Dirichlet series
| L(s) = 1 | + 256·2-s − 2.91e3·3-s + 4.09e4·4-s − 5.74e4·5-s − 7.46e5·6-s + 1.55e5·7-s + 5.24e6·8-s + 5.31e6·9-s − 1.47e7·10-s − 9.61e6·11-s − 1.19e8·12-s − 5.47e6·13-s + 3.97e7·14-s + 1.67e8·15-s + 5.87e8·16-s + 1.71e8·17-s + 1.36e9·18-s + 1.88e8·19-s − 2.35e9·20-s − 4.52e8·21-s − 2.46e9·22-s − 6.10e8·23-s − 1.52e10·24-s − 1.17e9·25-s − 1.40e9·26-s − 7.74e9·27-s + 6.35e9·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 2.30·3-s + 5·4-s − 1.64·5-s − 6.53·6-s + 0.498·7-s + 7.07·8-s + 10/3·9-s − 4.65·10-s − 1.63·11-s − 11.5·12-s − 0.314·13-s + 1.40·14-s + 3.79·15-s + 35/4·16-s + 1.72·17-s + 9.42·18-s + 0.917·19-s − 8.22·20-s − 1.15·21-s − 4.62·22-s − 0.859·23-s − 16.3·24-s − 0.959·25-s − 0.890·26-s − 3.84·27-s + 2.49·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(14-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+13/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 3^{4} \cdot 19^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.23305\times 10^{8}\) |
| Root analytic conductor: | \(11.0563\) |
| Motivic weight: | \(13\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{4} \cdot 19^{4} ,\ ( \ : 13/2, 13/2, 13/2, 13/2 ),\ 1 )\) |
Particular Values
| \(L(7)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{15}{2})\) | 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 | $C_1$ | \( ( 1 - p^{6} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 + p^{6} T )^{4} \) | |
| 19 | $C_1$ | \( ( 1 - p^{6} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 57444 T + 894104149 p T^{2} + 6693371075538 p^{2} T^{3} + 12148205602128792 p^{4} T^{4} + 6693371075538 p^{15} T^{5} + 894104149 p^{27} T^{6} + 57444 p^{39} T^{7} + p^{52} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 155142 T + 248765664797 T^{2} - 37851262071519546 T^{3} + \)\(41\!\cdots\!76\)\( p T^{4} - 37851262071519546 p^{13} T^{5} + 248765664797 p^{26} T^{6} - 155142 p^{39} T^{7} + p^{52} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 9611582 T + 12911262981687 p T^{2} + 7206154191797691762 p^{2} T^{3} + \)\(54\!\cdots\!92\)\( p^{3} T^{4} + 7206154191797691762 p^{15} T^{5} + 12911262981687 p^{27} T^{6} + 9611582 p^{39} T^{7} + p^{52} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 421514 p T + 895962309375248 T^{2} + \)\(37\!\cdots\!26\)\( p T^{3} + \)\(37\!\cdots\!54\)\( T^{4} + \)\(37\!\cdots\!26\)\( p^{14} T^{5} + 895962309375248 p^{26} T^{6} + 421514 p^{40} T^{7} + p^{52} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 171325678 T + 28230436437173313 T^{2} - \)\(33\!\cdots\!06\)\( T^{3} + \)\(40\!\cdots\!24\)\( T^{4} - \)\(33\!\cdots\!06\)\( p^{13} T^{5} + 28230436437173313 p^{26} T^{6} - 171325678 p^{39} T^{7} + p^{52} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 610385522 T + 1404561515274558300 T^{2} + \)\(73\!\cdots\!14\)\( T^{3} + \)\(10\!\cdots\!62\)\( T^{4} + \)\(73\!\cdots\!14\)\( p^{13} T^{5} + 1404561515274558300 p^{26} T^{6} + 610385522 p^{39} T^{7} + p^{52} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 2688902784 T + 14836439166516243116 T^{2} - \)\(40\!