Dirichlet series
| L(s) = 1 | + 3·4-s + 6·7-s + 6·13-s − 3·16-s + 24·19-s − 24·25-s + 18·28-s − 12·31-s + 6·37-s − 12·43-s + 39·49-s + 18·52-s − 48·61-s − 23·64-s − 66·67-s − 12·73-s + 72·76-s + 42·79-s + 36·91-s + 60·97-s − 72·100-s − 48·103-s + 24·109-s − 18·112-s + 39·121-s − 36·124-s + 127-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 2.26·7-s + 1.66·13-s − 3/4·16-s + 5.50·19-s − 4.79·25-s + 3.40·28-s − 2.15·31-s + 0.986·37-s − 1.82·43-s + 39/7·49-s + 2.49·52-s − 6.14·61-s − 2.87·64-s − 8.06·67-s − 1.40·73-s + 8.25·76-s + 4.72·79-s + 3.77·91-s + 6.09·97-s − 7.19·100-s − 4.72·103-s + 2.29·109-s − 1.70·112-s + 3.54·121-s − 3.23·124-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 | \( 1 - 3 T^{2} + 3 p^{2} T^{4} - 11 p T^{6} + 9 p^{3} T^{8} - 9 p^{4} T^{10} + 393 T^{12} - 9 p^{6} T^{14} + 9 p^{7} T^{16} - 11 p^{7} T^{18} + 3 p^{10} T^{20} - 3 p^{10} T^{22} + p^{12} T^{24} \) |
| 5 | \( 1 + 24 T^{2} + 354 T^{4} + 3731 T^{6} + 30717 T^{8} + 204813 T^{10} + 1124529 T^{12} + 204813 p^{2} T^{14} + 30717 p^{4} T^{16} + 3731 p^{6} T^{18} + 354 p^{8} T^{20} + 24 p^{10} T^{22} + p^{12} T^{24} \) | |
| 7 | \( ( 1 - 3 T - 6 T^{2} + 50 T^{3} - 99 T^{4} - 207 T^{5} + 1401 T^{6} - 207 p T^{7} - 99 p^{2} T^{8} + 50 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 11 | \( 1 - 39 T^{2} + 714 T^{4} - 7708 T^{6} + 63189 T^{8} - 610893 T^{10} + 7111041 T^{12} - 610893 p^{2} T^{14} + 63189 p^{4} T^{16} - 7708 p^{6} T^{18} + 714 p^{8} T^{20} - 39 p^{10} T^{22} + p^{12} T^{24} \) | |
| 13 | \( ( 1 - 3 T - 6 T^{2} + 50 T^{3} - 135 T^{4} - 189 T^{5} + 3273 T^{6} - 189 p T^{7} - 135 p^{2} T^{8} + 50 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 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 | \( 1 + 33 T^{2} + 2334 T^{4} + 47048 T^{6} + 2039913 T^{8} + 27989451 T^{10} + 1148943705 T^{12} + 27989451 p^{2} T^{14} + 2039913 p^{4} T^{16} + 47048 p^{6} T^{18} + 2334 p^{8} T^{20} + 33 p^{10} T^{22} + p^{12} T^{24} \) | |
| 29 | \( 1 - 3 T^{2} - 924 T^{4} + 32792 T^{6} + 252018 T^{8} - 13922199 T^{10} + 371650839 T^{12} - 13922199 p^{2} T^{14} + 252018 p^{4} T^{16} + 32792 p^{6} T^{18} - 924 p^{8} T^{20} - 3 p^{10} T^{22} + p^{12} T^{24} \) | |
| 31 | \( ( 1 + 6 T + 84 T^{2} + 644 T^{3} + 5202 T^{4} + 30042 T^{5} + 205341 T^{6} + 30042 p T^{7} + 5202 p^{2} T^{8} + 644 p^{3} T^{9} + 84 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 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 | \( 1 + 15 T^{2} - 2913 T^{4} + 6107 T^{6} + 4774446 T^{8} + 7741476 T^{10} - 3789535503 T^{12} + 7741476 p^{2} T^{14} + 4774446 p^{4} T^{16} + 6107 p^{6} T^{18} - 2913 p^{8} T^{20} + 15 p^{10} T^{22} + p^{12} T^{24} \) | |
| 43 | \( ( 1 + 6 T + 12 T^{2} + 194 T^{3} - 936 T^{4} - 19188 T^{5} - 81507 T^{6} - 19188 p T^{7} - 936 p^{2} T^{8} + 194 p^{3} T^{9} + 12 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 - 3 T^{2} + 1092 T^{4} - 122188 T^{6} - 3809682 T^{8} + 118839681 T^{10} + 201019479 T^{12} + 118839681 p^{2} T^{14} - 3809682 p^{4} T^{16} - 122188 p^{6} T^{18} + 1092 p^{8} T^{20} - 3 p^{10} T^{22} + p^{12} T^{24} \) | |
