Dirichlet series
| L(s) = 1 | + 12·3-s − 12·5-s + 2·7-s + 78·9-s − 5·11-s + 6·13-s − 144·15-s + 9·17-s − 11·19-s + 24·21-s + 19·23-s + 78·25-s + 364·27-s + 8·29-s − 4·31-s − 60·33-s − 24·35-s + 5·37-s + 72·39-s − 9·41-s − 14·43-s − 936·45-s − 6·47-s + 108·51-s − 20·53-s + 60·55-s − 132·57-s + ⋯ |
| L(s) = 1 | + 6.92·3-s − 5.36·5-s + 0.755·7-s + 26·9-s − 1.50·11-s + 1.66·13-s − 37.1·15-s + 2.18·17-s − 2.52·19-s + 5.23·21-s + 3.96·23-s + 78/5·25-s + 70.0·27-s + 1.48·29-s − 0.718·31-s − 10.4·33-s − 4.05·35-s + 0.821·37-s + 11.5·39-s − 1.40·41-s − 2.13·43-s − 139.·45-s − 0.875·47-s + 15.1·51-s − 2.74·53-s + 8.09·55-s − 17.4·57-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 5^{12} \cdot 67^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 5^{12} \cdot 67^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 3^{12} \cdot 5^{12} \cdot 67^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.19684\times 10^{18}\) |
| Root analytic conductor: | \(5.66567\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 3^{12} \cdot 5^{12} \cdot 67^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(273.3160706\) |
| \(L(\frac12)\) | \(\approx\) | \(273.3160706\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 + T )^{12} \) | |
| 67 | \( 1 - 3 T - 9 T^{2} + 945 T^{3} - 6138 T^{4} - 52035 T^{5} + 480724 T^{6} - 52035 p T^{7} - 6138 p^{2} T^{8} + 945 p^{3} T^{9} - 9 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 7 | \( 1 - 2 T + 4 T^{2} - 48 T^{4} + 107 T^{5} + 396 T^{6} - 1161 T^{7} + 2332 T^{8} - 8784 T^{9} - 6359 T^{10} - 549 T^{11} + 84688 T^{12} - 549 p T^{13} - 6359 p^{2} T^{14} - 8784 p^{3} T^{15} + 2332 p^{4} T^{16} - 1161 p^{5} T^{17} + 396 p^{6} T^{18} + 107 p^{7} T^{19} - 48 p^{8} T^{20} + 4 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) |
| 11 | \( 1 + 5 T - 10 T^{2} - 39 T^{3} + 71 T^{4} - 535 T^{5} - 156 p T^{6} - 1332 T^{7} - 1883 T^{8} + 75951 T^{9} - 168035 T^{10} - 92937 T^{11} + 7017904 T^{12} - 92937 p T^{13} - 168035 p^{2} T^{14} + 75951 p^{3} T^{15} - 1883 p^{4} T^{16} - 1332 p^{5} T^{17} - 156 p^{7} T^{18} - 535 p^{7} T^{19} + 71 p^{8} T^{20} - 39 p^{9} T^{21} - 10 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{6} \) | |
| 17 | \( 1 - 9 T + 4 T^{2} + 355 T^{3} - 1746 T^{4} - 1694 T^{5} + 47899 T^{6} - 146091 T^{7} - 390577 T^{8} + 271049 p T^{9} - 10879892 T^{10} - 42482959 T^{11} + 376180152 T^{12} - 42482959 p T^{13} - 10879892 p^{2} T^{14} + 271049 p^{4} T^{15} - 390577 p^{4} T^{16} - 146091 p^{5} T^{17} + 47899 p^{6} T^{18} - 1694 p^{7} T^{19} - 1746 p^{8} T^{20} + 355 p^{9} T^{21} + 4 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 11 T + 33 T^{2} - 388 T^{4} - 4838 T^{5} - 28432 T^{6} - 111029 T^{7} - 333428 T^{8} - 54312 T^{9} + 4625736 T^{10} + 47081975 T^{11} + 298793580 T^{12} + 47081975 p T^{13} + 4625736 p^{2} T^{14} - 54312 p^{3} T^{15} - 333428 p^{4} T^{16} - 111029 p^{5} T^{17} - 28432 p^{6} T^{18} - 4838 p^{7} T^{19} - 388 p^{8} T^{20} + 33 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 19 T + 120 T^{2} - 95 T^{3} - 1424 T^{4} - 2326 T^{5} + 61689 T^{6} - 352863 T^{7} + 2083525 T^{8} - 6539565 T^{9} - 1493768 T^{10} - 115618269 T^{11} + 1361354230 T^{12} - 115618269 p T^{13} - 1493768 p^{2} T^{14} - 6539565 p^{3} T^{15} + 2083525 p^{4} T^{16} - 352863 p^{5} T^{17} + 61689 p^{6} T^{18} - 2326 p^{7} T^{19} - 1424 p^{8} T^{20} - 95 p^{9} T^{21} + 120 p^{10} T^{22} - 19 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 8 T + 32 T^{2} - 368 T^{3} + 1206 T^{4} - 1655 T^{5} + 89070 T^{6} - 339779 T^{7} + 1355490 T^{8} - 22458518 T^{9} + 53699035 T^{10} - 227990417 T^{11} + 3614850538 T^{12} - 227990417 p