\cdots\!20\)\( T^{3} + \)\(22\!\cdots\!14\)\( T^{4} - \)\(40\!\cdots\!20\)\( p^{13} T^{5} + 14836439166516243116 p^{26} T^{6} - 2688902784 p^{39} T^{7} + p^{52} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 3232968622 T + 2506085409024872528 p T^{2} - \)\(22\!\cdots\!18\)\( T^{3} + \)\(25\!\cdots\!82\)\( T^{4} - \)\(22\!\cdots\!18\)\( p^{13} T^{5} + 2506085409024872528 p^{27} T^{6} - 3232968622 p^{39} T^{7} + p^{52} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 14825828778 T + \)\(64\!\cdots\!64\)\( T^{2} - \)\(63\!\cdots\!78\)\( T^{3} + \)\(19\!\cdots\!62\)\( T^{4} - \)\(63\!\cdots\!78\)\( p^{13} T^{5} + \)\(64\!\cdots\!64\)\( p^{26} T^{6} - 14825828778 p^{39} T^{7} + p^{52} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 21810935816 T + \)\(16\!\cdots\!64\)\( T^{2} - \)\(21\!\cdots\!92\)\( T^{3} + \)\(46\!\cdots\!06\)\( T^{4} - \)\(21\!\cdots\!92\)\( p^{13} T^{5} + \)\(16\!\cdots\!64\)\( p^{26} T^{6} + 21810935816 p^{39} T^{7} + p^{52} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 21901984594 T + \)\(39\!\cdots\!21\)\( T^{2} - \)\(10\!\cdots\!14\)\( T^{3} + \)\(89\!\cdots\!88\)\( T^{4} - \)\(10\!\cdots\!14\)\( p^{13} T^{5} + \)\(39\!\cdots\!21\)\( p^{26} T^{6} - 21901984594 p^{39} T^{7} + p^{52} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 17015241308 T + \)\(65\!\cdots\!21\)\( T^{2} + \)\(25\!\cdots\!22\)\( T^{3} + \)\(12\!\cdots\!16\)\( T^{4} + \)\(25\!\cdots\!22\)\( p^{13} T^{5} + \)\(65\!\cdots\!21\)\( p^{26} T^{6} + 17015241308 p^{39} T^{7} + p^{52} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 142153081216 T + \)\(82\!\cdots\!88\)\( T^{2} + \)\(94\!\cdots\!60\)\( T^{3} + \)\(30\!\cdots\!46\)\( T^{4} + \)\(94\!\cdots\!60\)\( p^{13} T^{5} + \)\(82\!\cdots\!88\)\( p^{26} T^{6} + 142153081216 p^{39} T^{7} + p^{52} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 1278325136 T + \)\(17\!\cdots\!32\)\( T^{2} - \)\(55\!\cdots\!44\)\( T^{3} + \)\(12\!\cdots\!02\)\( T^{4} - \)\(55\!\cdots\!44\)\( p^{13} T^{5} + \)\(17\!\cdots\!32\)\( p^{26} T^{6} + 1278325136 p^{39} T^{7} + p^{52} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 190142306854 T + \)\(53\!\cdots\!09\)\( T^{2} + \)\(77\!\cdots\!30\)\( T^{3} + \)\(12\!\cdots\!12\)\( T^{4} + \)\(77\!\cdots\!30\)\( p^{13} T^{5} + \)\(53\!\cdots\!09\)\( p^{26} T^{6} + 190142306854 p^{39} T^{7} + p^{52} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 35259143324 T + \)\(16\!\cdots\!16\)\( T^{2} + \)\(16\!\cdots\!96\)\( T^{3} + \)\(46\!\cdots\!06\)\( T^{4} + \)\(16\!\cdots\!96\)\( p^{13} T^{5} + \)\(16\!\cdots\!16\)\( p^{26} T^{6} + 35259143324 p^{39} T^{7} + p^{52} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 334864926160 T + \)\(24\!\cdots\!72\)\( T^{2} + \)\(15\!\cdots\!40\)\( T^{3} + \)\(33\!\cdots\!42\)\( T^{4} + \)\(15\!\cdots\!40\)\( p^{13} T^{5} + \)\(24\!\cdots\!72\)\( p^{26} T^{6} + 334864926160 p^{39} T^{7} + p^{52} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 1713440585934 T + \)\(52\!\cdots\!61\)\( T^{2} + \)\(53\!