Dirichlet series
| L(s) = 1 | + 2·5-s + 6·7-s − 4·11-s + 10·13-s − 4·17-s + 4·19-s − 8·23-s + 10·25-s + 2·29-s + 8·31-s + 12·35-s + 2·41-s − 2·43-s + 14·47-s + 30·49-s − 24·53-s − 8·55-s − 6·59-s + 14·61-s + 20·65-s + 4·67-s + 28·71-s + 60·73-s − 24·77-s + 16·79-s − 24·83-s − 8·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 2.26·7-s − 1.20·11-s + 2.77·13-s − 0.970·17-s + 0.917·19-s − 1.66·23-s + 2·25-s + 0.371·29-s + 1.43·31-s + 2.02·35-s + 0.312·41-s − 0.304·43-s + 2.04·47-s + 30/7·49-s − 3.29·53-s − 1.07·55-s − 0.781·59-s + 1.79·61-s + 2.48·65-s + 0.488·67-s + 3.32·71-s + 7.02·73-s − 2.73·77-s + 1.80·79-s − 2.63·83-s − 0.867·85-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{84} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{84} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{84} \cdot 3^{36}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.95078\times 10^{17}\) |
| Root analytic conductor: | \(5.25321\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{84} \cdot 3^{36} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(128.0747593\) |
| \(L(\frac12)\) | \(\approx\) | \(128.0747593\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | |
| good | 5 | \( 1 - 2 T - 6 T^{2} - 4 p T^{3} + 18 p T^{4} + 202 T^{5} - 172 T^{6} - 2034 T^{7} - 282 p T^{8} + 1844 p T^{9} + 26974 T^{10} - 29526 T^{11} - 138266 T^{12} - 29526 p T^{13} + 26974 p^{2} T^{14} + 1844 p^{4} T^{15} - 282 p^{5} T^{16} - 2034 p^{5} T^{17} - 172 p^{6} T^{18} + 202 p^{7} T^{19} + 18 p^{9} T^{20} - 4 p^{10} T^{21} - 6 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 - 6 T + 6 T^{2} + 16 T^{3} - 6 p T^{4} + 204 T^{5} - 20 p^{2} T^{6} + 3840 T^{7} - 7386 T^{8} - 6824 T^{9} + 49698 T^{10} - 149706 T^{11} + 474394 T^{12} - 149706 p T^{13} + 49698 p^{2} T^{14} - 6824 p^{3} T^{15} - 7386 p^{4} T^{16} + 3840 p^{5} T^{17} - 20 p^{8} T^{18} + 204 p^{7} T^{19} - 6 p^{9} T^{20} + 16 p^{9} T^{21} + 6 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 4 T - 19 T^{2} - 48 T^{3} + 331 T^{4} + 430 T^{5} - 2296 T^{6} - 2796 T^{7} - 14403 T^{8} + 19340 T^{9} + 552863 T^{10} - 26990 T^{11} - 7241418 T^{12} - 26990 p T^{13} + 552863 p^{2} T^{14} + 19340 p^{3} T^{15} - 14403 p^{4} T^{16} - 2796 p^{5} T^{17} - 2296 p^{6} T^{18} + 430 p^{7} T^{19} + 331 p^{8} T^{20} - 48 p^{9} T^{21} - 19 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 10 T + 30 T^{2} - 68 T^{3} + 486 T^{4} - 1390 T^{5} + 28 p T^{6} - 2754 T^{7} + 4410 T^{8} - 99148 T^{9} + 1849474 T^{10} - 6605286 T^{11} + 12214214 T^{12} - 6605286 p T^{13} + 1849474 p^{2} T^{14} - 99148 p^{3} T^{15} + 4410 p^{4} T^{16} - 2754 p^{5} T^{17} + 28 p^{7} T^{18} - 1390 p^{7} T^{19} + 486 p^{8} T^{20} - 68 p^{9} T^{21} + 30 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( ( 1 + 2 T + 19 T^{2} - 6 T^{3} + 115 