Dirichlet series
| L(s) = 1 | − 375·2-s − 2.62e4·3-s − 3.46e4·4-s − 1.54e5·5-s + 9.84e6·6-s − 2.30e7·7-s + 4.56e7·8-s + 4.30e8·9-s + 5.78e7·10-s − 1.45e9·11-s + 9.09e8·12-s + 1.79e9·13-s + 8.64e9·14-s + 4.04e9·15-s − 7.84e9·16-s + 6.85e10·17-s − 1.61e11·18-s − 2.57e9·19-s + 5.34e9·20-s + 6.05e11·21-s + 5.44e11·22-s + 6.49e11·23-s − 1.19e12·24-s − 1.86e12·25-s − 6.72e11·26-s − 5.64e12·27-s + 7.98e11·28-s + ⋯ |
| L(s) = 1 | − 1.03·2-s − 2.30·3-s − 0.264·4-s − 0.176·5-s + 2.39·6-s − 1.51·7-s + 0.963·8-s + 10/3·9-s + 0.182·10-s − 2.04·11-s + 0.610·12-s + 0.609·13-s + 1.56·14-s + 0.407·15-s − 0.456·16-s + 2.38·17-s − 3.45·18-s − 0.0348·19-s + 0.0466·20-s + 3.49·21-s + 2.11·22-s + 1.72·23-s − 2.22·24-s − 2.44·25-s − 0.631·26-s − 3.84·27-s + 0.399·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+17/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(194481\) = \(3^{4} \cdot 7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.19173\times 10^{6}\) |
| Root analytic conductor: | \(6.20295\) |
| Motivic weight: | \(17\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 194481,\ (\ :17/2, 17/2, 17/2, 17/2),\ 1)\) |
Particular Values
| \(L(9)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{19}{2})\) | 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 | 3 | $C_1$ | \( ( 1 + p^{8} T )^{4} \) |
| 7 | $C_1$ | \( ( 1 + p^{8} T )^{4} \) | |
| good | 2 | $C_2 \wr S_4$ | \( 1 + 375 T + 21909 p^{3} T^{2} + 1031883 p^{5} T^{3} + 17891023 p^{9} T^{4} + 1031883 p^{22} T^{5} + 21909 p^{37} T^{6} + 375 p^{51} T^{7} + p^{68} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 30828 p T + 75675184968 p^{2} T^{2} + 1238168756504964 p^{4} T^{3} + \)\(11\!\cdots\!78\)\( p^{6} T^{4} + 1238168756504964 p^{21} T^{5} + 75675184968 p^{36} T^{6} + 30828 p^{52} T^{7} + p^{68} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 1452022884 T + 2395275844663622496 T^{2} + \)\(18\!\cdots\!64\)\( p T^{3} + \)\(15\!\cdots\!90\)\( p^{2} T^{4} + \)\(18\!\cdots\!64\)\( p^{18} T^{5} + 2395275844663622496 p^{34} T^{6} + 1452022884 p^{51} T^{7} + p^{68} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 137966080 p T + 160314533419768588 p^{2} T^{2} - \)\(20\!\cdots\!56\)\( p^{2} T^{3} + \)\(14\!\cdots\!66\)\( p^{3} T^{4} - \)\(20\!\cdots\!56\)\( p^{19} T^{5} + 160314533419768588 p^{36} T^{6} - 137966080 p^{52} T^{7} + p^{68} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 68581449948 T + \)\(23\!\cdots\!76\)\( p T^{2} - \)\(16\!\cdots\!36\)\( T^{3} + \)\(51\!\cdots\!38\)\( T^{4} - \)\(16\!\cdots\!36\)\( p^{17} T^{5} + \)\(23\!\cdots\!76\)\( p^{35} T^{6} - 68581449948 p^{51} T^{7} + p^{68} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 2578000592 T + \)\(96\!