Dirichlet series
| L(s) = 1 | + 2·2-s + 6·3-s + 12·6-s + 25·9-s + 30·11-s + 8·16-s − 30·17-s + 50·18-s − 78·19-s + 60·22-s − 121·25-s + 62·27-s + 32·32-s + 180·33-s − 60·34-s − 156·38-s + 232·41-s − 200·43-s + 48·48-s − 242·50-s − 180·51-s + 124·54-s − 468·57-s + 110·59-s + 16·64-s + 360·66-s + 434·67-s + ⋯ |
| L(s) = 1 | + 2-s + 2·3-s + 2·6-s + 25/9·9-s + 2.72·11-s + 1/2·16-s − 1.76·17-s + 25/9·18-s − 4.10·19-s + 2.72·22-s − 4.83·25-s + 2.29·27-s + 32-s + 5.45·33-s − 1.76·34-s − 4.10·38-s + 5.65·41-s − 4.65·43-s + 48-s − 4.83·50-s − 3.52·51-s + 2.29·54-s − 8.21·57-s + 1.86·59-s + 1/4·64-s + 5.45·66-s + 6.47·67-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{36} \cdot 7^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.20522\times 10^{12}\) |
| Root analytic conductor: | \(3.26821\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{36} \cdot 7^{24} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(5.281433873\) |
| \(L(\frac12)\) | \(\approx\) | \(5.281433873\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{3} T^{4} - p^{5} T^{5} + 5 p^{4} T^{6} - p^{7} T^{7} + p^{7} T^{8} - p^{9} T^{9} + p^{10} T^{10} - p^{11} T^{11} + p^{12} T^{12} \) |
| 7 | \( 1 \) | |
| good | 3 | \( ( 1 - p T + T^{2} + 14 T^{3} - 65 T^{4} + 37 T^{5} + 514 T^{6} + 37 p^{2} T^{7} - 65 p^{4} T^{8} + 14 p^{6} T^{9} + p^{8} T^{10} - p^{11} T^{11} + p^{12} T^{12} )^{2} \) |
| 5 | \( 1 + 121 T^{2} + 7979 T^{4} + 380588 T^{6} + 14474897 T^{8} + 457025259 T^{10} + 12295779174 T^{12} + 457025259 p^{4} T^{14} + 14474897 p^{8} T^{16} + 380588 p^{12} T^{18} + 7979 p^{16} T^{20} + 121 p^{20} T^{22} + p^{24} T^{24} \) | |
| 11 | \( ( 1 - 15 T - 129 T^{2} + 84 p T^{3} + 35727 T^{4} - 38013 T^{5} - 5198650 T^{6} - 38013 p^{2} T^{7} + 35727 p^{4} T^{8} + 84 p^{7} T^{9} - 129 p^{8} T^{10} - 15 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 13 | \( ( 1 - 86 T^{2} + 60895 T^{4} - 4902788 T^{6} + 60895 p^{4} T^{8} - 86 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 17 | \( ( 1 + 15 T - 665 T^{2} - 3920 T^{3} + 23845 p T^{4} + 1221665 T^{5} - 124870762 T^{6} + 1221665 p^{2} T^{7} + 23845 p^{5} T^{8} - 3920 p^{6} T^{9} - 665 p^{8} T^{10} + 15 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 19 | \( ( 1 + 39 T + 151 T^{2} - 3964 T^{3} + 263147 T^{4} + 5820949 T^{5} + 41830430 T^{6} + 5820949 p^{2} T^{7} + 263147 p^{4} T^{8} - 3964 p^{6} T^{9} + 151 p^{8} T^{10} + 39 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 23 | \( 1 + 693 T^{2} - 15581 T^{4} - 315659500 T^{6} - 121343121715 T^{8} + 25399919636503 T^{10} + 46081428374834798 T^{12} + 25399919636503 p^{4} T^{14} - 121343121715 p^{8} T^{16} - 315659500 p^{12} T^{18} - 15581 p^{16} T^{20} + 693 p^{20} T^{22} + p^{24} T^{24} \) | |
| 29 | \( ( 1 - 3662 T^{2} + 6471151 T^{4} - 6858243380 T^{6} + 6471151 p^{4} T^{8} - 3662 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 31 | \( 1 + 3561 T^{2} + 5838879 T^{4} + 8280992264 T^{6} + 11856779149293 T^{8} + 13054196823637455 T^{10} + 12193464685853753718 T^{12} + 13054196823637455 p^{4} T^{14} + 11856779149293 p^{8} T^{16} + 8280992264 p^{12} T^{18} + 5838879 p^{16} T^{20} + 3561 p^{20} T^{22} + p^{24} T^{24} \) | |
| 37 | \( 1 + 5785 T^{2} + 17889827 T^{4} + 38047414052 T^{6} + 62842022068961 T^{8} + 88030006321881747 T^{10} + \)\(11\!\cdots\!14\)\( T^{12} + 88030006321881747 p^{4} T^{14} + 62842022068961 p^{8} T^{16} + 38047414052 p^{12} T^{18} + 17889827 p^{16} T^{20} + 5785 p^{20} T^{22} + p^{24} T^{24} \) | |
| 41 | \( ( 1 - 58 T + 3139 T^{2} - 88960 T^{3} + 3139 p^{2} T^{4} - 58 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
| 43 | \( ( 1 + 50 T + 4047 T^{2} + 107900 T^{3} + 4047 p^{2} T^{4} + 50 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
| 47 | \( 1 + 3905 T^{2} + 6537087 T^{4} + 2079696952 T^{6} - 28977286497379 T^{8} - 99900087904130553 T^{10} - \)\(24\!\cdots\!46\)\( T^{12} - 99900087904130553 p^{4} T^{14} - 28977286497379 p^{8} T^{16} + 2079696952 p^{12} T^{18} + 6537087 p^{16} T^{20} + 3905 p^{20} T^{22} + p^{24} T^{24} \) | |
