Dirichlet series
| L(s) = 1 | + 2-s − 4·5-s − 9·8-s − 22·9-s − 4·10-s − 24·13-s − 13·16-s − 4·17-s − 22·18-s + 98·25-s − 24·26-s − 72·29-s + 16·32-s − 4·34-s − 28·37-s + 36·40-s + 40·41-s + 88·45-s + 98·50-s − 92·53-s − 72·58-s − 164·61-s − 11·64-s + 96·65-s + 198·72-s − 132·73-s − 28·74-s + ⋯ |
| L(s) = 1 | + 1/2·2-s − 4/5·5-s − 9/8·8-s − 2.44·9-s − 2/5·10-s − 1.84·13-s − 0.812·16-s − 0.235·17-s − 1.22·18-s + 3.91·25-s − 0.923·26-s − 2.48·29-s + 1/2·32-s − 0.117·34-s − 0.756·37-s + 9/10·40-s + 0.975·41-s + 1.95·45-s + 1.95·50-s − 1.73·53-s − 1.24·58-s − 2.68·61-s − 0.171·64-s + 1.47·65-s + 11/4·72-s − 1.80·73-s − 0.378·74-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 7^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.38385\times 10^{8}\) |
| Root analytic conductor: | \(2.31097\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 7^{24} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.1487285403\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1487285403\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - T + T^{2} + p^{3} T^{3} - p^{2} T^{4} - p^{4} T^{5} + p^{7} T^{6} - p^{6} T^{7} - p^{6} T^{8} + p^{9} T^{9} + p^{8} T^{10} - p^{10} T^{11} + p^{12} T^{12} \) |
| 7 | \( 1 \) | |
| good | 3 | \( 1 + 22 T^{2} + 55 p T^{4} + 214 p T^{6} + 5786 T^{8} + 24398 T^{10} - 191939 T^{12} + 24398 p^{4} T^{14} + 5786 p^{8} T^{16} + 214 p^{13} T^{18} + 55 p^{17} T^{20} + 22 p^{20} T^{22} + p^{24} T^{24} \) |
| 5 | \( ( 1 + 2 T - 43 T^{2} - 34 p T^{3} + 154 p T^{4} + 2754 T^{5} - 9051 T^{6} + 2754 p^{2} T^{7} + 154 p^{5} T^{8} - 34 p^{7} T^{9} - 43 p^{8} T^{10} + 2 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 11 | \( 1 + 214 T^{2} - 5883 T^{4} - 1568254 T^{6} + 562937882 T^{8} + 399372222 p^{2} T^{10} - 189748115 p^{4} T^{12} + 399372222 p^{6} T^{14} + 562937882 p^{8} T^{16} - 1568254 p^{12} T^{18} - 5883 p^{16} T^{20} + 214 p^{20} T^{22} + p^{24} T^{24} \) | |
| 13 | \( ( 1 + 6 T + 11 p T^{2} + 316 T^{3} + 11 p^{3} T^{4} + 6 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
| 17 | \( ( 1 + 2 T - 747 T^{2} + 166 T^{3} + 345626 T^{4} - 199446 T^{5} - 114609011 T^{6} - 199446 p^{2} T^{7} + 345626 p^{4} T^{8} + 166 p^{6} T^{9} - 747 p^{8} T^{10} + 2 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 19 | \( 1 + 566 T^{2} - 38427 T^{4} - 44781470 T^{6} + 11825982362 T^{8} + 1146477008622 T^{10} - 2432024847236867 T^{12} + 1146477008622 p^{4} T^{14} + 11825982362 p^{8} T^{16} - 44781470 p^{12} T^{18} - 38427 p^{16} T^{20} + 566 p^{20} T^{22} + p^{24} T^{24} \) | |
| 23 | \( 1 + 2246 T^{2} + 2652597 T^{4} + 2233126834 T^{6} + 1526684099162 T^{8} + 923399067573438 T^{10} + 511264171862123245 T^{12} + 923399067573438 p^{4} T^{14} + 1526684099162 p^{8} T^{16} + 2233126834 p^{12} T^{18} + 2652597 p^{16} T^{20} + 2246 p^{20} T^{22} + p^{24} T^{24} \) | |
| 29 | \( ( 1 + 18 T + 1575 T^{2} + 35228 T^{3} + 1575 p^{2} T^{4} + 18 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
| 31 | \( 1 + 3718 T^{2} + 7253397 T^{4} + 9359563442 T^{6} + 9087192843866 T^{8} + 7551892717667070 T^{10} + 6682560385371321613 T^{12} + 7551892717667070 p^{4} T^{14} + 9087192843866 p^{8} T^{16} + 9359563442 p^{12} T^{18} + 7253397 p^{16} T^{20} + 3718 p^{20} T^{22} + p^{24} T^{24} \) | |
| 37 | \( ( 1 + 14 T - 2611 T^{2} - 46022 T^{3} + 3439226 T^{4} + 38386950 T^{5} - 4202699019 T^{6} + 38386950 p^{2} T^{7} + 3439226 p^{4} T^{8} - 46022 p^{6} T^{9} - 2611 p^{8} T^{10} + 14 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 41 | \( ( 1 - 10 T + 3647 T^{2} - 18252 T^{3} + 3647 p^{2} T^{4} - 10 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
| 43 | \( ( 1 - 6998 T^{2} + 25469023 T^{4} - 57500194868 T^{6} + 25469023 p^{4} T^{8} - 6998 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 47 | \( 1 + 7814 T^{2} + 26826837 T^{4} + 83638562866 T^{6} + 286477553787482 T^{8} + 722286368171723262 T^{10} + \)\(15\!\cdots\!65\)\( T^{12} + 722286368171723262 p^{4} T^{14} + 286477553787482 p^{8} T^{16} + 83638562866 p^{12} T^{18} + 26826837 p^{16} T^{20} + 7814 p^{20} T^{22} + p^{24} T^{24} \) | |
