Dirichlet series
| L(s) = 1 | − 3·3-s + 4·5-s + 3·7-s + 9·9-s + 13·11-s − 12·15-s − 11·17-s + 15·19-s − 9·21-s − 15·23-s − 6·25-s − 10·27-s − 9·29-s + 6·31-s − 39·33-s + 12·35-s + 14·37-s − 20·41-s − 10·43-s + 36·45-s + 20·47-s + 12·49-s + 33·51-s + 2·53-s + 52·55-s − 45·57-s + 50·59-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.78·5-s + 1.13·7-s + 3·9-s + 3.91·11-s − 3.09·15-s − 2.66·17-s + 3.44·19-s − 1.96·21-s − 3.12·23-s − 6/5·25-s − 1.92·27-s − 1.67·29-s + 1.07·31-s − 6.78·33-s + 2.02·35-s + 2.30·37-s − 3.12·41-s − 1.52·43-s + 5.36·45-s + 2.91·47-s + 12/7·49-s + 4.62·51-s + 0.274·53-s + 7.01·55-s − 5.96·57-s + 6.50·59-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{36} \cdot 13^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.50637\times 10^{12}\) |
| Root analytic conductor: | \(3.28569\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{36} \cdot 13^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(58.51196840\) |
| \(L(\frac12)\) | \(\approx\) | \(58.51196840\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T - 17 T^{3} - p^{3} T^{4} + 22 T^{5} + 164 T^{6} + 71 p T^{7} - 205 T^{8} - 1102 T^{9} - 953 T^{10} + 1733 T^{11} + 6091 T^{12} + 1733 p T^{13} - 953 p^{2} T^{14} - 1102 p^{3} T^{15} - 205 p^{4} T^{16} + 71 p^{6} T^{17} + 164 p^{6} T^{18} + 22 p^{7} T^{19} - p^{11} T^{20} - 17 p^{9} T^{21} + p^{12} T^{23} + p^{12} T^{24} \) |
| 5 | \( ( 1 - 2 T + 9 T^{2} - 27 T^{3} + 53 T^{4} - 107 T^{5} + 322 T^{6} - 107 p T^{7} + 53 p^{2} T^{8} - 27 p^{3} T^{9} + 9 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 7 | \( 1 - 3 T - 3 T^{2} - 40 T^{3} + 107 T^{4} + 199 T^{5} + 145 p T^{6} - 1167 T^{7} - 7262 T^{8} - 21786 T^{9} - 6319 T^{10} + 111159 T^{11} + 390755 T^{12} + 111159 p T^{13} - 6319 p^{2} T^{14} - 21786 p^{3} T^{15} - 7262 p^{4} T^{16} - 1167 p^{5} T^{17} + 145 p^{7} T^{18} + 199 p^{7} T^{19} + 107 p^{8} T^{20} - 40 p^{9} T^{21} - 3 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 13 T + 62 T^{2} - 147 T^{3} + 393 T^{4} - 1172 T^{5} - 838 T^{6} + 911 p T^{7} - 8005 T^{8} + 116132 T^{9} - 964795 T^{10} + 3049927 T^{11} - 7859321 T^{12} + 3049927 p T^{13} - 964795 p^{2} T^{14} + 116132 p^{3} T^{15} - 8005 p^{4} T^{16} + 911 p^{6} T^{17} - 838 p^{6} T^{18} - 1172 p^{7} T^{19} + 393 p^{8} T^{20} - 147 p^{9} T^{21} + 62 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 11 T - 10 T^{2} - 435 T^{3} + 553 T^{4} + 16442 T^{5} + 24 p T^{6} - 298601 T^{7} + 345935 T^{8} + 5256326 T^{9} - 5052263 T^{10} - 21105805 T^{11} + 202190453 T^{12} - 21105805 p T^{13} - 5052263 p^{2} T^{14} + 5256326 p^{3} T^{15} + 345935 p^{4} T^{16} - 298601 p^{5} T^{17} + 24 p^{7} T^{18} + 16442 p^{7} T^{19} + 553 p^{8} T^{20} - 435 p^{9} T^{21} - 10 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 15 T + 70 