Dirichlet series
| L(s) = 1 | − 3-s − 5-s + 3·7-s + 9·9-s + 6·11-s − 4·13-s + 15-s + 6·17-s − 2·19-s − 3·21-s + 5·23-s + 15·25-s − 16·27-s − 30·29-s − 8·31-s − 6·33-s − 3·35-s + 16·37-s + 4·39-s + 8·41-s + 48·43-s − 9·45-s + 47-s + 6·49-s − 6·51-s + 32·53-s − 6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.13·7-s + 3·9-s + 1.80·11-s − 1.10·13-s + 0.258·15-s + 1.45·17-s − 0.458·19-s − 0.654·21-s + 1.04·23-s + 3·25-s − 3.07·27-s − 5.57·29-s − 1.43·31-s − 1.04·33-s − 0.507·35-s + 2.63·37-s + 0.640·39-s + 1.24·41-s + 7.31·43-s − 1.34·45-s + 0.145·47-s + 6/7·49-s − 0.840·51-s + 4.39·53-s − 0.809·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{12} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{12} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{36} \cdot 7^{12} \cdot 17^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.72364\times 10^{10}\) |
| Root analytic conductor: | \(2.75712\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{36} \cdot 7^{12} \cdot 17^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(10.61148371\) |
| \(L(\frac12)\) | \(\approx\) | \(10.61148371\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 3 T + 3 T^{2} - 27 T^{3} + 24 T^{4} + 12 T^{5} + 412 T^{6} + 12 p T^{7} + 24 p^{2} T^{8} - 27 p^{3} T^{9} + 3 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) | |
| 17 | \( ( 1 - T + T^{2} )^{6} \) | |
| good | 3 | \( 1 + T - 8 T^{2} - T^{3} + 31 T^{4} - 22 T^{5} - 25 p T^{6} + 44 T^{7} + 167 T^{8} + 151 T^{9} - 770 T^{10} - 413 T^{11} + 3088 T^{12} - 413 p T^{13} - 770 p^{2} T^{14} + 151 p^{3} T^{15} + 167 p^{4} T^{16} + 44 p^{5} T^{17} - 25 p^{7} T^{18} - 22 p^{7} T^{19} + 31 p^{8} T^{20} - p^{9} T^{21} - 8 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 + T - 14 T^{2} - 11 T^{3} + 4 p^{2} T^{4} + 77 T^{5} - 369 T^{6} - 114 p T^{7} + 66 T^{8} + 3148 T^{9} + 7458 T^{10} - 7426 T^{11} - 50654 T^{12} - 7426 p T^{13} + 7458 p^{2} T^{14} + 3148 p^{3} T^{15} + 66 p^{4} T^{16} - 114 p^{6} T^{17} - 369 p^{6} T^{18} + 77 p^{7} T^{19} + 4 p^{10} T^{20} - 11 p^{9} T^{21} - 14 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 6 T - 20 T^{2} + 134 T^{3} + 420 T^{4} - 1770 T^{5} - 7791 T^{6} + 18136 T^{7} + 103828 T^{8} - 136770 T^{9} - 1116376 T^{10} + 577308 T^{11} + 11392176 T^{12} + 577308 p T^{13} - 1116376 p^{2} T^{14} - 136770 p^{3} T^{15} + 103828 p^{4} T^{16} + 18136 p^{5} T^{17} - 7791 p^{6} T^{18} - 1770 p^{7} T^{19} + 420 p^{8} T^{20} + 134 p^{9} T^{21} - 20 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( ( 1 + 2 T + 4 p T^{2} + 83 T^{3} + 1336 T^{4} + 1862 T^{5} + 21608 T^{6} + 1862 p T^{7} + 1336 p^{2} T^{8} + 83 p^{3} T^{9} + 4 p^{5} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( 1 + 2 T - 50 T^{2} - 180 T^{3} + 1206 T^{4} + 7002 T^{5} - 508 p T^{6} - 176350 T^{7} - 343566 T^{8} + 2983932 T^{9} + 17152794 T^{10} - 23321014 T^{11} - 409748074 T^{12} - 23321014 p T^{13} + 17152794 p^{2} T^{14} + 2983932 p^{3} T^{15} - 343566 p^{4} T^{16} - 176350 p^{5} T^{17} - 508 p^{7} T^{18} + 7002 p^{7} T^{19} + 1206 p^{8} T^{20} - 180 p^{9} T^{21} - 50 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 5 T - 95 T^{2} + 430 T^{3} + 5434 T^{4} - 20200 T^{5} - 230589 T^{6} + 593265 T^{7} + 8146332 T^{8} - 11463155 T^{9} - 242812683 T^{10} + 4385860 p T^{11} + 6110969188 T^{12} + 4385860 p^{2} T^{13} - 242812683 p^{2} T^{14} - 11463155 p^{3} T^{15} + 8146332 p^{4} T^{16} + 593265 p^{5} T^{17} - 230589 p^{6} T^{18} - 20200 p^{7} T^{19} + 5434 p^{8} T^{20} + 430 p^{9} T^{21} - 95 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( ( 1 + 15 T + 219 T^{2} + 1988 T^{3} + 16973 T^{4} + 109573 T^{5} + 665822 T^{6} + 109573 p T^{7} + 16973 p^{2} T^{8} + 1988 p^{3} T^{9} + 219 