T^{13} + 53699035 p^{2} T^{14} - 22458518 p^{3} T^{15} + 1355490 p^{4} T^{16} - 339779 p^{5} T^{17} + 89070 p^{6} T^{18} - 1655 p^{7} T^{19} + 1206 p^{8} T^{20} - 368 p^{9} T^{21} + 32 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 4 T - 109 T^{2} - 606 T^{3} + 5492 T^{4} + 36266 T^{5} - 184734 T^{6} - 1057922 T^{7} + 6973790 T^{8} + 13628888 T^{9} - 332452362 T^{10} - 40779865 T^{11} + 12559806528 T^{12} - 40779865 p T^{13} - 332452362 p^{2} T^{14} + 13628888 p^{3} T^{15} + 6973790 p^{4} T^{16} - 1057922 p^{5} T^{17} - 184734 p^{6} T^{18} + 36266 p^{7} T^{19} + 5492 p^{8} T^{20} - 606 p^{9} T^{21} - 109 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 5 T - 35 T^{2} + 4 p T^{3} + 910 T^{4} + 4694 T^{5} + 48184 T^{6} - 753797 T^{7} - 2440934 T^{8} + 39099266 T^{9} + 52006098 T^{10} - 569825299 T^{11} - 258124762 T^{12} - 569825299 p T^{13} + 52006098 p^{2} T^{14} + 39099266 p^{3} T^{15} - 2440934 p^{4} T^{16} - 753797 p^{5} T^{17} + 48184 p^{6} T^{18} + 4694 p^{7} T^{19} + 910 p^{8} T^{20} + 4 p^{10} T^{21} - 35 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 9 T - 112 T^{2} - 853 T^{3} + 9803 T^{4} + 50521 T^{5} - 565952 T^{6} - 1758476 T^{7} + 24242199 T^{8} + 40372655 T^{9} - 791971767 T^{10} - 501901089 T^{11} + 26593085586 T^{12} - 501901089 p T^{13} - 791971767 p^{2} T^{14} + 40372655 p^{3} T^{15} + 24242199 p^{4} T^{16} - 1758476 p^{5} T^{17} - 565952 p^{6} T^{18} + 50521 p^{7} T^{19} + 9803 p^{8} T^{20} - 853 p^{9} T^{21} - 112 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( ( 1 + 7 T + 202 T^{2} + 1304 T^{3} + 19362 T^{4} + 102444 T^{5} + 1076954 T^{6} + 102444 p T^{7} + 19362 p^{2} T^{8} + 1304 p^{3} T^{9} + 202 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 + 6 T - 23 T^{2} - 1160 T^{3} - 6572 T^{4} + 16868 T^{5} + 616228 T^{6} + 3244950 T^{7} - 6853044 T^{8} - 219512470 T^{9} - 865976942 T^{10} + 3383470903 T^{11} + 68645946092 T^{12} + 3383470903 p T^{13} - 865976942 p^{2} T^{14} - 219512470 p^{3} T^{15} - 6853044 p^{4} T^{16} + 3244950 p^{5} T^{17} + 616228 p^{6} T^{18} + 16868 p^{7} T^{19} - 6572 p^{8} T^{20} - 1160 p^{9} T^{21} - 23 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( ( 1 + 10 T + 292 T^{2} + 2399 T^{3} + 37181 T^{4} + 241012 T^{5} + 2598106 T^{6} + 241012 p T^{7} + 37181 p^{2} T^{8} + 2399 p^{3} T^{9} + 292 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( ( 1 - 3 T + 70 T^{2} - 98 T^{3} + 1610 T^{4} - 9168 T^{5} + 82294 T^{6} - 9168 p T^{7} + 1610 p^{2} T^{8} - 98 p^{3} T^{9} + 70 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 61 | \( 1 - 27 T + 151 T^{2} + 1702 T^{3} - 6246 T^{4} - 320230 T^{5} + 2445536 T^{6} + 10370959 T^{7} - 96143610 T^{8} - 1233847964 T^{9} + 14003131220 T^{10} - 9804089745 T^{11} - 438692150218 T^{12} - 9804089745 p T^{13} + 14003131220 p^{2} T^{14} - 1233847964 p^{3} T^{15} - 96143610 p^{4} T^{16} + 10370959 p^{5} T^{17} + 2445536 p^{6} T^{18} - 320230 p^{7} T^{19} - 6246 p^{8} T^{20} + 1702 p^{9} T^{21} + 151 p^{10} T^{22} - 27 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 25 T + 66 T^{2} + 2427 T^{3} - 921 T^{4} - 287069 T^{5} - 349824 T^{6} + 22779568 T^{7} + 109032831 T^{8} - 1504395045 T^{9} - 10745043831 T^{10} + 19122129269 T^{11} + 1224386041720 T^{12} + 19122129269 p T^{13} - 10745043831 p^{2} T^{14} - 1504395045 p^{3} T^{15} + 109032831 p^{4} T^{16} + 22779568 p^{5} T^{17} - 349824 p^{6} T^{18} - 287069 p^{7} T^{19} - 921 p^{8} T^{20} + 2427 p^{9} T^{21} + 66 p^{10} T^{22} - 25 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 8 T - 13 T^{2} - 410 T^{3} - 9332 T^{4} - 91230 T^{5} + 69722 T^{6} + 6044840 T^{7} + 55975786 T^{8} + 340615830 