Dirichlet series
| L(s) = 1 | + 2-s + 2·4-s + 2·5-s − 3·8-s + 2·10-s − 6·13-s + 16-s + 20·17-s + 4·20-s + 77·25-s − 6·26-s − 22·29-s + 2·32-s + 20·34-s + 108·37-s − 6·40-s + 92·41-s − 114·49-s + 77·50-s − 12·52-s − 232·53-s − 22·58-s − 54·61-s − 67·64-s − 12·65-s + 40·68-s − 156·73-s + ⋯ |
| L(s) = 1 | + 1/2·2-s + 1/2·4-s + 2/5·5-s − 3/8·8-s + 1/5·10-s − 0.461·13-s + 1/16·16-s + 1.17·17-s + 1/5·20-s + 3.07·25-s − 0.230·26-s − 0.758·29-s + 1/16·32-s + 0.588·34-s + 2.91·37-s − 0.149·40-s + 2.24·41-s − 2.32·49-s + 1.53·50-s − 0.230·52-s − 4.37·53-s − 0.379·58-s − 0.885·61-s − 1.04·64-s − 0.184·65-s + 0.588·68-s − 2.13·73-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 3^{48}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.24161\times 10^{11}\) |
| Root analytic conductor: | \(2.97125\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 3^{48} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.04647488866\) |
| \(L(\frac12)\) | \(\approx\) | \(0.04647488866\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - T - T^{2} + 3 p T^{3} - p^{3} T^{4} - p^{3} T^{5} + 7 p^{4} T^{6} - p^{5} T^{7} - p^{7} T^{8} + 3 p^{7} T^{9} - p^{8} T^{10} - p^{10} T^{11} + p^{12} T^{12} \) |
| 3 | \( 1 \) | |
| good | 5 | \( ( 1 - T - 37 T^{2} + 36 p T^{3} + 77 p T^{4} - 2867 T^{5} + 6406 T^{6} - 2867 p^{2} T^{7} + 77 p^{5} T^{8} + 36 p^{7} T^{9} - 37 p^{8} T^{10} - p^{10} T^{11} + p^{12} T^{12} )^{2} \) |
| 7 | \( 1 + 114 T^{2} + 771 p T^{4} + 56438 T^{6} - 6661734 T^{8} - 425700582 T^{10} - 21918600819 T^{12} - 425700582 p^{4} T^{14} - 6661734 p^{8} T^{16} + 56438 p^{12} T^{18} + 771 p^{17} T^{20} + 114 p^{20} T^{22} + p^{24} T^{24} \) | |
| 11 | \( 1 + 210 T^{2} + 6309 T^{4} - 1603594 T^{6} - 126807270 T^{8} - 7112331078 T^{10} - 1334567686275 T^{12} - 7112331078 p^{4} T^{14} - 126807270 p^{8} T^{16} - 1603594 p^{12} T^{18} + 6309 p^{16} T^{20} + 210 p^{20} T^{22} + p^{24} T^{24} \) | |
| 13 | \( ( 1 + 3 T - 333 T^{2} + 980 T^{3} + 59097 T^{4} - 253239 T^{5} - 10369818 T^{6} - 253239 p^{2} T^{7} + 59097 p^{4} T^{8} + 980 p^{6} T^{9} - 333 p^{8} T^{10} + 3 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 17 | \( ( 1 - 5 T + 710 T^{2} - 2657 T^{3} + 710 p^{2} T^{4} - 5 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
| 19 | \( ( 1 - 690 T^{2} + 175551 T^{4} - 20841212 T^{6} + 175551 p^{4} T^{8} - 690 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 23 | \( 1 + 306 T^{2} - 294315 T^{4} - 101578186 T^{6} + 8236830234 T^{8} + 252722985690 T^{10} + 6510830262749901 T^{12} + 252722985690 p^{4} T^{14} + 8236830234 p^{8} T^{16} - 101578186 p^{12} T^{18} - 294315 p^{16} T^{20} + 306 p^{20} T^{22} + p^{24} T^{24} \) | |
| 29 | \( ( 1 + 11 T - 541 T^{2} + 35316 T^{3} + 35689 T^{4} - 13747487 T^{5} + 931978726 T^{6} - 13747487 p^{2} T^{7} + 35689 p^{4} T^{8} + 35316 p^{6} T^{9} - 541 p^{8} T^{10} + 11 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 31 | \( 1 + 390 T^{2} - 211563 T^{4} - 3351258190 T^{6} - 884751558054 T^{8} + 705817384212990 T^{10} + 4373878248652847373 T^{12} + 705817384212990 p^{4} T^{14} - 884751558054 p^{8} T^{16} - 3351258190 p^{12} T^{18} - 211563 p^{16} T^{20} + 390 p^{20} T^{22} + p^{24} T^{24} \) | |
| 37 | \( ( 1 - 27 T + 78 T^{2} + 70681 T^{3} + 78 p^{2} T^{4} - 27 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
| 41 | \( ( 1 - 46 T - 1915 T^{2} + 79926 T^{3} + 3369754 T^{4} - 1009750 p T^{5} - 3682739 p^{2} T^{6} - 1009750 p^{3} T^{7} + 3369754 p^{4} T^{8} + 79926 p^{6} T^{9} - 1915 p^{8} T^{10} - 46 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 43 | \( 1 + 4722 T^{2} + 9868965 T^{4} + 6464505302 T^{6} - 11323018434534 T^{8} - 11664390218838 p^{2} T^{10} - 7894504504179 p^{4} T^{12} - 11664390218838 p^{6} T^{14} - 11323018434534 p^{8} T^{16} + 6464505302 p^{12} T^{18} + 9868965 p^{16} T^{20} + 4722 p^{20} T^{22} + p^{24} T^{24} \) | |
| 47 | \( 1 + 5046 T^{2} + 15715797 T^{4} + 19923312194 T^{6} - 26382629404518 T^{8} - 217240801372523922 T^{10} - \)\(62\!\cdots\!35\)\( T^{12} - 217240801372523922 p^{4} T^{14} - 26382629404518 p^{8} T^{16} + 19923312194 p^{12} T^{18} + 15715797 p^{16} T^{20} + 5046 p^{20} T^{22} + p^{24} T^{24} \) | |