T^{4} + 456 T^{5} + 4566 T^{6} + 456 p T^{7} + 115 p^{2} T^{8} - 6 p^{3} T^{9} + 19 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( ( 1 - 2 T + 49 T^{2} - 110 T^{3} + 1635 T^{4} - 3108 T^{5} + 37062 T^{6} - 3108 p T^{7} + 1635 p^{2} T^{8} - 110 p^{3} T^{9} + 49 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 23 | \( 1 + 8 T - 10 T^{2} - 252 T^{3} - 650 T^{4} + 58 p T^{5} + 7724 T^{6} - 18798 T^{7} - 55530 T^{8} + 2562028 T^{9} + 14851874 T^{10} - 40741060 T^{11} - 563728326 T^{12} - 40741060 p T^{13} + 14851874 p^{2} T^{14} + 2562028 p^{3} T^{15} - 55530 p^{4} T^{16} - 18798 p^{5} T^{17} + 7724 p^{6} T^{18} + 58 p^{8} T^{19} - 650 p^{8} T^{20} - 252 p^{9} T^{21} - 10 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 2 T - 86 T^{2} + 244 T^{3} + 2642 T^{4} - 9254 T^{5} - 80308 T^{6} + 141174 T^{7} + 4438542 T^{8} - 3069156 T^{9} - 146032362 T^{10} + 64819794 T^{11} + 3526923078 T^{12} + 64819794 p T^{13} - 146032362 p^{2} T^{14} - 3069156 p^{3} T^{15} + 4438542 p^{4} T^{16} + 141174 p^{5} T^{17} - 80308 p^{6} T^{18} - 9254 p^{7} T^{19} + 2642 p^{8} T^{20} + 244 p^{9} T^{21} - 86 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 8 T - 50 T^{2} + 116 T^{3} + 3034 T^{4} + 10138 T^{5} - 131348 T^{6} + 10566 T^{7} + 1346874 T^{8} - 5966220 T^{9} - 99477774 T^{10} - 7938012 T^{11} + 6918069882 T^{12} - 7938012 p T^{13} - 99477774 p^{2} T^{14} - 5966220 p^{3} T^{15} + 1346874 p^{4} T^{16} + 10566 p^{5} T^{17} - 131348 p^{6} T^{18} + 10138 p^{7} T^{19} + 3034 p^{8} T^{20} + 116 p^{9} T^{21} - 50 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( ( 1 + 162 T^{2} + 68 T^{3} + 12531 T^{4} + 5820 T^{5} + 584916 T^{6} + 5820 p T^{7} + 12531 p^{2} T^{8} + 68 p^{3} T^{9} + 162 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 - 2 T - 161 T^{2} - 38 T^{3} + 14483 T^{4} + 23188 T^{5} - 832960 T^{6} - 2293860 T^{7} + 34705461 T^{8} + 105740430 T^{9} - 1089960255 T^{10} - 2027388630 T^{11} + 37240793262 T^{12} - 2027388630 p T^{13} - 1089960255 p^{2} T^{14} + 105740430 p^{3} T^{15} + 34705461 p^{4} T^{16} - 2293860 p^{5} T^{17} - 832960 p^{6} T^{18} + 23188 p^{7} T^{19} + 14483 p^{8} T^{20} - 38 p^{9} T^{21} - 161 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 2 T - 155 T^{2} - 50 T^{3} + 11803 T^{4} - 11110 T^{5} - 685952 T^{6} + 480720 T^{7} + 37074837 T^{8} + 4517562 T^{9} - 1917932577 T^{10} - 322637328 T^{11} + 89159638086 T^{12} - 322637328 p T^{13} - 1917932577 p^{2} T^{14} + 4517562 p^{3} T^{15} + 37074837 p^{4} T^{16} + 480720 p^{5} T^{17} - 685952 p^{6} T^{18} - 11110 p^{7} T^{19} + 11803 p^{8} T^{20} - 50 p^{9} T^{21} - 155 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 14 T - 66 T^{2} + 1624 T^{3} + 2274 T^{4} - 94616 T^{5} - 180484 T^{6} + 3216588 T^{7} + 25557282 T^{8} - 52812440 T^{9} - 2374780958 T^{10} + 