| 53 | \( 1 + 10561 T^{2} + 59733731 T^{4} + 209884512884 T^{6} + 494098524300977 T^{8} + 757507316642438811 T^{10} + \)\(12\!\cdots\!10\)\( T^{12} + 757507316642438811 p^{4} T^{14} + 494098524300977 p^{8} T^{16} + 209884512884 p^{12} T^{18} + 59733731 p^{16} T^{20} + 10561 p^{20} T^{22} + p^{24} T^{24} \) | |
| 59 | \( ( 1 - 55 T - 7367 T^{2} + 181142 T^{3} + 51119807 T^{4} - 598293727 T^{5} - 186579818926 T^{6} - 598293727 p^{2} T^{7} + 51119807 p^{4} T^{8} + 181142 p^{6} T^{9} - 7367 p^{8} T^{10} - 55 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 61 | \( 1 + 9201 T^{2} + 56941411 T^{4} + 154999649300 T^{6} - 75695642240335 T^{8} - 3911957905117164149 T^{10} - \)\(19\!\cdots\!14\)\( T^{12} - 3911957905117164149 p^{4} T^{14} - 75695642240335 p^{8} T^{16} + 154999649300 p^{12} T^{18} + 56941411 p^{16} T^{20} + 9201 p^{20} T^{22} + p^{24} T^{24} \) | |
| 67 | \( ( 1 - 217 T + 18053 T^{2} - 1666970 T^{3} + 209974835 T^{4} - 15078221277 T^{5} + 815515698066 T^{6} - 15078221277 p^{2} T^{7} + 209974835 p^{4} T^{8} - 1666970 p^{6} T^{9} + 18053 p^{8} T^{10} - 217 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 71 | \( ( 1 - 23062 T^{2} + 245626031 T^{4} - 1559141837940 T^{6} + 245626031 p^{4} T^{8} - 23062 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 73 | \( ( 1 + 51 T - 9165 T^{2} - 579660 T^{3} + 45654729 T^{4} + 1993134921 T^{5} - 160997144218 T^{6} + 1993134921 p^{2} T^{7} + 45654729 p^{4} T^{8} - 579660 p^{6} T^{9} - 9165 p^{8} T^{10} + 51 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 79 | \( 1 + 16693 T^{2} + 149251283 T^{4} + 640487711012 T^{6} - 978609215845699 T^{8} - 44625107831251066425 T^{10} - \)\(37\!\cdots\!74\)\( T^{12} - 44625107831251066425 p^{4} T^{14} - 978609215845699 p^{8} T^{16} + 640487711012 p^{12} T^{18} + 149251283 p^{16} T^{20} + 16693 p^{20} T^{22} + p^{24} T^{24} \) | |
| 83 | \( ( 1 - 134 T + 22583 T^{2} - 1661172 T^{3} + 22583 p^{2} T^{4} - 134 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
| 89 | \( ( 1 + 107 T + 1395 T^{2} - 497084 T^{3} - 62702935 T^{4} - 2422278015 T^{5} + 93422604838 T^{6} - 2422278015 p^{2} T^{7} - 62702935 p^{4} T^{8} - 497084 p^{6} T^{9} + 1395 p^{8} T^{10} + 107 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 97 | \( ( 1 - 38 T + 25191 T^{2} - 597344 T^{3} + 25191 p^{2} T^{4} - 38 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
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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
−3.60200047011411358575192801801, −3.55669884443040275769153141062, −3.42950558381249588808895329904, −3.30495663046000457454151834557, −3.28729843869003149060095528458, −2.89943258167116643089374700676, −2.81502567479588521232887815195, −2.70248540661583144048292182329, −2.56959022304154785556906304290, −2.53201058509185907802973865612, −2.26173099050145286406535832937, −2.11619680820326200646658251052, −2.08572570824559889981556375703, −2.08427014658395446894495394240, −1.92872444509069939457528347960, −1.90670107170903811493783003213, −1.78987662684924012620950189231, −1.54302782920484941221545471472, −1.52009368061662089569858036566, −1.02400597019138794760947404753, −0.981323193860938873469372587055, −0.72515083703637123421572838954, −0.60392361461948579892764315472, −0.46561418950275031136336016445, −0.07279213038137324653108910375, 0.07279213038137324653108910375, 0.46561418950275031136336016445, 0.60392361461948579892764315472, 0.72515083703637123421572838954, 0.981323193860938873469372587055, 1.02400597019138794760947404753, 1.52009368061662089569858036566, 1.54302782920484941221545471472, 1.78987662684924012620950189231, 1.90670107170903811493783003213, 1.92872444509069939457528347960, 2.08427014658395446894495394240, 2.08572570824559889981556375703, 2.11619680820326200646658251052, 2.26173099050145286406535832937, 2.53201058509185907802973865612, 2.56959022304154785556906304290, 2.70248540661583144048292182329, 2.81502567479588521232887815195, 2.89943258167116643089374700676, 3.28729843869003149060095528458, 3.30495663046000457454151834557, 3.42950558381249588808895329904, 3.55669884443040275769153141062, 3.60200047011411358575192801801
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.