Dirichlet series
| L(s) = 1 | − 4·2-s + 4·3-s + 2·4-s − 16·6-s + 12·8-s + 8·9-s − 4·11-s + 8·12-s − 19·16-s − 8·17-s − 32·18-s + 16·22-s + 12·23-s + 48·24-s + 16·27-s + 12·29-s + 8·32-s − 16·33-s + 32·34-s + 16·36-s + 12·37-s − 24·41-s − 16·43-s − 8·44-s − 48·46-s − 76·48-s − 32·51-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 2.30·3-s + 4-s − 6.53·6-s + 4.24·8-s + 8/3·9-s − 1.20·11-s + 2.30·12-s − 4.75·16-s − 1.94·17-s − 7.54·18-s + 3.41·22-s + 2.50·23-s + 9.79·24-s + 3.07·27-s + 2.22·29-s + 1.41·32-s − 2.78·33-s + 5.48·34-s + 8/3·36-s + 1.97·37-s − 3.74·41-s − 2.43·43-s − 1.20·44-s − 7.07·46-s − 10.9·48-s − 4.48·51-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(5^{24} \cdot 17^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.33341\times 10^{6}\) |
| Root analytic conductor: | \(1.84218\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 5^{24} \cdot 17^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.050526180\) |
| \(L(\frac12)\) | \(\approx\) | \(1.050526180\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 + 8 T + 22 T^{2} - 48 T^{3} - 377 T^{4} + 744 T^{5} + 10004 T^{6} + 744 p T^{7} - 377 p^{2} T^{8} - 48 p^{3} T^{9} + 22 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 2 | \( ( 1 + p T + 5 T^{2} + p^{3} T^{3} + 15 T^{4} + 9 p T^{5} + 29 T^{6} + 9 p^{2} T^{7} + 15 p^{2} T^{8} + p^{6} T^{9} + 5 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} )^{2} \) |
| 3 | \( 1 - 4 T + 8 T^{2} - 16 T^{3} + 22 T^{4} - 16 T^{5} + 16 T^{6} - 28 T^{7} - 29 T^{8} + 304 T^{9} - 760 T^{10} + 1928 T^{11} - 4304 T^{12} + 1928 p T^{13} - 760 p^{2} T^{14} + 304 p^{3} T^{15} - 29 p^{4} T^{16} - 28 p^{5} T^{17} + 16 p^{6} T^{18} - 16 p^{7} T^{19} + 22 p^{8} T^{20} - 16 p^{9} T^{21} + 8 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 - 4 p T^{3} + 6 p T^{4} + 212 T^{5} + 8 p^{2} T^{6} - 928 T^{7} - 3125 T^{8} + 6744 T^{9} + 31992 T^{10} + 888 p T^{11} - 376 p^{3} T^{12} + 888 p^{2} T^{13} + 31992 p^{2} T^{14} + 6744 p^{3} T^{15} - 3125 p^{4} T^{16} - 928 p^{5} T^{17} + 8 p^{8} T^{18} + 212 p^{7} T^{19} + 6 p^{9} T^{20} - 4 p^{10} T^{21} + p^{12} T^{24} \) | |
| 11 | \( 1 + 4 T + 8 T^{2} + 64 T^{3} + 74 T^{4} - 72 p T^{5} - 1712 T^{6} - 13364 T^{7} - 46833 T^{8} + 42384 T^{9} + 129224 T^{10} + 1266064 T^{11} + 10886032 T^{12} + 1266064 p T^{13} + 129224 p^{2} T^{14} + 42384 p^{3} T^{15} - 46833 p^{4} T^{16} - 13364 p^{5} T^{17} - 1712 p^{6} T^{18} - 72 p^{8} T^{19} + 74 p^{8} T^{20} + 64 p^{9} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 40 T^{2} + 70 p T^{4} - 17952 T^{6} + 312111 T^{8} - 4903032 T^{10} + 69035060 T^{12} - 4903032 p^{2} T^{14} + 312111 p^{4} T^{16} - 17952 p^{6} T^{18} + 70 p^{9} T^{20} - 40 p^{10} T^{22} + p^{12} T^{24} \) | |
| 19 | \( 1 - 92 T^{2} + 4626 T^{4} - 167116 T^{6} + 4842975 T^{8} - 117605944 T^{10} + 2422842556 T^{12} - 117605944 p^{2} T^{14} + 4842975 p^{4} T^{16} - 167116 p^{6} T^{18} + 4626 p^{8} T^{20} - 92 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( 1 - 12 T + 72 T^{2} - 400 T^{3} + 2878 T^{4} - 18936 T^{5} + 100016 T^{6} - 521948 T^{7} + 2709363 T^{8} - 13154744 T^{9} + 63001992 T^{10} - 314746192 T^{11} + 1561862208 T^{12} - 314746192 p T^{13} + 63001992 p^{2} T^{14} - 13154744 p^{3} T^{15} + 2709363 p^{4} T^{16} - 521948 p^{5} T^{17} + 100016 p^{6} T^{18} - 18936 p^{7} T^{19} + 2878 p^{8} T^{20} - 400 p^{9} T^{21} + 72 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 12 T + 72 T^{2} - 708 T^{3} + 5562 T^{4} - 21252 T^{5} + 105192 T^{6} - 726444 T^{7} + 804479 T^{8} + 8873352 T^{9} - 9334512 T^{10} + 305351640 T^{11} - 3778100564 T^{12} + 