| 53 | \( ( 1 + 58 T + 8567 T^{2} + 320716 T^{3} + 8567 p^{2} T^{4} + 58 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
| 59 | \( 1 + 7494 T^{2} + 23269317 T^{4} + 14290664786 T^{6} - 186774179935398 T^{8} - 1054613632918700418 T^{10} - \)\(43\!\cdots\!15\)\( T^{12} - 1054613632918700418 p^{4} T^{14} - 186774179935398 p^{8} T^{16} + 14290664786 p^{12} T^{18} + 23269317 p^{16} T^{20} + 7494 p^{20} T^{22} + p^{24} T^{24} \) | |
| 61 | \( ( 1 + 27 T - 5613 T^{2} - 28252 T^{3} + 13929321 T^{4} - 267915615 T^{5} - 46766410362 T^{6} - 267915615 p^{2} T^{7} + 13929321 p^{4} T^{8} - 28252 p^{6} T^{9} - 5613 p^{8} T^{10} + 27 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 67 | \( 1 + 12786 T^{2} + 67742277 T^{4} + 199597066262 T^{6} + 686986653965466 T^{8} + 4900908833643326682 T^{10} + \)\(27\!\cdots\!01\)\( T^{12} + 4900908833643326682 p^{4} T^{14} + 686986653965466 p^{8} T^{16} + 199597066262 p^{12} T^{18} + 67742277 p^{16} T^{20} + 12786 p^{20} T^{22} + p^{24} T^{24} \) | |
| 71 | \( ( 1 - 21714 T^{2} + 225392559 T^{4} - 1424221410044 T^{6} + 225392559 p^{4} T^{8} - 21714 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 73 | \( ( 1 + 39 T + 7086 T^{2} - 32189 T^{3} + 7086 p^{2} T^{4} + 39 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
| 79 | \( 1 + 29682 T^{2} + 478317045 T^{4} + 5552299732022 T^{6} + 51491498910113946 T^{8} + \)\(40\!\cdots\!78\)\( T^{10} + \)\(27\!\cdots\!81\)\( T^{12} + \)\(40\!\cdots\!78\)\( p^{4} T^{14} + 51491498910113946 p^{8} T^{16} + 5552299732022 p^{12} T^{18} + 478317045 p^{16} T^{20} + 29682 p^{20} T^{22} + p^{24} T^{24} \) | |
| 83 | \( 1 + 6438 T^{2} - 50128155 T^{4} - 217903183822 T^{6} + 1440876227339226 T^{8} - 4486982814512549538 T^{10} - \)\(11\!\cdots\!39\)\( T^{12} - 4486982814512549538 p^{4} T^{14} + 1440876227339226 p^{8} T^{16} - 217903183822 p^{12} T^{18} - 50128155 p^{16} T^{20} + 6438 p^{20} T^{22} + p^{24} T^{24} \) | |
| 89 | \( ( 1 - 125 T + 19286 T^{2} - 1872473 T^{3} + 19286 p^{2} T^{4} - 125 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
| 97 | \( ( 1 - 102 T - 13323 T^{2} + 680174 T^{3} + 181606458 T^{4} - 268269918 T^{5} - 2149095910131 T^{6} - 268269918 p^{2} T^{7} + 181606458 p^{4} T^{8} + 680174 p^{6} T^{9} - 13323 p^{8} T^{10} - 102 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
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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
−3.48389549776101379338759064972, −3.44406308773068595190708125800, −3.41712546629884416637601065328, −3.26356869833129403931175475176, −3.24226506328250683745324420240, −3.19639043704215978629381936050, −3.04472548610937927951439817680, −3.00234042130811057591528672656, −2.67086312485450852165375643143, −2.52855259991543319987058553361, −2.47275236567736160436918655319, −2.42694053300384852348994950769, −2.39840201765888921482566479590, −2.11945943851689825718430254664, −2.10217957478837978210832754921, −1.66584185086688782797132152978, −1.62487422570878887240032525536, −1.47731365879822312960422143281, −1.32796393825092792372964507763, −1.26801998830538892209240012513, −1.05143984768758679868936125512, −0.65634448172124427371109602450, −0.63844003524280464224281280942, −0.57474437267353539388077003098, −0.01048425907924287681321207820, 0.01048425907924287681321207820, 0.57474437267353539388077003098, 0.63844003524280464224281280942, 0.65634448172124427371109602450, 1.05143984768758679868936125512, 1.26801998830538892209240012513, 1.32796393825092792372964507763, 1.47731365879822312960422143281, 1.62487422570878887240032525536, 1.66584185086688782797132152978, 2.10217957478837978210832754921, 2.11945943851689825718430254664, 2.39840201765888921482566479590, 2.42694053300384852348994950769, 2.47275236567736160436918655319, 2.52855259991543319987058553361, 2.67086312485450852165375643143, 3.00234042130811057591528672656, 3.04472548610937927951439817680, 3.19639043704215978629381936050, 3.24226506328250683745324420240, 3.26356869833129403931175475176, 3.41712546629884416637601065328, 3.44406308773068595190708125800, 3.48389549776101379338759064972
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.