T^{2} - 129 T^{3} + 709 T^{4} - 360 T^{5} - 50726 T^{6} + 292767 T^{7} - 572965 T^{8} + 1100124 T^{9} - 2018375 T^{10} - 72663183 T^{11} + 582440783 T^{12} - 72663183 p T^{13} - 2018375 p^{2} T^{14} + 1100124 p^{3} T^{15} - 572965 p^{4} T^{16} + 292767 p^{5} T^{17} - 50726 p^{6} T^{18} - 360 p^{7} T^{19} + 709 p^{8} T^{20} - 129 p^{9} T^{21} + 70 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 15 T + 44 T^{2} - 457 T^{3} - 2620 T^{4} + 9075 T^{5} + 103400 T^{6} + 232255 T^{7} - 517932 T^{8} - 8998093 T^{9} - 43515684 T^{10} + 137227707 T^{11} + 1934656334 T^{12} + 137227707 p T^{13} - 43515684 p^{2} T^{14} - 8998093 p^{3} T^{15} - 517932 p^{4} T^{16} + 232255 p^{5} T^{17} + 103400 p^{6} T^{18} + 9075 p^{7} T^{19} - 2620 p^{8} T^{20} - 457 p^{9} T^{21} + 44 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 9 T + T^{2} - 436 T^{3} - 2839 T^{4} - 4629 T^{5} + 63311 T^{6} + 595599 T^{7} + 1747220 T^{8} - 90558 p T^{9} - 31802103 T^{10} - 140688215 T^{11} - 765987465 T^{12} - 140688215 p T^{13} - 31802103 p^{2} T^{14} - 90558 p^{4} T^{15} + 1747220 p^{4} T^{16} + 595599 p^{5} T^{17} + 63311 p^{6} T^{18} - 4629 p^{7} T^{19} - 2839 p^{8} T^{20} - 436 p^{9} T^{21} + p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( ( 1 - 3 T + 132 T^{2} - 422 T^{3} + 8311 T^{4} - 24619 T^{5} + 320496 T^{6} - 24619 p T^{7} + 8311 p^{2} T^{8} - 422 p^{3} T^{9} + 132 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 37 | \( 1 - 14 T - 67 T^{2} + 1120 T^{3} + 9878 T^{4} - 83636 T^{5} - 812908 T^{6} + 4033568 T^{7} + 52459469 T^{8} - 134898960 T^{9} - 2632836577 T^{10} + 1608385450 T^{11} + 112472550501 T^{12} + 1608385450 p T^{13} - 2632836577 p^{2} T^{14} - 134898960 p^{3} T^{15} + 52459469 p^{4} T^{16} + 4033568 p^{5} T^{17} - 812908 p^{6} T^{18} - 83636 p^{7} T^{19} + 9878 p^{8} T^{20} + 1120 p^{9} T^{21} - 67 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 20 T + 78 T^{2} - 1098 T^{3} - 8579 T^{4} + 34656 T^{5} + 560905 T^{6} + 1898453 T^{7} - 2194646 T^{8} - 111813678 T^{9} - 843544973 T^{10} + 3218298927 T^{11} + 64940986812 T^{12} + 3218298927 p T^{13} - 843544973 p^{2} T^{14} - 111813678 p^{3} T^{15} - 2194646 p^{4} T^{16} + 1898453 p^{5} T^{17} + 560905 p^{6} T^{18} + 34656 p^{7} T^{19} - 8579 p^{8} T^{20} - 1098 p^{9} T^{21} + 78 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 10 T - 107 T^{2} - 1274 T^{3} + 6388 T^{4} + 78994 T^{5} - 381001 T^{6} - 3438948 T^{7} + 23161095 T^{8} + 100509358 T^{9} - 1383399972 T^{10} - 30422488 p T^{11} + 69730072257 T^{12} - 30422488 p^{2} T^{13} - 1383399972 p^{2} T^{14} + 100509358 p^{3} T^{15} + 23161095 p^{4} T^{16} - 3438948 p^{5} T^{17} - 381001 p^{6} T^{18} + 78994 p^{7} T^{19} + 6388 p^{8} T^{20} - 1274 p^{9} T^{21} - 107 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( ( 