Dirichlet series
| L(s) = 1 | + 64·2-s − 324·3-s + 2.56e3·4-s − 739·5-s − 2.07e4·6-s + 2.41e3·7-s + 8.19e4·8-s + 6.56e4·9-s − 4.72e4·10-s − 4.27e4·11-s − 8.29e5·12-s + 7.70e4·13-s + 1.54e5·14-s + 2.39e5·15-s + 2.29e6·16-s − 3.10e5·17-s + 4.19e6·18-s − 5.21e5·19-s − 1.89e6·20-s − 7.82e5·21-s − 2.73e6·22-s − 7.07e5·23-s − 2.65e7·24-s − 2.42e6·25-s + 4.93e6·26-s − 1.06e7·27-s + 6.18e6·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 2.30·3-s + 5·4-s − 0.528·5-s − 6.53·6-s + 0.380·7-s + 7.07·8-s + 10/3·9-s − 1.49·10-s − 0.881·11-s − 11.5·12-s + 0.748·13-s + 1.07·14-s + 1.22·15-s + 35/4·16-s − 0.900·17-s + 9.42·18-s − 0.917·19-s − 2.64·20-s − 0.877·21-s − 2.49·22-s − 0.527·23-s − 16.3·24-s − 1.24·25-s + 2.11·26-s − 3.84·27-s + 1.90·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 3^{4} \cdot 19^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.18841\times 10^{7}\) |
| Root analytic conductor: | \(7.66251\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{4} \cdot 19^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(21.83254830\) |
| \(L(\frac12)\) | \(\approx\) | \(21.83254830\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | |
|---|---|---|---|
| bad | 2 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) | |
| 19 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 739 T + 2968688 T^{2} + 109382213 p T^{3} + 272853420142 p^{2} T^{4} + 109382213 p^{10} T^{5} + 2968688 p^{18} T^{6} + 739 p^{27} T^{7} + p^{36} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 345 p T + 424432 p^{2} T^{2} - 89570205 p^{3} T^{3} - 176622108546 p^{4} T^{4} - 89570205 p^{12} T^{5} + 424432 p^{20} T^{6} - 345 p^{28} T^{7} + p^{36} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 42793 T + 5700121994 T^{2} + 225107377437601 T^{3} + 19179250208779898410 T^{4} + 225107377437601 p^{9} T^{5} + 5700121994 p^{18} T^{6} + 42793 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 77078 T + 41087573776 T^{2} - 2400727661927714 T^{3} + \)\(64\!\cdots\!74\)\( T^{4} - 2400727661927714 p^{9} T^{5} + 41087573776 p^{18} T^{6} - 77078 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 310011 T + 247653596210 T^{2} + 99437577154000821 T^{3} + \)\(32\!\cdots\!30\)\( T^{4} + 99437577154000821 p^{9} T^{5} + 247653596210 p^{18} T^{6} + 310011 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 707668 T + 3692208150992 T^{2} + 4906858440128498884 T^{3} + \)\(70\!\cdots\!38\)\( T^{4} + 4906858440128498884 p^{9} T^{5} + 3692208150992 p^{18} T^{6} + 707668 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 6554110 T + 41486701826792 T^{2} - \)\(13\!\cdots\!74\)\( T^{3} + \)\(61\!\cdots\!06\)\( T^{4} - \)\(13\!\cdots\!74\)\( p^{9} T^{5} + 41486701826792 p^{18} T^{6} - 6554110 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 10803616 T + 104548535781040 T^{2} - \)\(65\!\cdots\!44\)\( T^{3} + \)\(40\!\cdots\!14\)\( T^{4} - \)\(65\!\cdots\!44\)\( p^{9} T^{5} + 104548535781040 p^{18} T^{6} - 10803616 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 13889584 T + 398826672910048 T^{2} - \)\(13\!\cdots\!92\)\( p T^{3} + \)\(71\!\cdots\!34\)\( T^{4} - \)\(13\!\cdots\!92\)\( p^{10} T^{5} + 398826672910048 p^{18} T^{6} - 13889584 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 14088040 T + 1316938980111104 T^{2} - \)\(13\!\cdots\!20\)\( T^{3} + \)\(64\!\cdots\!46\)\( T^{4} - \)\(13\!\cdots\!20\)\( p^{9} T^{5} + 1316938980111104 p^{18} T^{6} - 14088040 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 4064503 T + 471843526636972 T^{2} - \)\(12\!\cdots\!79\)\( T^{3} + \)\(95\!\cdots\!98\)\( T^{4} - \)\(12\!\cdots\!79\)\( p^{9} T^{5} + 471843526636972 p^{18} T^{6} - 4064503 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 14919731 T + 1262589912704540 T^{2} - \)\(26\!\cdots\!11\)\( T^{3} + \)\(29\!\cdots\!70\)\( T^{4} - \)\(26\!\cdots\!11\)\( p^{9} T^{5} + 1262589912704540 p^{18} T^{6} - 14919731 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 109486706 T + 14708849354187032 T^{2} - \)\(10\!\cdots\!34\)\( T^{3} + \)\(74\!\cdots\!34\)\( T^{4} - \)\(10\!