210204606 T^{11} + 138811342666 T^{12} + 210204606 p T^{13} - 2374780958 p^{2} T^{14} - 52812440 p^{3} T^{15} + 25557282 p^{4} T^{16} + 3216588 p^{5} T^{17} - 180484 p^{6} T^{18} - 94616 p^{7} T^{19} + 2274 p^{8} T^{20} + 1624 p^{9} T^{21} - 66 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( ( 1 + 12 T + 294 T^{2} + 2664 T^{3} + 36195 T^{4} + 256476 T^{5} + 2484460 T^{6} + 256476 p T^{7} + 36195 p^{2} T^{8} + 2664 p^{3} T^{9} + 294 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 + 6 T - 183 T^{2} - 1094 T^{3} + 14871 T^{4} + 71610 T^{5} - 1176532 T^{6} - 2457504 T^{7} + 109864125 T^{8} + 122046958 T^{9} - 7583261313 T^{10} - 4376692308 T^{11} + 430621860766 T^{12} - 4376692308 p T^{13} - 7583261313 p^{2} T^{14} + 122046958 p^{3} T^{15} + 109864125 p^{4} T^{16} - 2457504 p^{5} T^{17} - 1176532 p^{6} T^{18} + 71610 p^{7} T^{19} + 14871 p^{8} T^{20} - 1094 p^{9} T^{21} - 183 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 14 T - 166 T^{2} + 2452 T^{3} + 23486 T^{4} - 256682 T^{5} - 2904116 T^{6} + 18433410 T^{7} + 310892370 T^{8} - 1035224932 T^{9} - 25169458330 T^{10} + 25123222006 T^{11} + 1689393499046 T^{12} + 25123222006 p T^{13} - 25169458330 p^{2} T^{14} - 1035224932 p^{3} T^{15} + 310892370 p^{4} T^{16} + 18433410 p^{5} T^{17} - 2904116 p^{6} T^{18} - 256682 p^{7} T^{19} + 23486 p^{8} T^{20} + 2452 p^{9} T^{21} - 166 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 4 T - 139 T^{2} + 440 T^{3} + 8819 T^{4} + 10838 T^{5} + 22768 T^{6} - 7726476 T^{7} - 28246803 T^{8} + 772234804 T^{9} + 2282110511 T^{10} - 26367816958 T^{11} - 108360067642 T^{12} - 26367816958 p T^{13} + 2282110511 p^{2} T^{14} + 772234804 p^{3} T^{15} - 28246803 p^{4} T^{16} - 7726476 p^{5} T^{17} + 22768 p^{6} T^{18} + 10838 p^{7} T^{19} + 8819 p^{8} T^{20} + 440 p^{9} T^{21} - 139 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( ( 1 - 14 T + 354 T^{2} - 3362 T^{3} + 50451 T^{4} - 366980 T^{5} + 4312564 T^{6} - 366980 p T^{7} + 50451 p^{2} T^{8} - 3362 p^{3} T^{9} + 354 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 73 | \( ( 1 - 30 T + 663 T^{2} - 10462 T^{3} + 1887 p T^{4} - 1485216 T^{5} + 13863318 T^{6} - 1485216 p T^{7} + 1887 p^{3} T^{8} - 10462 p^{3} T^{9} + 663 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 79 | \( 1 - 16 T - 150 T^{2} + 1036 T^{3} + 48534 T^{4} - 92602 T^{5} - 4789796 T^{6} - 25294038 T^{7} + 421747950 T^{8} + 2410897964 T^{9} - 6318240602 T^{10} - 147604045356 T^{11} + 127084960154 T^{12} - 147604045356 p T^{13} - 6318240602 p^{2} T^{14} + 2410897964 p^{3} T^{15} + 421747950 p^{4} T^{16} - 25294038 p^{5} T^{17} - 4789796 p^{6} T^{18} - 92602 p^{7} T^{19} + 48534 p^{8} T^{20} + 1036 p^{9} T^{21} - 150 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 24 T + 36 T^{2} - 4140 T^{3} - 28128 T^{4} + 434994 T^{5} + 