305351640 p T^{13} - 9334512 p^{2} T^{14} + 8873352 p^{3} T^{15} + 804479 p^{4} T^{16} - 726444 p^{5} T^{17} + 105192 p^{6} T^{18} - 21252 p^{7} T^{19} + 5562 p^{8} T^{20} - 708 p^{9} T^{21} + 72 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 156 T^{3} - 2754 T^{4} + 452 T^{5} + 12168 T^{6} + 245112 T^{7} + 3961447 T^{8} + 323440 T^{9} - 4624648 T^{10} - 168645768 T^{11} - 4098407752 T^{12} - 168645768 p T^{13} - 4624648 p^{2} T^{14} + 323440 p^{3} T^{15} + 3961447 p^{4} T^{16} + 245112 p^{5} T^{17} + 12168 p^{6} T^{18} + 452 p^{7} T^{19} - 2754 p^{8} T^{20} - 156 p^{9} T^{21} + p^{12} T^{24} \) | |
| 37 | \( 1 - 12 T + 72 T^{2} - 308 T^{3} + 2250 T^{4} - 13364 T^{5} + 45800 T^{6} + 79636 T^{7} - 905 p^{2} T^{8} + 18417192 T^{9} - 170081904 T^{10} + 1483635480 T^{11} - 9533058292 T^{12} + 1483635480 p T^{13} - 170081904 p^{2} T^{14} + 18417192 p^{3} T^{15} - 905 p^{6} T^{16} + 79636 p^{5} T^{17} + 45800 p^{6} T^{18} - 13364 p^{7} T^{19} + 2250 p^{8} T^{20} - 308 p^{9} T^{21} + 72 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 24 T + 288 T^{2} + 2680 T^{3} + 20998 T^{4} + 135832 T^{5} + 803744 T^{6} + 4720696 T^{7} + 28651119 T^{8} + 187625072 T^{9} + 744256 p^{2} T^{10} + 199495088 p T^{11} + 52761079188 T^{12} + 199495088 p^{2} T^{13} + 744256 p^{4} T^{14} + 187625072 p^{3} T^{15} + 28651119 p^{4} T^{16} + 4720696 p^{5} T^{17} + 803744 p^{6} T^{18} + 135832 p^{7} T^{19} + 20998 p^{8} T^{20} + 2680 p^{9} T^{21} + 288 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( ( 1 + 8 T + 124 T^{2} + 436 T^{3} + 5195 T^{4} - 1788 T^{5} + 132228 T^{6} - 1788 p T^{7} + 5195 p^{2} T^{8} + 436 p^{3} T^{9} + 124 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 - 208 T^{2} + 22622 T^{4} - 1802760 T^{6} + 119815519 T^{8} - 6923192744 T^{10} + 349170604820 T^{12} - 6923192744 p^{2} T^{14} + 119815519 p^{4} T^{16} - 1802760 p^{6} T^{18} + 22622 p^{8} T^{20} - 208 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( ( 1 + 102 T^{2} - 360 T^{3} + 8079 T^{4} - 33592 T^{5} + 448980 T^{6} - 33592 p T^{7} + 8079 p^{2} T^{8} - 360 p^{3} T^{9} + 102 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 - 468 T^{2} + 102802 T^{4} - 14182372 T^{6} + 1399420159 T^{8} - 107717762600 T^{10} + 6886886586172 T^{12} - 107717762600 p^{2} T^{14} + 1399420159 p^{4} T^{16} - 14182372 p^{6} T^{18} + 102802 p^{8} T^{20} - 468 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( 1 - 40 T + 800 T^{2} - 12504 T^{3} + 184598 T^{4} - 40056 p T^{5} + 28233248 T^{6} - 305052840 T^{7} + 3158111167 T^{8} - 30151790672 T^{9} + 267303164864 T^{10} - 2283320825520 T^{11} + 18527414094644 T^{12} - 2283320825520 p T^{13} + 267303164864 p^{2} T^{14} - 30151790672 p^{3} T^{15} + 3158111167 p^{4} T^{16} - 305052840 p^{5} T^{17} + 28233248 p^{6} T^{18} - 40056 p^{8} T^{19} + 184598 p^{8} T^{20} - 12504 p^{9} T^{21} + 800 p^{10} T^{22} - 40 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 696 T^{2} + 227166 T^{4} - 46066448 T^{6} + 6474527599 T^{8} - 665187758008 T^{10} + 51260175275796 T^{12} - 665187758008 p^{2} T^{14} + 6474527599 p^{4} T^{16} - 46066448 p^{6} T^{18} + 227166 p^{8} T^{20} - 696 p^{10} T^{22} + p^{12} T^{24} \) | |