1 - 10 T + 227 T^{2} - 1899 T^{3} + 23541 T^{4} - 160969 T^{5} + 1420310 T^{6} - 160969 p T^{7} + 23541 p^{2} T^{8} - 1899 p^{3} T^{9} + 227 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 53 | \( ( 1 - T + 4 p T^{2} - 64 T^{3} + 21695 T^{4} + 2801 T^{5} + 1397776 T^{6} + 2801 p T^{7} + 21695 p^{2} T^{8} - 64 p^{3} T^{9} + 4 p^{5} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 - 50 T + 1118 T^{2} - 17378 T^{3} + 255563 T^{4} - 3524420 T^{5} + 41060409 T^{6} - 436661831 T^{7} + 4567447342 T^{8} - 43730611404 T^{9} + 378763826469 T^{10} - 3206826029205 T^{11} + 25920439680244 T^{12} - 3206826029205 p T^{13} + 378763826469 p^{2} T^{14} - 43730611404 p^{3} T^{15} + 4567447342 p^{4} T^{16} - 436661831 p^{5} T^{17} + 41060409 p^{6} T^{18} - 3524420 p^{7} T^{19} + 255563 p^{8} T^{20} - 17378 p^{9} T^{21} + 1118 p^{10} T^{22} - 50 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 231 T^{2} - 378 T^{3} + 27834 T^{4} + 75222 T^{5} - 2173332 T^{6} - 7832160 T^{7} + 124748853 T^{8} + 467691840 T^{9} - 5750294109 T^{10} - 12237165234 T^{11} + 294705743969 T^{12} - 12237165234 p T^{13} - 5750294109 p^{2} T^{14} + 467691840 p^{3} T^{15} + 124748853 p^{4} T^{16} - 7832160 p^{5} T^{17} - 2173332 p^{6} T^{18} + 75222 p^{7} T^{19} + 27834 p^{8} T^{20} - 378 p^{9} T^{21} - 231 p^{10} T^{22} + p^{12} T^{24} \) | |
| 67 | \( 1 - 6 T - 127 T^{2} + 1610 T^{3} + 918 T^{4} - 103050 T^{5} + 1053089 T^{6} - 4296428 T^{7} - 51859833 T^{8} + 1018435014 T^{9} - 3548303380 T^{10} - 36787522794 T^{11} + 557444038493 T^{12} - 36787522794 p T^{13} - 3548303380 p^{2} T^{14} + 1018435014 p^{3} T^{15} - 51859833 p^{4} T^{16} - 4296428 p^{5} T^{17} + 1053089 p^{6} T^{18} - 103050 p^{7} T^{19} + 918 p^{8} T^{20} + 1610 p^{9} T^{21} - 127 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 9 T - 173 T^{2} + 1042 T^{3} + 17225 T^{4} - 33185 T^{5} - 1457025 T^{6} + 4515343 T^{7} + 81761736 T^{8} - 447430558 T^{9} - 7117889445 T^{10} + 10645956407 T^{11} + 722144834231 T^{12} + 10645956407 p T^{13} - 7117889445 p^{2} T^{14} - 447430558 p^{3} T^{15} + 81761736 p^{4} T^{16} + 4515343 p^{5} T^{17} - 1457025 p^{6} T^{18} - 33185 p^{7} T^{19} + 17225 p^{8} T^{20} + 1042 p^{9} T^{21} - 173 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( ( 1 - 6 T + 117 T^{2} - 992 T^{3} + 11187 T^{4} - 54556 T^{5} + 881901 T^{6} - 54556 p T^{7} + 11187 p^{2} T^{8} - 992 p^{3} T^{9} + 117 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 79 | \( ( 1 - 13 T + 300 T^{2} - 3664 T^{3} + 49685 T^{4} - 504247 T^{5} + 4845092 T^{6} - 504247 p T^{7} + 49685 p^{2} T^{8} - 3664 p^{3} T^{9} + 300 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( ( 1 + 36 T + 855 T^{2} + 14546 T^{3} + 204485 T^{4} + 2355532 T^{5} + 23378677 T^{6} + 2355532 p T^{7} + 204485 p^{2} T^{8} + 