\cdots\!06\)\( T^{3} + \)\(10\!\cdots\!92\)\( T^{4} + \)\(53\!\cdots\!06\)\( p^{13} T^{5} + \)\(52\!\cdots\!61\)\( p^{26} T^{6} + 1713440585934 p^{39} T^{7} + p^{52} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 4422429367218 T + \)\(15\!\cdots\!80\)\( T^{2} + \)\(41\!\cdots\!02\)\( T^{3} + \)\(93\!\cdots\!50\)\( T^{4} + \)\(41\!\cdots\!02\)\( p^{13} T^{5} + \)\(15\!\cdots\!80\)\( p^{26} T^{6} + 4422429367218 p^{39} T^{7} + p^{52} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 6950294502760 T + \)\(41\!\cdots\!20\)\( T^{2} + \)\(14\!\cdots\!76\)\( T^{3} + \)\(50\!\cdots\!42\)\( T^{4} + \)\(14\!\cdots\!76\)\( p^{13} T^{5} + \)\(41\!\cdots\!20\)\( p^{26} T^{6} + 6950294502760 p^{39} T^{7} + p^{52} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 5787805692368 T + \)\(50\!\cdots\!44\)\( T^{2} + \)\(16\!\cdots\!28\)\( T^{3} + \)\(11\!\cdots\!90\)\( T^{4} + \)\(16\!\cdots\!28\)\( p^{13} T^{5} + \)\(50\!\cdots\!44\)\( p^{26} T^{6} + 5787805692368 p^{39} T^{7} + p^{52} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 35161422084052 T + \)\(64\!\cdots\!96\)\( T^{2} + \)\(78\!\cdots\!68\)\( T^{3} + \)\(73\!\cdots\!46\)\( T^{4} + \)\(78\!\cdots\!68\)\( p^{13} T^{5} + \)\(64\!\cdots\!96\)\( p^{26} T^{6} + 35161422084052 p^{39} T^{7} + p^{52} T^{8} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.898702120894019552507821771027, −7.53521722521927384773499943125, −7.18340385652531526853548851973, −7.07665107195752794567494946986, −7.02056578655713572071017439602, −6.18467636407457147307389863560, −6.14683547526793713404045120132, −5.90684612614640324726190469670, −5.80451452103514753309468976683, −5.27442260959895390272149337967, −5.07758212028744826599628538450, −4.99629638440616004267547912468, −4.85474201324406274701728774552, −4.16328976202102977966875162625, −4.11935359588519952413163592212, −4.00315558118885099326549577377, −3.70531864749515899232579501119, −3.18471448388488317821871869175, −2.77775793263874924310608498561, −2.60381231735861356960155930044, −2.57805558595953512259283924163, −1.52060771981106036759989399254, −1.44722284033311216679595894113, −1.31305894399112658563009841477, −1.11044361037725981529281116961, 0, 0, 0, 0, 1.11044361037725981529281116961, 1.31305894399112658563009841477, 1.44722284033311216679595894113, 1.52060771981106036759989399254, 2.57805558595953512259283924163, 2.60381231735861356960155930044, 2.77775793263874924310608498561, 3.18471448388488317821871869175, 3.70531864749515899232579501119, 4.00315558118885099326549577377, 4.11935359588519952413163592212, 4.16328976202102977966875162625, 4.85474201324406274701728774552, 4.99629638440616004267547912468, 5.07758212028744826599628538450, 5.27442260959895390272149337967, 5.80451452103514753309468976683, 5.90684612614640324726190469670, 6.14683547526793713404045120132, 6.18467636407457147307389863560, 7.02056578655713572071017439602, 7.07665107195752794567494946986, 7.18340385652531526853548851973, 7.53521722521927384773499943125, 7.898702120894019552507821771027