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 31 | \( 1 + 8 T - 21 T^{2} - 46 T^{3} + 1234 T^{4} - 2438 T^{5} - 8334 T^{6} - 112844 T^{7} - 1966941 T^{8} + 3079576 T^{9} + 29778441 T^{10} - 91027054 T^{11} - 310270845 T^{12} - 91027054 p T^{13} + 29778441 p^{2} T^{14} + 3079576 p^{3} T^{15} - 1966941 p^{4} T^{16} - 112844 p^{5} T^{17} - 8334 p^{6} T^{18} - 2438 p^{7} T^{19} + 1234 p^{8} T^{20} - 46 p^{9} T^{21} - 21 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 16 T + 15 T^{2} + 1018 T^{3} - 85 p T^{4} - 39312 T^{5} + 136327 T^{6} + 1193837 T^{7} - 2595129 T^{8} - 40892987 T^{9} + 89797697 T^{10} + 897572325 T^{11} - 6642966758 T^{12} + 897572325 p T^{13} + 89797697 p^{2} T^{14} - 40892987 p^{3} T^{15} - 2595129 p^{4} T^{16} + 1193837 p^{5} T^{17} + 136327 p^{6} T^{18} - 39312 p^{7} T^{19} - 85 p^{9} T^{20} + 1018 p^{9} T^{21} + 15 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( ( 1 - 4 T + 141 T^{2} + 13 T^{3} + 6809 T^{4} + 30719 T^{5} + 232690 T^{6} + 30719 p T^{7} + 6809 p^{2} T^{8} + 13 p^{3} T^{9} + 141 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 43 | \( ( 1 - 24 T + 392 T^{2} - 4744 T^{3} + 47200 T^{4} - 389136 T^{5} + 2769910 T^{6} - 389136 p T^{7} + 47200 p^{2} T^{8} - 4744 p^{3} T^{9} + 392 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 - T - 120 T^{2} - 669 T^{3} + 8944 T^{4} + 92563 T^{5} - 3264 p T^{6} - 7843281 T^{7} - 22020624 T^{8} + 360933383 T^{9} + 2938662232 T^{10} - 8036307045 T^{11} - 169329181826 T^{12} - 8036307045 p T^{13} + 2938662232 p^{2} T^{14} + 360933383 p^{3} T^{15} - 22020624 p^{4} T^{16} - 7843281 p^{5} T^{17} - 3264 p^{7} T^{18} + 92563 p^{7} T^{19} + 8944 p^{8} T^{20} - 669 p^{9} T^{21} - 120 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 32 T + 396 T^{2} - 2354 T^{3} + 10328 T^{4} - 98016 T^{5} + 576081 T^{6} + 3229428 T^{7} - 47648196 T^{8} - 29970810 T^{9} + 2903220448 T^{10} - 17780965700 T^{11} + 77591351084 T^{12} - 17780965700 p T^{13} + 2903220448 p^{2} T^{14} - 29970810 p^{3} T^{15} - 47648196 p^{4} T^{16} + 3229428 p^{5} T^{17} + 576081 p^{6} T^{18} - 98016 p^{7} T^{19} + 10328 p^{8} T^{20} - 2354 p^{9} T^{21} + 396 p^{10} T^{22} - 32 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 6 T - 209 T^{2} + 968 T^{3} + 23707 T^{4} - 76172 T^{5} - 2097561 T^{6} + 5935095 T^{7} + 152313997 T^{8} - 349734165 T^{9} - 10271601527 T^{10} + 139296535 p T^{11} + 11081565504 p T^{12} + 139296535 p^{2} T^{13} - 10271601527 p^{2} T^{14} - 349734165 p^{3} T^{15} + 152313997 p^{4} T^{16} + 5935095 p^{5} T^{17} - 2097561 p^{6} T^{18} - 76172 p^{7} T^{19} + 23707 p^{8} T^{20} + 968 p^{9} T^{21} - 209 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 10 T - 77 T^{2} - 1420 T^{3} - 5586 T^{4} + 14162 T^{5} + 344458 T^{6} + 4266644 T^{7} + 37330677 T^{8} + 35550014 T^{9} - 2001218363 T^{10} - 10217596212 T^{11} - 5179684075 T^{12} - 10217596212 p T^{13} - 2001218363 p^{2} T^{14} + 35550014 p^{3} T^{15} + 37330677 p^{4} T^{16} + 4266644 p^{5} T^{17} + 344458 p^{6} T^{18} + 14162 p^{7} T^{19} - 5586 p^{8} T^{20} - 1420 p^{9} T^{21} - 77 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 27 T + 176 T^{2} - 15 T^{3} + 14580 T^{4} + 158387 T^{5} - 1389979 T^{6} - 14056700 T^{7} + 51725028 T^{8} - 87203642 T^{9} - 9974451028 T^{10} - 10263959440 T^{11} + 539523429068 T^{12} - 10263959440 p T^{13} - 9974451028 p^{2} T^{14} - 87203642 p^{3} T^{15} + 51725028 p^{4} T^{16} - 14056700 p^{5} T^{17} - 1389979 p^{6} T^{18} + 158387 p^{7} T^{19} + 14580 p^{8} T^{20} - 15 p^{9} T^{21} + 176 p^{10} T^{22} + 27 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( ( 1 + 12 T + 401 T^{2} + 3752 T^{3} + 68542 T^{4} + 501020 T^{5} + 6402797 T^{6} + 501020 p T^{7} + 68542 p^{2} T^{8} + 3752 p^{3} T^{9} + 