Dirichlet series
| L(s) = 1 | + 3·4-s + 18·11-s − 8·13-s + 3·16-s − 4·17-s + 12·19-s + 6·23-s + 18·25-s + 10·29-s − 6·37-s + 24·41-s + 26·43-s + 54·44-s + 3·49-s − 24·52-s − 36·53-s − 6·59-s − 28·61-s − 2·64-s − 42·67-s − 12·68-s − 48·71-s + 36·76-s + 44·79-s − 12·89-s + 18·92-s + 60·97-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 5.42·11-s − 2.21·13-s + 3/4·16-s − 0.970·17-s + 2.75·19-s + 1.25·23-s + 18/5·25-s + 1.85·29-s − 0.986·37-s + 3.74·41-s + 3.96·43-s + 8.14·44-s + 3/7·49-s − 3.32·52-s − 4.94·53-s − 0.781·59-s − 3.58·61-s − 1/4·64-s − 5.13·67-s − 1.45·68-s − 5.69·71-s + 4.12·76-s + 4.95·79-s − 1.27·89-s + 1.87·92-s + 6.09·97-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{24} \cdot 7^{12} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.50664\times 10^{13}\) |
| Root analytic conductor: | \(3.61655\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{24} \cdot 7^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(36.26786651\) |
| \(L(\frac12)\) | \(\approx\) | \(36.26786651\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{3} \) |
| 3 | \( 1 \) | |
| 7 | \( ( 1 - T^{2} + T^{4} )^{3} \) | |
| 13 | \( 1 + 8 T + 15 T^{2} - 32 T^{3} + 32 T^{4} + 1488 T^{5} + 7393 T^{6} + 1488 p T^{7} + 32 p^{2} T^{8} - 32 p^{3} T^{9} + 15 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 5 | \( 1 - 18 T^{2} + 193 T^{4} - 1678 T^{6} + 12542 T^{8} - 77234 T^{10} + 409781 T^{12} - 77234 p^{2} T^{14} + 12542 p^{4} T^{16} - 1678 p^{6} T^{18} + 193 p^{8} T^{20} - 18 p^{10} T^{22} + p^{12} T^{24} \) |
| 11 | \( 1 - 18 T + 184 T^{2} - 1368 T^{3} + 8069 T^{4} - 39396 T^{5} + 162968 T^{6} - 577878 T^{7} + 1764242 T^{8} - 4642542 T^{9} + 10608360 T^{10} - 2059908 p T^{11} + 5412543 p T^{12} - 2059908 p^{2} T^{13} + 10608360 p^{2} T^{14} - 4642542 p^{3} T^{15} + 1764242 p^{4} T^{16} - 577878 p^{5} T^{17} + 162968 p^{6} T^{18} - 39396 p^{7} T^{19} + 8069 p^{8} T^{20} - 1368 p^{9} T^{21} + 184 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 4 T - 47 T^{2} - 92 T^{3} + 1422 T^{4} - 244 T^{5} - 29817 T^{6} + 52188 T^{7} + 486272 T^{8} - 966556 T^{9} - 5402563 T^{10} + 7740196 T^{11} + 69577228 T^{12} + 7740196 p T^{13} - 5402563 p^{2} T^{14} - 966556 p^{3} T^{15} + 486272 p^{4} T^{16} + 52188 p^{5} T^{17} - 29817 p^{6} T^{18} - 244 p^{7} T^{19} + 1422 p^{8} T^{20} - 92 p^{9} T^{21} - 47 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 12 T + 86 T^{2} - 24 p T^{3} + 1257 T^{4} + 2160 T^{5} - 49958 T^{6} + 362412 T^{7} - 1543666 T^{8} + 3330684 T^{9} + 8277174 T^{10} - 127836384 T^{11} + 719714221 T^{12} - 127836384 p T^{13} + 8277174 p^{2} T^{14} + 3330684 p^{3} T^{15} - 1543666 p^{4} T^{16} + 362412 p^{5} T^{17} - 49958 p^{6} T^{18} + 2160 p^{7} T^{19} + 1257 p^{8} T^{20} - 24 p^{10} T^{21} + 86 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 