| 53 | \( ( 1 + 46 T - 6227 T^{2} - 116006 T^{3} + 34675130 T^{4} + 291746278 T^{5} - 105443791915 T^{6} + 291746278 p^{2} T^{7} + 34675130 p^{4} T^{8} - 116006 p^{6} T^{9} - 6227 p^{8} T^{10} + 46 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 59 | \( 1 + 15670 T^{2} + 128488133 T^{4} + 802302449954 T^{6} + 4198724008338842 T^{8} + 18251742692672878446 T^{10} + \)\(67\!\cdots\!57\)\( T^{12} + 18251742692672878446 p^{4} T^{14} + 4198724008338842 p^{8} T^{16} + 802302449954 p^{12} T^{18} + 128488133 p^{16} T^{20} + 15670 p^{20} T^{22} + p^{24} T^{24} \) | |
| 61 | \( ( 1 + 82 T - 5019 T^{2} - 242458 T^{3} + 44781602 T^{4} + 1046675826 T^{5} - 139939679627 T^{6} + 1046675826 p^{2} T^{7} + 44781602 p^{4} T^{8} - 242458 p^{6} T^{9} - 5019 p^{8} T^{10} + 82 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 67 | \( 1 + 6870 T^{2} - 57353 p T^{4} - 189758463166 T^{6} - 279743014976230 T^{8} + 2448928913075902798 T^{10} + \)\(17\!\cdots\!25\)\( T^{12} + 2448928913075902798 p^{4} T^{14} - 279743014976230 p^{8} T^{16} - 189758463166 p^{12} T^{18} - 57353 p^{17} T^{20} + 6870 p^{20} T^{22} + p^{24} T^{24} \) | |
| 71 | \( ( 1 - 25318 T^{2} + 288430255 T^{4} - 1875969873364 T^{6} + 288430255 p^{4} T^{8} - 25318 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 73 | \( ( 1 + 66 T - 11291 T^{2} - 366778 T^{3} + 115262906 T^{4} + 1816577770 T^{5} - 636089874979 T^{6} + 1816577770 p^{2} T^{7} + 115262906 p^{4} T^{8} - 366778 p^{6} T^{9} - 11291 p^{8} T^{10} + 66 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 79 | \( 1 + 31494 T^{2} + 545651157 T^{4} + 6859757092018 T^{6} + 68456322522879066 T^{8} + \)\(55\!\cdots\!38\)\( T^{10} + \)\(37\!\cdots\!81\)\( T^{12} + \)\(55\!\cdots\!38\)\( p^{4} T^{14} + 68456322522879066 p^{8} T^{16} + 6859757092018 p^{12} T^{18} + 545651157 p^{16} T^{20} + 31494 p^{20} T^{22} + p^{24} T^{24} \) | |
| 83 | \( ( 1 - 15734 T^{2} + 12104959 T^{4} + 672850782796 T^{6} + 12104959 p^{4} T^{8} - 15734 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 89 | \( ( 1 - 174 T + 5317 T^{2} - 71626 T^{3} + 53653562 T^{4} + 4579593658 T^{5} - 1279148987395 T^{6} + 4579593658 p^{2} T^{7} + 53653562 p^{4} T^{8} - 71626 p^{6} T^{9} + 5317 p^{8} T^{10} - 174 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 97 | \( ( 1 + 126 T + 29295 T^{2} + 2174884 T^{3} + 29295 p^{2} T^{4} + 126 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
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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
−4.04041910026584859634134816601, −3.87234206792481639975825366758, −3.83736759478703208251734075247, −3.69051768984352026688504400370, −3.54803300798335900030899205303, −3.45723138965212881566370065257, −3.39830541055806213066296113559, −3.24252984289398177213146770128, −3.01903053753895238866667472216, −2.96814130925843878979888738953, −2.71843518718973713286411233715, −2.71484182420210674034671325541, −2.62495754107741372528709234994, −2.62127922394830348919698206798, −2.48452151951877950719591663791, −2.11153477445241696341886495085, −1.98756610177178629955229592967, −1.85953888828360039959162501390, −1.58069092327661275841665264058, −1.29161015802956793776793078658, −1.25012588911083003842807652993, −0.903104502286298490203799695991, −0.54831601448311635429837092677, −0.21292756149006271899920101026, −0.10728883242624134993046558173, 0.10728883242624134993046558173, 0.21292756149006271899920101026, 0.54831601448311635429837092677, 0.903104502286298490203799695991, 1.25012588911083003842807652993, 1.29161015802956793776793078658, 1.58069092327661275841665264058, 1.85953888828360039959162501390, 1.98756610177178629955229592967, 2.11153477445241696341886495085, 2.48452151951877950719591663791, 2.62127922394830348919698206798, 2.62495754107741372528709234994, 2.71484182420210674034671325541, 2.71843518718973713286411233715, 2.96814130925843878979888738953, 3.01903053753895238866667472216, 3.24252984289398177213146770128, 3.39830541055806213066296113559, 3.45723138965212881566370065257, 3.54803300798335900030899205303, 3.69051768984352026688504400370, 3.83736759478703208251734075247, 3.87234206792481639975825366758, 4.04041910026584859634134816601
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.