\cdots\!34\)\( p^{9} T^{5} + 14708849354187032 p^{18} T^{6} - 109486706 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 239457264 T + 30280411848065756 T^{2} - \)\(24\!\cdots\!12\)\( T^{3} + \)\(19\!\cdots\!70\)\( T^{4} - \)\(24\!\cdots\!12\)\( p^{9} T^{5} + 30280411848065756 p^{18} T^{6} - 239457264 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 460155485 T + 110165094568714978 T^{2} - \)\(17\!\cdots\!03\)\( T^{3} + \)\(21\!\cdots\!86\)\( T^{4} - \)\(17\!\cdots\!03\)\( p^{9} T^{5} + 110165094568714978 p^{18} T^{6} - 460155485 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 372675648 T + 123273073289029756 T^{2} - \)\(28\!\cdots\!48\)\( T^{3} + \)\(51\!\cdots\!14\)\( T^{4} - \)\(28\!\cdots\!48\)\( p^{9} T^{5} + 123273073289029756 p^{18} T^{6} - 372675648 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 142314272 T + 142668081019179308 T^{2} - \)\(17\!\cdots\!64\)\( T^{3} + \)\(90\!\cdots\!10\)\( T^{4} - \)\(17\!\cdots\!64\)\( p^{9} T^{5} + 142668081019179308 p^{18} T^{6} - 142314272 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 117256851 T + 231122104364989318 T^{2} - \)\(19\!\cdots\!25\)\( T^{3} + \)\(20\!\cdots\!26\)\( T^{4} - \)\(19\!\cdots\!25\)\( p^{9} T^{5} + 231122104364989318 p^{18} T^{6} - 117256851 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 27604034 T + 248589331102757332 T^{2} - \)\(34\!\cdots\!02\)\( T^{3} + \)\(34\!\cdots\!10\)\( T^{4} - \)\(34\!\cdots\!02\)\( p^{9} T^{5} + 248589331102757332 p^{18} T^{6} - 27604034 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 179588886 T + 531812228127016532 T^{2} + \)\(58\!\cdots\!70\)\( T^{3} + \)\(13\!\cdots\!98\)\( T^{4} + \)\(58\!\cdots\!70\)\( p^{9} T^{5} + 531812228127016532 p^{18} T^{6} + 179588886 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 647752302 T + 637168184006679824 T^{2} - \)\(12\!\cdots\!78\)\( T^{3} + \)\(12\!\cdots\!30\)\( T^{4} - \)\(12\!\cdots\!78\)\( p^{9} T^{5} + 637168184006679824 p^{18} T^{6} - 647752302 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 24288428 T + 1954394877480145588 T^{2} - \)\(11\!\cdots\!92\)\( T^{3} + \)\(19\!\cdots\!14\)\( T^{4} - \)\(11\!\cdots\!92\)\( p^{9} T^{5} + 1954394877480145588 p^{18} T^{6} + 24288428 p^{27} T^{7} + p^{36} T^{8} \) | |
| show more | |||
| show less | |||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.014986175026559973174957811267, −7.62551856010946560112284066351, −7.29513745614721510974945656502, −6.99453262481539363594995083442, −6.79852219909604983256374839978, −6.38374687430022876419933490053, −6.16965642649990414139394659978, −5.99887020941468019840395241024, −5.95260611215878373018901183965, −5.23749967670399694380086078708, −5.02304689508346665037884241325, −4.91184525613634325586000732149, −4.87790239002935964137334841409, −4.06460991212002798800887149434, −3.97447057150704649386613913677, −3.82487740560405372028511916912, −3.77663435833465638180167111153, −2.70090870737813537661590261772, −2.48099686128736666011566449320, −2.23877329841675100163344341379, −2.13904489892671712961664849667, −1.11402706289541074739012304971, −1.09310588696533147010388280451, −0.66859311110531784339814499548, −0.39997854969023231889782153071, 0.39997854969023231889782153071, 0.66859311110531784339814499548, 1.09310588696533147010388280451, 1.11402706289541074739012304971, 2.13904489892671712961664849667, 2.23877329841675100163344341379, 2.48099686128736666011566449320, 2.70090870737813537661590261772, 3.77663435833465638180167111153, 3.82487740560405372028511916912, 3.97447057150704649386613913677, 4.06460991212002798800887149434, 4.87790239002935964137334841409, 4.91184525613634325586000732149, 5.02304689508346665037884241325, 5.23749967670399694380086078708, 5.95260611215878373018901183965, 5.99887020941468019840395241024, 6.16965642649990414139394659978, 6.38374687430022876419933490053, 6.79852219909604983256374839978, 6.99453262481539363594995083442, 7.29513745614721510974945656502, 7.62551856010946560112284066351, 8.014986175026559973174957811267