| 71 | \( 1 - 28 T + 392 T^{2} - 4432 T^{3} + 35314 T^{4} - 151184 T^{5} + 211376 T^{6} + 3536900 T^{7} - 37488993 T^{8} + 144764696 T^{9} - 1069498168 T^{10} + 21338963608 T^{11} - 259039666848 T^{12} + 21338963608 p T^{13} - 1069498168 p^{2} T^{14} + 144764696 p^{3} T^{15} - 37488993 p^{4} T^{16} + 3536900 p^{5} T^{17} + 211376 p^{6} T^{18} - 151184 p^{7} T^{19} + 35314 p^{8} T^{20} - 4432 p^{9} T^{21} + 392 p^{10} T^{22} - 28 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 48 T + 1152 T^{2} - 19520 T^{3} + 277734 T^{4} - 3608128 T^{5} + 43755776 T^{6} - 498961648 T^{7} + 5387964143 T^{8} - 54979433440 T^{9} + 528636621952 T^{10} - 4817379931648 T^{11} + 41994385065940 T^{12} - 4817379931648 p T^{13} + 528636621952 p^{2} T^{14} - 54979433440 p^{3} T^{15} + 5387964143 p^{4} T^{16} - 498961648 p^{5} T^{17} + 43755776 p^{6} T^{18} - 3608128 p^{7} T^{19} + 277734 p^{8} T^{20} - 19520 p^{9} T^{21} + 1152 p^{10} T^{22} - 48 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 8 T + 32 T^{2} + 1220 T^{3} + 5350 T^{4} - 114868 T^{5} + 1491944 T^{6} + 12162584 T^{7} - 38174937 T^{8} - 233812952 T^{9} + 16545882392 T^{10} + 38512918912 T^{11} - 355031479032 T^{12} + 38512918912 p T^{13} + 16545882392 p^{2} T^{14} - 233812952 p^{3} T^{15} - 38174937 p^{4} T^{16} + 12162584 p^{5} T^{17} + 1491944 p^{6} T^{18} - 114868 p^{7} T^{19} + 5350 p^{8} T^{20} + 1220 p^{9} T^{21} + 32 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( ( 1 - 20 T + 588 T^{2} - 7984 T^{3} + 131023 T^{4} - 1299868 T^{5} + 14803132 T^{6} - 1299868 p T^{7} + 131023 p^{2} T^{8} - 7984 p^{3} T^{9} + 588 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 89 | \( ( 1 + 12 T + 354 T^{2} + 4000 T^{3} + 61415 T^{4} + 604868 T^{5} + 6702448 T^{6} + 604868 p T^{7} + 61415 p^{2} T^{8} + 4000 p^{3} T^{9} + 354 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 - 4 T + 8 T^{2} + 20 T^{3} - 17958 T^{4} + 32004 T^{5} + 15848 T^{6} - 4133236 T^{7} + 241837519 T^{8} - 226090952 T^{9} - 316274672 T^{10} + 41141248808 T^{11} - 2546305955668 T^{12} + 41141248808 p T^{13} - 316274672 p^{2} T^{14} - 226090952 p^{3} T^{15} + 241837519 p^{4} T^{16} - 4133236 p^{5} T^{17} + 15848 p^{6} T^{18} + 32004 p^{7} T^{19} - 17958 p^{8} T^{20} + 20 p^{9} T^{21} + 8 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.53613934528276098622350321927, −3.51378102878526639251273976008, −3.50509690590673187321830945410, −3.46295834246923675864236430135, −3.42438861093988020296998631481, −3.29141961486623829100884903608, −2.97781075854264258684584369763, −2.82928287996642233286160044854, −2.81361156951762403440720537421, −2.70579551212770429953246702767, −2.55678634657549231479934132362, −2.48932592373620066336713470820, −2.36863978652721378554944774354, −2.16087705179541949479471917526, −2.13712550154946702973558987740, −2.12906626830843853465727323463, −1.78117834909953377483252862248, −1.69240938387166907994526178753, −1.52469581997684026580208538731, −1.21466502385630849746773851360, −0.941469971165889785601834573679, −0.78719022233881782145754660744, −0.63860757598510477260640501578, −0.54893131291800664605927751306, −0.42759242668684285855792047530, 0.42759242668684285855792047530, 0.54893131291800664605927751306, 0.63860757598510477260640501578, 0.78719022233881782145754660744, 0.941469971165889785601834573679, 1.21466502385630849746773851360, 1.52469581997684026580208538731, 1.69240938387166907994526178753, 1.78117834909953377483252862248, 2.12906626830843853465727323463, 2.13712550154946702973558987740, 2.16087705179541949479471917526, 2.36863978652721378554944774354, 2.48932592373620066336713470820, 2.55678634657549231479934132362, 2.70579551212770429953246702767, 2.81361156951762403440720537421, 2.82928287996642233286160044854, 2.97781075854264258684584369763, 3.29141961486623829100884903608, 3.42438861093988020296998631481, 3.46295834246923675864236430135, 3.50509690590673187321830945410, 3.51378102878526639251273976008, 3.53613934528276098622350321927
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.