5173688 T^{6} - 26364546 T^{7} - 558947316 T^{8} + 862732332 T^{9} + 46488493284 T^{10} + 3735835668 T^{11} - 3451975469358 T^{12} + 3735835668 p T^{13} + 46488493284 p^{2} T^{14} + 862732332 p^{3} T^{15} - 558947316 p^{4} T^{16} - 26364546 p^{5} T^{17} + 5173688 p^{6} T^{18} + 434994 p^{7} T^{19} - 28128 p^{8} T^{20} - 4140 p^{9} T^{21} + 36 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( ( 1 - 24 T + 690 T^{2} - 10612 T^{3} + 171027 T^{4} - 1875108 T^{5} + 20918900 T^{6} - 1875108 p T^{7} + 171027 p^{2} T^{8} - 10612 p^{3} T^{9} + 690 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 + 14 T - 153 T^{2} - 2150 T^{3} + 12699 T^{4} + 84500 T^{5} - 1669976 T^{6} - 9414804 T^{7} + 97057365 T^{8} + 1065322190 T^{9} - 8753923919 T^{10} - 19649592630 T^{11} + 1679085585182 T^{12} - 19649592630 p T^{13} - 8753923919 p^{2} T^{14} + 1065322190 p^{3} T^{15} + 97057365 p^{4} T^{16} - 9414804 p^{5} T^{17} - 1669976 p^{6} T^{18} + 84500 p^{7} T^{19} + 12699 p^{8} T^{20} - 2150 p^{9} T^{21} - 153 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.45089098934296330522996859389, −2.39580414457560962054156025723, −2.33722959735180337329104812007, −2.33127462883853417175066959254, −2.21098969327389018916254579120, −2.17960120351476311381839466399, −2.15147069885598079822527056250, −2.13857212323365777783307262103, −2.01149016307354946715447224776, −1.81838853885631611700121528397, −1.68508336391776050089633126595, −1.59962297806342393843315085207, −1.55396887748376147132102527140, −1.39805559204521092093884337549, −1.31466366077842552458454229784, −1.17911881230769671766816772585, −1.05982438616860946576956151134, −1.01564498948338351786456906851, −0.898384773944383416893129553017, −0.892605079808426276219323521603, −0.76304439616245481127212434305, −0.66354931429544461399890342167, −0.37896173648944961791902943566, −0.32572919905919034117850353267, −0.31577555682606964237574623359, 0.31577555682606964237574623359, 0.32572919905919034117850353267, 0.37896173648944961791902943566, 0.66354931429544461399890342167, 0.76304439616245481127212434305, 0.892605079808426276219323521603, 0.898384773944383416893129553017, 1.01564498948338351786456906851, 1.05982438616860946576956151134, 1.17911881230769671766816772585, 1.31466366077842552458454229784, 1.39805559204521092093884337549, 1.55396887748376147132102527140, 1.59962297806342393843315085207, 1.68508336391776050089633126595, 1.81838853885631611700121528397, 2.01149016307354946715447224776, 2.13857212323365777783307262103, 2.15147069885598079822527056250, 2.17960120351476311381839466399, 2.21098969327389018916254579120, 2.33127462883853417175066959254, 2.33722959735180337329104812007, 2.39580414457560962054156025723, 2.45089098934296330522996859389
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.