| 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 | \( 1 + 330 T^{2} + 60555 T^{4} + 7676267 T^{6} + 749272185 T^{8} + 58783546035 T^{10} + 3809084348598 T^{12} + 58783546035 p^{2} T^{14} + 749272185 p^{4} T^{16} + 7676267 p^{6} T^{18} + 60555 p^{8} T^{20} + 330 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( ( 1 + 24 T + 336 T^{2} + 3614 T^{3} + 37692 T^{4} + 350622 T^{5} + 2940891 T^{6} + 350622 p T^{7} + 37692 p^{2} T^{8} + 3614 p^{3} T^{9} + 336 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( ( 1 + 33 T + 471 T^{2} + 4055 T^{3} + 35100 T^{4} + 421686 T^{5} + 4248357 T^{6} + 421686 p T^{7} + 35100 p^{2} T^{8} + 4055 p^{3} T^{9} + 471 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 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 | \( ( 1 - 21 T + 228 T^{2} - 1066 T^{3} + 8001 T^{4} - 158805 T^{5} + 2124699 T^{6} - 158805 p T^{7} + 8001 p^{2} T^{8} - 1066 p^{3} T^{9} + 228 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 + 285 T^{2} + 46506 T^{4} + 5672048 T^{6} + 639517149 T^{8} + 64854517383 T^{10} + 5837408562489 T^{12} + 64854517383 p^{2} T^{14} + 639517149 p^{4} T^{16} + 5672048 p^{6} T^{18} + 46506 p^{8} T^{20} + 285 p^{10} T^{22} + p^{12} T^{24} \) | |
| 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 | \( ( 1 - 30 T + 417 T^{2} - 2947 T^{3} - 2511 T^{4} + 410049 T^{5} - 5825766 T^{6} + 410049 p T^{7} - 2511 p^{2} T^{8} - 2947 p^{3} T^{9} + 417 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
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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
−3.35396172975016057350882524017, −3.25652511580256522864492522189, −3.23049362361987352917731275373, −3.06847818781508330570251347320, −3.00686719217122568772108687612, −2.93225009246355363228501668370, −2.87006991808792782396662685596, −2.60181095069118374609292220189, −2.54861434003666281299829179126, −2.34869775546885080896416483899, −2.23383950060896805318108107847, −2.16737279158926257665121310871, −2.05309941515337270880963107204, −1.82673490893327176674171682593, −1.78278160196351368155497694082, −1.75889105885540921563889366668, −1.68886000283347128355313421753, −1.57945028510744542175089542199, −1.32651575353131340359931315930, −1.28031084662150670435868966997, −1.11119735248990956893665897811, −1.07841329206853991397378572291, −0.68922420167391306017141400706, −0.39157925542589686518637285951, −0.26608268006152217468197900714, 0.26608268006152217468197900714, 0.39157925542589686518637285951, 0.68922420167391306017141400706, 1.07841329206853991397378572291, 1.11119735248990956893665897811, 1.28031084662150670435868966997, 1.32651575353131340359931315930, 1.57945028510744542175089542199, 1.68886000283347128355313421753, 1.75889105885540921563889366668, 1.78278160196351368155497694082, 1.82673490893327176674171682593, 2.05309941515337270880963107204, 2.16737279158926257665121310871, 2.23383950060896805318108107847, 2.34869775546885080896416483899, 2.54861434003666281299829179126, 2.60181095069118374609292220189, 2.87006991808792782396662685596, 2.93225009246355363228501668370, 3.00686719217122568772108687612, 3.06847818781508330570251347320, 3.23049362361987352917731275373, 3.25652511580256522864492522189, 3.35396172975016057350882524017
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.