Dirichlet series
| L(s) = 1 | + 3·4-s − 12·7-s − 30·13-s + 15·16-s + 24·19-s + 21·25-s − 36·28-s + 24·31-s + 6·37-s − 12·43-s + 66·49-s − 90·52-s + 60·61-s + 31·64-s + 78·67-s − 12·73-s + 72·76-s + 6·79-s + 360·91-s + 6·97-s + 63·100-s + 78·103-s + 24·109-s − 180·112-s + 3·121-s + 72·124-s + 127-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 4.53·7-s − 8.32·13-s + 15/4·16-s + 5.50·19-s + 21/5·25-s − 6.80·28-s + 4.31·31-s + 0.986·37-s − 1.82·43-s + 66/7·49-s − 12.4·52-s + 7.68·61-s + 31/8·64-s + 9.52·67-s − 1.40·73-s + 8.25·76-s + 0.675·79-s + 37.7·91-s + 0.609·97-s + 6.29·100-s + 7.68·103-s + 2.29·109-s − 17.0·112-s + 3/11·121-s + 6.46·124-s + 0.0887·127-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{72}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{72}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{72}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.51375\times 10^{9}\) |
| Root analytic conductor: | \(2.41269\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{72} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(12.77406038\) |
| \(L(\frac12)\) | \(\approx\) | \(12.77406038\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 3 T^{2} - 3 p T^{4} + p^{5} T^{6} - 27 T^{8} - 81 T^{10} + 285 T^{12} - 81 p^{2} T^{14} - 27 p^{4} T^{16} + p^{11} T^{18} - 3 p^{9} T^{20} - 3 p^{10} T^{22} + p^{12} T^{24} \) |
| 5 | \( 1 - 21 T^{2} + 219 T^{4} - 1669 T^{6} + 11178 T^{8} - 67284 T^{10} + 357729 T^{12} - 67284 p^{2} T^{14} + 11178 p^{4} T^{16} - 1669 p^{6} T^{18} + 219 p^{8} T^{20} - 21 p^{10} T^{22} + p^{12} T^{24} \) | |
| 7 | \( ( 1 + 6 T + 3 p T^{2} + 11 p T^{3} + 207 T^{4} + 531 T^{5} + 1590 T^{6} + 531 p T^{7} + 207 p^{2} T^{8} + 11 p^{4} T^{9} + 3 p^{5} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 11 | \( 1 - 3 T^{2} - 150 T^{4} + 824 T^{6} - 153 p T^{8} - 3771 p T^{10} + 1596345 T^{12} - 3771 p^{3} T^{14} - 153 p^{5} T^{16} + 824 p^{6} T^{18} - 150 p^{8} T^{20} - 3 p^{10} T^{22} + p^{12} T^{24} \) | |
| 13 | \( ( 1 + 15 T + 120 T^{2} + 698 T^{3} + 3303 T^{4} + 13401 T^{5} + 49605 T^{6} + 13401 p T^{7} + 3303 p^{2} T^{8} + 698 p^{3} T^{9} + 120 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 17 | \( 1 - 21 T^{2} - 141 T^{4} + 3236 T^{6} + 8217 T^{8} + 812097 T^{10} - 26711418 T^{12} + 812097 p^{2} T^{14} + 8217 p^{4} T^{16} + 3236 p^{6} T^{18} - 141 p^{8} T^{20} - 21 p^{10} T^{22} + p^{12} T^{24} \) | |
| 19 | \( ( 1 - 12 T + 48 T^{2} - 202 T^{3} + 1692 T^{4} - 324 p T^{5} + 645 p T^{6} - 324 p^{2} T^{7} + 1692 p^{2} T^{8} - 202 p^{3} T^{9} + 48 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 23 | \( 1 - 111 T^{2} + 4566 T^{4} - 68944 T^{6} - 380871 T^{8} + 19499067 T^{10} - 239021103 T^{12} + 19499067 p^{2} T^{14} - 380871 p^{4} T^{16} - 68944 p^{6} T^{18} + 4566 p^{8} T^{20} - 111 p^{10} T^{22} + p^{12} T^{24} \) | |
| 29 | \( 1 + 24 T^{2} + 1110 T^{4} + 52043 T^{6} + 1259505 T^{8} + 45123381 T^{10} + 1264081353 T^{12} + 45123381 p^{2} T^{14} + 1259505 p^{4} T^{16} + 52043 p^{6} T^{18} + 1110 p^{8} T^{20} + 24 p^{10} T^{22} + p^{12} T^{24} \) | |
| 31 | \( ( 1 - 12 T + 30 T^{2} + 428 T^{3} - 2970 T^{4} - 5238 T^{5} + 118293 T^{6} - 5238 p T^{7} - 2970 p^{2} T^{8} + 428 p^{3} T^{9} + 30 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 37 | \( ( 1 - 3 T - 96 T^{2} + 95 T^{3} + 6525 T^{4} - 2322 T^{5} - 276915 T^{6} - 2322 p T^{7} + 6525 p^{2} T^{8} + 95 p^{3} T^{9} - 96 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 + 69 T^{2} + 6708 T^{4} + 404276 T^{6} + 519678 p T^{8} + 26727957 p T^{10} + 43305921951 T^{12} + 26727957 p^{3} T^{14} + 519678 p^{5} T^{16} + 404276 p^{6} T^{18} + 6708 p^{8} T^{20} + 69 p^{10} T^{22} + p^{12} T^{24} \) | |
