^{7} T^{18} + 194 p^{9} T^{20} - 64 p^{10} T^{22} + p^{12} T^{24} \) | |
| 13 | \( 1 - 40 T^{2} + 966 T^{4} - 14056 T^{6} + 142227 T^{8} - 855872 T^{10} + 6120396 T^{12} - 855872 p^{2} T^{14} + 142227 p^{4} T^{16} - 14056 p^{6} T^{18} + 966 p^{8} T^{20} - 40 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( ( 1 - 2 T + 38 T^{2} - 66 T^{3} + 803 T^{4} - 980 T^{5} + 12572 T^{6} - 980 p T^{7} + 803 p^{2} T^{8} - 66 p^{3} T^{9} + 38 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 23 | \( 1 - 76 T^{2} + 3658 T^{4} - 129564 T^{6} + 3585471 T^{8} - 3794216 p T^{10} + 2016083884 T^{12} - 3794216 p^{3} T^{14} + 3585471 p^{4} T^{16} - 129564 p^{6} T^{18} + 3658 p^{8} T^{20} - 76 p^{10} T^{22} + p^{12} T^{24} \) | |
| 29 | \( 1 - 8 p T^{2} + 878 p T^{4} - 1777704 T^{6} + 89791731 T^{8} - 3532351936 T^{10} + 112869552172 T^{12} - 3532351936 p^{2} T^{14} + 89791731 p^{4} T^{16} - 1777704 p^{6} T^{18} + 878 p^{9} T^{20} - 8 p^{11} T^{22} + p^{12} T^{24} \) | |
| 31 | \( ( 1 + 6 T + 134 T^{2} + 674 T^{3} + 8691 T^{4} + 36020 T^{5} + 337484 T^{6} + 36020 p T^{7} + 8691 p^{2} T^{8} + 674 p^{3} T^{9} + 134 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 37 | \( 1 - 88 T^{2} + 6966 T^{4} - 424840 T^{6} + 22510035 T^{8} - 1008193712 T^{10} + 39632581164 T^{12} - 1008193712 p^{2} T^{14} + 22510035 p^{4} T^{16} - 424840 p^{6} T^{18} + 6966 p^{8} T^{20} - 88 p^{10} T^{22} + p^{12} T^{24} \) | |
| 41 | \( 1 - 284 T^{2} + 41118 T^{4} - 3993452 T^{6} + 289087587 T^{8} - 16408240408 T^{10} + 748005955452 T^{12} - 16408240408 p^{2} T^{14} + 289087587 p^{4} T^{16} - 3993452 p^{6} T^{18} + 41118 p^{8} T^{20} - 284 p^{10} T^{22} + p^{12} T^{24} \) | |
| 43 | \( 1 - 112 T^{2} + 10902 T^{4} - 706960 T^{6} + 39797523 T^{8} - 1952687312 T^{10} + 85963943532 T^{12} - 1952687312 p^{2} T^{14} + 39797523 p^{4} T^{16} - 706960 p^{6} T^{18} + 10902 p^{8} T^{20} - 112 p^{10} T^{22} + p^{12} T^{24} \) | |
| 47 | \( 1 - 380 T^{2} + 69930 T^{4} - 8342156 T^{6} + 724428447 T^{8} - 48406756792 T^{10} + 2552197195116 T^{12} - 48406756792 p^{2} T^{14} + 724428447 p^{4} T^{16} - 8342156 p^{6} T^{18} + 69930 p^{8} T^{20} - 380 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( 1 - 364 T^{2} + 68218 T^{4} - 8607516 T^{6} + 810449967 T^{8} - 59748149848 T^{10} + 3527294199052 T^{12} - 59748149848 p^{2} T^{14} + 810449967 p^{4} T^{16} - 8607516 p^{6} T^{18} + 68218 p^{8} T^{20} - 364 p^{10} T^{22} + p^{12} T^{24} \) | |
| 59 | \( ( 1 - 26 T + 426 T^{2} - 5110 T^{3} + 52087 T^{4} - 451748 T^{5} + 3628076 T^{6} - 451748 p T^{7} + 52087 p^{2} T^{8} - 5110 p^{3} T^{9} + 426 p^{4} T^{10} - 26 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 61 | \( ( 1 + 28 T + 534 T^{2} + 7468 T^{3} + 86523 T^{4} + 13736 p T^{5} + 7078476 T^{6} + 13736 p^{2} T^{7} + 86523 p^{2} T^{8} + 7468 p^{3} T^{9} + 534 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( ( 1 + 16 T + 282 T^{2} + 2920 T^{3} + 33663 T^{4} + 303512 T^{5} + 2786268 T^{6} + 303512 p T^{7} + 33663 p^{2} T^{8} + 2920 p^{3} T^{9} + 282 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( ( 1 - 4 T + 330 T^{2} - 980 T^{3} + 49975 T^{4} - 117328 T^{5} + 4494428 T^{6} - 