6 T - 34 T^{2} - 16 T^{3} + 4 p^{2} T^{4} + 3942 T^{5} - 22472 T^{6} - 380418 T^{7} - 103496 T^{8} + 7831728 T^{9} + 38620982 T^{10} - 4658482 p T^{11} - 973324338 T^{12} - 4658482 p^{2} T^{13} + 38620982 p^{2} T^{14} + 7831728 p^{3} T^{15} - 103496 p^{4} T^{16} - 380418 p^{5} T^{17} - 22472 p^{6} T^{18} + 3942 p^{7} T^{19} + 4 p^{10} T^{20} - 16 p^{9} T^{21} - 34 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 10 T - 35 T^{2} + 446 T^{3} + 1530 T^{4} - 6782 T^{5} - 96021 T^{6} - 11838 T^{7} + 4070624 T^{8} - 1776506 T^{9} - 86164135 T^{10} + 41320190 T^{11} + 1727300488 T^{12} + 41320190 p T^{13} - 86164135 p^{2} T^{14} - 1776506 p^{3} T^{15} + 4070624 p^{4} T^{16} - 11838 p^{5} T^{17} - 96021 p^{6} T^{18} - 6782 p^{7} T^{19} + 1530 p^{8} T^{20} + 446 p^{9} T^{21} - 35 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 284 T^{2} + 37878 T^{4} - 3169036 T^{6} + 187500191 T^{8} - 8369451480 T^{10} + 291872222164 T^{12} - 8369451480 p^{2} T^{14} + 187500191 p^{4} T^{16} - 3169036 p^{6} T^{18} + 37878 p^{8} T^{20} - 284 p^{10} T^{22} + p^{12} T^{24} \) | |
| 37 | \( 1 + 6 T + 85 T^{2} + 438 T^{3} + 2402 T^{4} + 17130 T^{5} - 35653 T^{6} + 395970 T^{7} - 938584 T^{8} + 16019286 T^{9} + 226015905 T^{10} + 1002132438 T^{11} + 15959668344 T^{12} + 1002132438 p T^{13} + 226015905 p^{2} T^{14} + 16019286 p^{3} T^{15} - 938584 p^{4} T^{16} + 395970 p^{5} T^{17} - 35653 p^{6} T^{18} + 17130 p^{7} T^{19} + 2402 p^{8} T^{20} + 438 p^{9} T^{21} + 85 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 24 T + 421 T^{2} - 5496 T^{3} + 60646 T^{4} - 577944 T^{5} + 4899619 T^{6} - 915336 p T^{7} + 264046160 T^{8} - 1732798824 T^{9} + 10900859329 T^{10} - 67457612616 T^{11} + 426112495412 T^{12} - 67457612616 p T^{13} + 10900859329 p^{2} T^{14} - 1732798824 p^{3} T^{15} + 264046160 p^{4} T^{16} - 915336 p^{6} T^{17} + 4899619 p^{6} T^{18} - 577944 p^{7} T^{19} + 60646 p^{8} T^{20} - 5496 p^{9} T^{21} + 421 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 26 T + 204 T^{2} - 388 T^{3} + 6469 T^{4} - 107912 T^{5} + 123284 T^{6} + 2376138 T^{7} + 25645138 T^{8} - 205458190 T^{9} - 752652116 T^{10} - 1747770144 T^{11} + 94649183413 T^{12} - 1747770144 p T^{13} - 752652116 p^{2} T^{14} - 205458190 p^{3} T^{15} + 25645138 p^{4} T^{16} + 2376138 p^{5} T^{17} + 123284 p^{6} T^{18} - 107912 p^{7} T^{19} + 6469 p^{8} T^{20} - 388 p^{9} T^{21} + 204 p^{10} T^{22} - 26 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 292 T^{2} + 42370 T^{4} - 4163860 T^{6} + 314054063 T^{8} - 19250228296 T^{10} + 985546087964 T^{12} - 19250228296 p^{2} T^{14} + 314054063 p^{4} T^{16} - 4163860 p^{6} T^{18} + 42370 p^{8} T^{20} - 292 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( ( 1 + 18 T + 267 T^{2} + 3114 T^{3} + 31467 T^{4} + 273420 