T^{9} + 20999164 T^{10} - 25331655699 T^{11} - 349115977166 T^{12} - 25331655699 p T^{13} + 20999164 p^{2} T^{14} + 340615830 p^{3} T^{15} + 55975786 p^{4} T^{16} + 6044840 p^{5} T^{17} + 69722 p^{6} T^{18} - 91230 p^{7} T^{19} - 9332 p^{8} T^{20} - 410 p^{9} T^{21} - 13 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 2 T - 254 T^{2} + 354 T^{3} + 30644 T^{4} - 104573 T^{5} - 2255531 T^{6} + 2220835 T^{7} + 156777095 T^{8} + 785447464 T^{9} - 19907349612 T^{10} - 43426772841 T^{11} + 2128421622846 T^{12} - 43426772841 p T^{13} - 19907349612 p^{2} T^{14} + 785447464 p^{3} T^{15} + 156777095 p^{4} T^{16} + 2220835 p^{5} T^{17} - 2255531 p^{6} T^{18} - 104573 p^{7} T^{19} + 30644 p^{8} T^{20} + 354 p^{9} T^{21} - 254 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 20 T + 131 T^{2} - 1714 T^{3} + 14638 T^{4} + 93416 T^{5} - 900524 T^{6} + 12623066 T^{7} - 198497842 T^{8} + 47393114 T^{9} + 2206202642 T^{10} - 56276030963 T^{11} + 1636885001364 T^{12} - 56276030963 p T^{13} + 2206202642 p^{2} T^{14} + 47393114 p^{3} T^{15} - 198497842 p^{4} T^{16} + 12623066 p^{5} T^{17} - 900524 p^{6} T^{18} + 93416 p^{7} T^{19} + 14638 p^{8} T^{20} - 1714 p^{9} T^{21} + 131 p^{10} T^{22} - 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( ( 1 - 6 T + 278 T^{2} - 1201 T^{3} + 41023 T^{4} - 169406 T^{5} + 4354474 T^{6} - 169406 p T^{7} + 41023 p^{2} T^{8} - 1201 p^{3} T^{9} + 278 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 + T - 74 T^{2} - 3085 T^{3} + 512 T^{4} + 161294 T^{5} + 5154103 T^{6} + 4425829 T^{7} - 201336043 T^{8} - 7008527843 T^{9} - 13501932242 T^{10} + 195913282487 T^{11} + 7566580882136 T^{12} + 195913282487 p T^{13} - 13501932242 p^{2} T^{14} - 7008527843 p^{3} T^{15} - 201336043 p^{4} T^{16} + 4425829 p^{5} T^{17} + 5154103 p^{6} T^{18} + 161294 p^{7} T^{19} + 512 p^{8} T^{20} - 3085 p^{9} T^{21} - 74 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.68829723742412189884675165932, −2.61809757908892663690688341140, −2.57950061268242404232005944878, −2.23073447934802708067028940135, −2.14783630828241307783737962103, −2.13440519239245292889334571323, −2.06817664326389638220960801811, −2.05604471370584324771719207221, −2.05187266393029350505778134849, −1.92029377951126167351488275453, −1.70427064142130569852306902862, −1.51340084635741512286458989984, −1.49001197364102746795453468155, −1.43381447325651215886322517179, −1.40666229629579026213972900674, −1.23126512197581024446080974818, −1.21368651235110826426924833502, −1.07282434126902277154638842149, −0.935880450309957144400092918412, −0.69511195426649139188586618242, −0.64699454257182999173659111662, −0.62368182913077868489132600036, −0.58806087892235202257402338256, −0.27252326291244688547156151484, −0.20898437402749892898902862000, 0.20898437402749892898902862000, 0.27252326291244688547156151484, 0.58806087892235202257402338256, 0.62368182913077868489132600036, 0.64699454257182999173659111662, 0.69511195426649139188586618242, 0.935880450309957144400092918412, 1.07282434126902277154638842149, 1.21368651235110826426924833502, 1.23126512197581024446080974818, 1.40666229629579026213972900674, 1.43381447325651215886322517179, 1.49001197364102746795453468155, 1.51340084635741512286458989984, 1.70427064142130569852306902862, 1.92029377951126167351488275453, 2.05187266393029350505778134849, 2.05604471370584324771719207221, 2.06817664326389638220960801811, 2.13440519239245292889334571323, 2.14783630828241307783737962103, 2.23073447934802708067028940135, 2.57950061268242404232005944878, 2.61809757908892663690688341140, 2.68829723742412189884675165932
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.