14546 p^{3} T^{9} + 855 p^{4} T^{10} + 36 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 89 | \( 1 + 18 T - 303 T^{2} - 4750 T^{3} + 97746 T^{4} + 1016282 T^{5} - 19380659 T^{6} - 120805208 T^{7} + 3236278667 T^{8} + 10923147598 T^{9} - 396786409312 T^{10} - 339577796574 T^{11} + 40481280431893 T^{12} - 339577796574 p T^{13} - 396786409312 p^{2} T^{14} + 10923147598 p^{3} T^{15} + 3236278667 p^{4} T^{16} - 120805208 p^{5} T^{17} - 19380659 p^{6} T^{18} + 1016282 p^{7} T^{19} + 97746 p^{8} T^{20} - 4750 p^{9} T^{21} - 303 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 14 T - 245 T^{2} - 4410 T^{3} + 24770 T^{4} + 561988 T^{5} - 2937837 T^{6} - 37565010 T^{7} + 639814181 T^{8} + 1769720722 T^{9} - 100762361854 T^{10} - 57137895850 T^{11} + 11162693703189 T^{12} - 57137895850 p T^{13} - 100762361854 p^{2} T^{14} + 1769720722 p^{3} T^{15} + 639814181 p^{4} T^{16} - 37565010 p^{5} T^{17} - 2937837 p^{6} T^{18} + 561988 p^{7} T^{19} + 24770 p^{8} T^{20} - 4410 p^{9} T^{21} - 245 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.98140050214161031695373456927, −2.96355116452832526094856472373, −2.85777735532755054895717407841, −2.74376337775326936961543791121, −2.63523808289949090053635400035, −2.29279080638248033133253888716, −2.26984910510909112624571231955, −2.18222599498017188103828351709, −2.14108265949266520555264449380, −2.13416589499746918877184821188, −2.06059369087118308350972211733, −1.86153357868776843249962482730, −1.75307977466688193217157326048, −1.72376051483870409941248136398, −1.68579101824133637535726997189, −1.58747000104384375533250032890, −1.38351915400693253933241216593, −1.33882877456787664701347717718, −1.04644817732864179258752136917, −0.894938745679424168778136181273, −0.73699734970712824252412794865, −0.73202944895959238811610155361, −0.64598043814842331615983247282, −0.60665703124421403529408242421, −0.33883242902753506870901051610, 0.33883242902753506870901051610, 0.60665703124421403529408242421, 0.64598043814842331615983247282, 0.73202944895959238811610155361, 0.73699734970712824252412794865, 0.894938745679424168778136181273, 1.04644817732864179258752136917, 1.33882877456787664701347717718, 1.38351915400693253933241216593, 1.58747000104384375533250032890, 1.68579101824133637535726997189, 1.72376051483870409941248136398, 1.75307977466688193217157326048, 1.86153357868776843249962482730, 2.06059369087118308350972211733, 2.13416589499746918877184821188, 2.14108265949266520555264449380, 2.18222599498017188103828351709, 2.26984910510909112624571231955, 2.29279080638248033133253888716, 2.63523808289949090053635400035, 2.74376337775326936961543791121, 2.85777735532755054895717407841, 2.96355116452832526094856472373, 2.98140050214161031695373456927
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.