T^{5} + 2178178 T^{6} + 273420 p T^{7} + 31467 p^{2} T^{8} + 3114 p^{3} T^{9} + 267 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 + 6 T + 288 T^{2} + 1656 T^{3} + 45304 T^{4} + 245166 T^{5} + 4749560 T^{6} + 24446682 T^{7} + 373366952 T^{8} + 1863645816 T^{9} + 24290241760 T^{10} + 120302765010 T^{11} + 1452403081778 T^{12} + 120302765010 p T^{13} + 24290241760 p^{2} T^{14} + 1863645816 p^{3} T^{15} + 373366952 p^{4} T^{16} + 24446682 p^{5} T^{17} + 4749560 p^{6} T^{18} + 245166 p^{7} T^{19} + 45304 p^{8} T^{20} + 1656 p^{9} T^{21} + 288 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 28 T + 349 T^{2} + 4268 T^{3} + 59969 T^{4} + 666584 T^{5} + 6515682 T^{6} + 65752208 T^{7} + 636141881 T^{8} + 5695939476 T^{9} + 48615514181 T^{10} + 407224627676 T^{11} + 3289754106258 T^{12} + 407224627676 p T^{13} + 48615514181 p^{2} T^{14} + 5695939476 p^{3} T^{15} + 636141881 p^{4} T^{16} + 65752208 p^{5} T^{17} + 6515682 p^{6} T^{18} + 666584 p^{7} T^{19} + 59969 p^{8} T^{20} + 4268 p^{9} T^{21} + 349 p^{10} T^{22} + 28 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 42 T + 1144 T^{2} + 23352 T^{3} + 399749 T^{4} + 5988756 T^{5} + 80541464 T^{6} + 986711982 T^{7} + 11104853666 T^{8} + 115726637814 T^{9} + 1122346599720 T^{10} + 10161311081820 T^{11} + 86018599956165 T^{12} + 10161311081820 p T^{13} + 1122346599720 p^{2} T^{14} + 115726637814 p^{3} T^{15} + 11104853666 p^{4} T^{16} + 986711982 p^{5} T^{17} + 80541464 p^{6} T^{18} + 5988756 p^{7} T^{19} + 399749 p^{8} T^{20} + 23352 p^{9} T^{21} + 1144 p^{10} T^{22} + 42 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 48 T + 1416 T^{2} + 31104 T^{3} + 560104 T^{4} + 8672640 T^{5} + 119324132 T^{6} + 1487912928 T^{7} + 17062448120 T^{8} + 181475231616 T^{9} + 1800797358808 T^{10} + 16719062739024 T^{11} + 145499346265574 T^{12} + 16719062739024 p T^{13} + 1800797358808 p^{2} T^{14} + 181475231616 p^{3} T^{15} + 17062448120 p^{4} T^{16} + 1487912928 p^{5} T^{17} + 119324132 p^{6} T^{18} + 8672640 p^{7} T^{19} + 560104 p^{8} T^{20} + 31104 p^{9} T^{21} + 1416 p^{10} T^{22} + 48 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 266 T^{2} + 43311 T^{4} - 5373610 T^{6} + 548198363 T^{8} - 48209574060 T^{10} + 3719753139946 T^{12} - 48209574060 p^{2} T^{14} + 548198363 p^{4} T^{16} - 5373610 p^{6} T^{18} + 43311 p^{8} T^{20} - 266 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( ( 1 - 22 T + 536 T^{2} - 7106 T^{3} + 100727 T^{4} - 974028 T^{5} + 10218624 T^{6} - 974028 p T^{7} + 100727 p^{2} T^{8} - 7106 p^{3} T^{9} + 536 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 - 532 T^{2} + 133678 T^{4} - 21920644 T^{6} + 2768318687 T^{8} - 293232724168 T^{10} + 26452486990532 T^{12} - 293232724168 p^{2} T^{14} + 2768318687 p^{4} T^{16} - 21920644 p^{6} T^{18} + 133678 p^{8} T^{20} - 