401 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 73 | \( 1 + 15 T - 133 T^{2} - 2862 T^{3} + 7635 T^{4} + 235343 T^{5} - 865921 T^{6} - 12501131 T^{7} + 140052390 T^{8} + 347155666 T^{9} - 18535046137 T^{10} + 528205637 T^{11} + 1723163792699 T^{12} + 528205637 p T^{13} - 18535046137 p^{2} T^{14} + 347155666 p^{3} T^{15} + 140052390 p^{4} T^{16} - 12501131 p^{5} T^{17} - 865921 p^{6} T^{18} + 235343 p^{7} T^{19} + 7635 p^{8} T^{20} - 2862 p^{9} T^{21} - 133 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 21 T - 114 T^{2} + 3303 T^{3} + 37293 T^{4} - 494214 T^{5} - 6516373 T^{6} + 48997440 T^{7} + 871936011 T^{8} - 3274327377 T^{9} - 92978802450 T^{10} + 92593185489 T^{11} + 8252667493116 T^{12} + 92593185489 p T^{13} - 92978802450 p^{2} T^{14} - 3274327377 p^{3} T^{15} + 871936011 p^{4} T^{16} + 48997440 p^{5} T^{17} - 6516373 p^{6} T^{18} - 494214 p^{7} T^{19} + 37293 p^{8} T^{20} + 3303 p^{9} T^{21} - 114 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( ( 1 + 20 T + 489 T^{2} + 6117 T^{3} + 91479 T^{4} + 879945 T^{5} + 9837342 T^{6} + 879945 p T^{7} + 91479 p^{2} T^{8} + 6117 p^{3} T^{9} + 489 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 89 | \( 1 - 33 T + 228 T^{2} + 2763 T^{3} - 9014 T^{4} - 704901 T^{5} + 5162806 T^{6} + 2385789 T^{7} + 432182474 T^{8} - 4102809699 T^{9} - 30671282872 T^{10} - 293595560679 T^{11} + 10596889073042 T^{12} - 293595560679 p T^{13} - 30671282872 p^{2} T^{14} - 4102809699 p^{3} T^{15} + 432182474 p^{4} T^{16} + 2385789 p^{5} T^{17} + 5162806 p^{6} T^{18} - 704901 p^{7} T^{19} - 9014 p^{8} T^{20} + 2763 p^{9} T^{21} + 228 p^{10} T^{22} - 33 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( ( 1 + 9 T + 430 T^{2} + 2940 T^{3} + 82765 T^{4} + 454215 T^{5} + 9821344 T^{6} + 454215 p T^{7} + 82765 p^{2} T^{8} + 2940 p^{3} T^{9} + 430 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
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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
−3.26902100995378665439882702664, −3.26239944702805467516507266794, −2.95730196567661144124430864001, −2.91636061502607868946642260350, −2.84313257722502932580384551044, −2.70563653844097251972201180949, −2.48812211949100397201095433068, −2.39653109381948132085982251542, −2.27515060097215314380782484239, −2.26438146784422909114032311264, −2.24882346042668803998549825463, −2.15787723687262319289660578474, −2.12634771903526808017828454593, −1.93183039339880972826295090291, −1.46774813832205506567659017899, −1.39694629458540188444161227433, −1.36867342216342568529262968399, −1.30325260158676225893708563472, −1.29722822343005428944554550801, −1.25862600626418184365933588698, −1.09130746549026195178049142002, −0.819983301912439871191497744037, −0.66445145490562832473971621429, −0.44267403686974045941366001186, −0.20886288698742158653416748317, 0.20886288698742158653416748317, 0.44267403686974045941366001186, 0.66445145490562832473971621429, 0.819983301912439871191497744037, 1.09130746549026195178049142002, 1.25862600626418184365933588698, 1.29722822343005428944554550801, 1.30325260158676225893708563472, 1.36867342216342568529262968399, 1.39694629458540188444161227433, 1.46774813832205506567659017899, 1.93183039339880972826295090291, 2.12634771903526808017828454593, 2.15787723687262319289660578474, 2.24882346042668803998549825463, 2.26438146784422909114032311264, 2.27515060097215314380782484239, 2.39653109381948132085982251542, 2.48812211949100397201095433068, 2.70563653844097251972201180949, 2.84313257722502932580384551044, 2.91636061502607868946642260350, 2.95730196567661144124430864001, 3.26239944702805467516507266794, 3.26902100995378665439882702664
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.