| 43 | \( ( 1 + 6 T + 48 T^{2} + 410 T^{3} + 3852 T^{4} + 23670 T^{5} + 136761 T^{6} + 23670 p T^{7} + 3852 p^{2} T^{8} + 410 p^{3} T^{9} + 48 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 - 12 T^{2} - 1266 T^{4} - 103153 T^{6} + 2848257 T^{8} + 113611329 T^{10} + 2736443409 T^{12} + 113611329 p^{2} T^{14} + 2848257 p^{4} T^{16} - 103153 p^{6} T^{18} - 1266 p^{8} T^{20} - 12 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( ( 1 + 210 T^{2} + 21831 T^{4} + 1416508 T^{6} + 21831 p^{2} T^{8} + 210 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 - 255 T^{2} + 35850 T^{4} - 3942508 T^{6} + 358774245 T^{8} - 26868726405 T^{10} + 1703627519073 T^{12} - 26868726405 p^{2} T^{14} + 358774245 p^{4} T^{16} - 3942508 p^{6} T^{18} + 35850 p^{8} T^{20} - 255 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( ( 1 - 30 T + 372 T^{2} - 2002 T^{3} - 5904 T^{4} + 226944 T^{5} - 2382213 T^{6} + 226944 p T^{7} - 5904 p^{2} T^{8} - 2002 p^{3} T^{9} + 372 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( ( 1 - 39 T + 660 T^{2} - 5476 T^{3} + 1494 T^{4} + 528939 T^{5} - 6511575 T^{6} + 528939 p T^{7} + 1494 p^{2} T^{8} - 5476 p^{3} T^{9} + 660 p^{4} T^{10} - 39 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( 1 - 354 T^{2} + 68853 T^{4} - 9662278 T^{6} + 1075804938 T^{8} - 98985561066 T^{10} + 7655275368861 T^{12} - 98985561066 p^{2} T^{14} + 1075804938 p^{4} T^{16} - 9662278 p^{6} T^{18} + 68853 p^{8} T^{20} - 354 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( ( 1 + 6 T - 114 T^{2} - 1030 T^{3} + 5760 T^{4} + 45324 T^{5} - 125085 T^{6} + 45324 p T^{7} + 5760 p^{2} T^{8} - 1030 p^{3} T^{9} - 114 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 79 | \( ( 1 - 3 T - 132 T^{2} + 1040 T^{3} - 2709 T^{4} - 61083 T^{5} + 1139253 T^{6} - 61083 p T^{7} - 2709 p^{2} T^{8} + 1040 p^{3} T^{9} - 132 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 - 363 T^{2} + 54534 T^{4} - 3717580 T^{6} - 58850559 T^{8} + 41822364519 T^{10} - 4845832961079 T^{12} + 41822364519 p^{2} T^{14} - 58850559 p^{4} T^{16} - 3717580 p^{6} T^{18} + 54534 p^{8} T^{20} - 363 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( 1 - 147 T^{2} + 24420 T^{4} - 2440777 T^{6} + 239936211 T^{8} - 19780581708 T^{10} + 1611896402121 T^{12} - 19780581708 p^{2} T^{14} + 239936211 p^{4} T^{16} - 2440777 p^{6} T^{18} + 24420 p^{8} T^{20} - 147 p^{10} T^{22} + p^{12} T^{24} \) | |
| 97 | \( ( 1 - 3 T + 48 T^{2} + 428 T^{3} + 387 T^{4} - 38925 T^{5} + 1101489 T^{6} - 38925 p T^{7} + 387 p^{2} T^{8} + 428 p^{3} T^{9} + 48 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.39014910536183546339529289560, −3.25451310757297421444214225362, −3.11320807913761258177775993431, −3.09258351734180223961451967150, −2.98226978638077791889148029052, −2.86533934677251544714233457288, −2.72334442719390605343825038448, −2.71350252133016895899776913971, −2.67015658190047962158852866762, −2.45761220700379642066382227535, −2.42153069496494517170434371515, −2.24214692403022444161756114451, −2.18353243566683180971786423464, −2.17268295707162211961773773007, −2.15171588119276474888451464654, −1.89084738641495041672478950555, −1.71993521062899748954086536967, −1.36235173811013166272104627780, −0.966454563550970335119922011037, −0.927255857591062732309210297914, −0.856851872084023224857245447313, −0.77307404679102834944936487064, −0.75673667439372483522270822151, −0.59359622012803051811145355827, −0.37255505142840396154959470318, 0.37255505142840396154959470318, 0.59359622012803051811145355827, 0.75673667439372483522270822151, 0.77307404679102834944936487064, 0.856851872084023224857245447313, 0.927255857591062732309210297914, 0.966454563550970335119922011037, 1.36235173811013166272104627780, 1.71993521062899748954086536967, 1.89084738641495041672478950555, 2.15171588119276474888451464654, 2.17268295707162211961773773007, 2.18353243566683180971786423464, 2.24214692403022444161756114451, 2.42153069496494517170434371515, 2.45761220700379642066382227535, 2.67015658190047962158852866762, 2.71350252133016895899776913971, 2.72334442719390605343825038448, 2.86533934677251544714233457288, 2.98226978638077791889148029052, 3.09258351734180223961451967150, 3.11320807913761258177775993431, 3.25451310757297421444214225362, 3.39014910536183546339529289560
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.