532 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( 1 + 12 T + 398 T^{2} + 4200 T^{3} + 74725 T^{4} + 604368 T^{5} + 8843594 T^{6} + 55476156 T^{7} + 849122282 T^{8} + 5326866180 T^{9} + 88164449030 T^{10} + 572764108704 T^{11} + 8789695427309 T^{12} + 572764108704 p T^{13} + 88164449030 p^{2} T^{14} + 5326866180 p^{3} T^{15} + 849122282 p^{4} T^{16} + 55476156 p^{5} T^{17} + 8843594 p^{6} T^{18} + 604368 p^{7} T^{19} + 74725 p^{8} T^{20} + 4200 p^{9} T^{21} + 398 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 60 T + 2086 T^{2} - 53160 T^{3} + 1093109 T^{4} - 19256400 T^{5} + 301606130 T^{6} - 4303052940 T^{7} + 56755920650 T^{8} - 697207127220 T^{9} + 8014248965502 T^{10} - 86468019551520 T^{11} + 877593761269437 T^{12} - 86468019551520 p T^{13} + 8014248965502 p^{2} T^{14} - 697207127220 p^{3} T^{15} + 56755920650 p^{4} T^{16} - 4303052940 p^{5} T^{17} + 301606130 p^{6} T^{18} - 19256400 p^{7} T^{19} + 1093109 p^{8} T^{20} - 53160 p^{9} T^{21} + 2086 p^{10} T^{22} - 60 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.87202064354737723262237851712, −2.86084337402300595353017255172, −2.78713441193942321906303362060, −2.74883803919799337556115608951, −2.57009290462463113762655907093, −2.44773306211082300296879890891, −2.39991897578885041132813792532, −2.28012032687777358265253611413, −2.16079708030022018615363977058, −2.09592991444404758346045938817, −1.99903119263241178157146622856, −1.79492925072440445174809807495, −1.63691278720637472048026839068, −1.56385712193918364956182058835, −1.54754253100856410646313792759, −1.37158764093879750724689664774, −1.35678539728457361497009440933, −1.32438888429897324348160615643, −1.05992034725063246260820329733, −0.884122202246548482612109342596, −0.867642148442995386340932855182, −0.842534623739461893395076219275, −0.69917211380920880250532149126, −0.44772477620575003666346420125, −0.16023752689218488783500413660, 0.16023752689218488783500413660, 0.44772477620575003666346420125, 0.69917211380920880250532149126, 0.842534623739461893395076219275, 0.867642148442995386340932855182, 0.884122202246548482612109342596, 1.05992034725063246260820329733, 1.32438888429897324348160615643, 1.35678539728457361497009440933, 1.37158764093879750724689664774, 1.54754253100856410646313792759, 1.56385712193918364956182058835, 1.63691278720637472048026839068, 1.79492925072440445174809807495, 1.99903119263241178157146622856, 2.09592991444404758346045938817, 2.16079708030022018615363977058, 2.28012032687777358265253611413, 2.39991897578885041132813792532, 2.44773306211082300296879890891, 2.57009290462463113762655907093, 2.74883803919799337556115608951, 2.78713441193942321906303362060, 2.86084337402300595353017255172, 2.87202064354737723262237851712
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.