117328 p T^{7} + 49975 p^{2} T^{8} - 980 p^{3} T^{9} + 330 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 73 | \( ( 1 - 16 T + 394 T^{2} - 3768 T^{3} + 54471 T^{4} - 373912 T^{5} + 4520764 T^{6} - 373912 p T^{7} + 54471 p^{2} T^{8} - 3768 p^{3} T^{9} + 394 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 79 | \( ( 1 + 14 T + 138 T^{2} + 794 T^{3} + 8943 T^{4} + 14476 T^{5} + 53964 T^{6} + 14476 p T^{7} + 8943 p^{2} T^{8} + 794 p^{3} T^{9} + 138 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 - 616 T^{2} + 194482 T^{4} - 40827432 T^{6} + 6291690159 T^{8} - 745855504816 T^{10} + 69562353246940 T^{12} - 745855504816 p^{2} T^{14} + 6291690159 p^{4} T^{16} - 40827432 p^{6} T^{18} + 194482 p^{8} T^{20} - 616 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( 1 - 604 T^{2} + 157342 T^{4} - 21524796 T^{6} + 1310341923 T^{8} + 40008531704 T^{10} - 11522524660868 T^{12} + 40008531704 p^{2} T^{14} + 1310341923 p^{4} T^{16} - 21524796 p^{6} T^{18} + 157342 p^{8} T^{20} - 604 p^{10} T^{22} + p^{12} T^{24} \) | |
| 97 | \( 1 - 664 T^{2} + 229302 T^{4} - 53558632 T^{6} + 9319581603 T^{8} - 1265176645904 T^{10} + 136977060412524 T^{12} - 1265176645904 p^{2} T^{14} + 9319581603 p^{4} T^{16} - 53558632 p^{6} T^{18} + 229302 p^{8} T^{20} - 664 p^{10} T^{22} + p^{12} T^{24} \) | |
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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
−2.50037576391664910341053757270, −2.24401920465956172736261466822, −2.23005330504443654257071007524, −2.19190780554518216611595382977, −2.13879139194613047259927194601, −2.12720914766792318351834810417, −2.10211499043116180198414601239, −1.95306712868375394459204726142, −1.94264308656866910591113843198, −1.90800418806490057072467061592, −1.88351526168905699059237923634, −1.60719624337733606535935343754, −1.56944844115943200835952621447, −1.51190529404512094523002961167, −1.48031241635663842769156391165, −1.43003570956903891449096492048, −1.20173725167901652543687023370, −1.17877005594296514101043141272, −1.00127510323370610131652399167, −0.888499468461333493772975611178, −0.68191853862854572113044407504, −0.64528560512640388277834728070, −0.63382430360681609956808479713, −0.63143365513192384401251120722, −0.28758736628474856367518198037, 0.28758736628474856367518198037, 0.63143365513192384401251120722, 0.63382430360681609956808479713, 0.64528560512640388277834728070, 0.68191853862854572113044407504, 0.888499468461333493772975611178, 1.00127510323370610131652399167, 1.17877005594296514101043141272, 1.20173725167901652543687023370, 1.43003570956903891449096492048, 1.48031241635663842769156391165, 1.51190529404512094523002961167, 1.56944844115943200835952621447, 1.60719624337733606535935343754, 1.88351526168905699059237923634, 1.90800418806490057072467061592, 1.94264308656866910591113843198, 1.95306712868375394459204726142, 2.10211499043116180198414601239, 2.12720914766792318351834810417, 2.13879139194613047259927194601, 2.19190780554518216611595382977, 2.23005330504443654257071007524, 2.24401920465956172736261466822, 2.50037576391664910341053757270
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.