\cdots\!44\)\( T^{2} + \)\(46\!\cdots\!52\)\( T^{3} + \)\(51\!\cdots\!50\)\( T^{4} + \)\(46\!\cdots\!52\)\( p^{17} T^{5} + \)\(96\!\cdots\!44\)\( p^{34} T^{6} + 2578000592 p^{51} T^{7} + p^{68} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 649239533556 T + \)\(58\!\cdots\!52\)\( T^{2} - \)\(24\!\cdots\!72\)\( T^{3} + \)\(12\!\cdots\!42\)\( T^{4} - \)\(24\!\cdots\!72\)\( p^{17} T^{5} + \)\(58\!\cdots\!52\)\( p^{34} T^{6} - 649239533556 p^{51} T^{7} + p^{68} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 3258162962760 T + \)\(14\!\cdots\!80\)\( T^{2} + \)\(35\!\cdots\!56\)\( T^{3} + \)\(13\!\cdots\!38\)\( T^{4} + \)\(35\!\cdots\!56\)\( p^{17} T^{5} + \)\(14\!\cdots\!80\)\( p^{34} T^{6} + 3258162962760 p^{51} T^{7} + p^{68} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 2414795507136 T + \)\(35\!\cdots\!60\)\( T^{2} + \)\(41\!\cdots\!84\)\( T^{3} + \)\(60\!\cdots\!62\)\( T^{4} + \)\(41\!\cdots\!84\)\( p^{17} T^{5} + \)\(35\!\cdots\!60\)\( p^{34} T^{6} - 2414795507136 p^{51} T^{7} + p^{68} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 23164217924208 T + \)\(17\!\cdots\!40\)\( T^{2} - \)\(27\!\cdots\!92\)\( T^{3} + \)\(11\!\cdots\!10\)\( T^{4} - \)\(27\!\cdots\!92\)\( p^{17} T^{5} + \)\(17\!\cdots\!40\)\( p^{34} T^{6} - 23164217924208 p^{51} T^{7} + p^{68} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 117516076237164 T + \)\(11\!\cdots\!12\)\( T^{2} + \)\(79\!\cdots\!72\)\( T^{3} + \)\(45\!\cdots\!22\)\( T^{4} + \)\(79\!\cdots\!72\)\( p^{17} T^{5} + \)\(11\!\cdots\!12\)\( p^{34} T^{6} + 117516076237164 p^{51} T^{7} + p^{68} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 175144919320720 T + \)\(16\!\cdots\!12\)\( T^{2} + \)\(62\!\cdots\!76\)\( T^{3} + \)\(23\!\cdots\!02\)\( T^{4} + \)\(62\!\cdots\!76\)\( p^{17} T^{5} + \)\(16\!\cdots\!12\)\( p^{34} T^{6} + 175144919320720 p^{51} T^{7} + p^{68} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 392013343869480 T + \)\(79\!\cdots\!12\)\( T^{2} + \)\(13\!\cdots\!52\)\( T^{3} + \)\(25\!\cdots\!94\)\( T^{4} + \)\(13\!\cdots\!52\)\( p^{17} T^{5} + \)\(79\!\cdots\!12\)\( p^{34} T^{6} + 392013343869480 p^{51} T^{7} + p^{68} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 86395731614976 T + \)\(18\!\cdots\!64\)\( T^{2} - \)\(42\!\cdots\!00\)\( T^{3} + \)\(38\!\cdots\!06\)\( T^{4} - \)\(42\!\cdots\!00\)\( p^{17} T^{5} + \)\(18\!\cdots\!64\)\( p^{34} T^{6} + 86395731614976 p^{51} T^{7} + p^{68} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 23145424510920 T + \)\(31\!\cdots\!76\)\( T^{2} + \)\(76\!\cdots\!96\)\( T^{3} + \)\(48\!\cdots\!66\)\( T^{4} + \)\(76\!\cdots\!96\)\( p^{17} T^{5} + \)\(31\!\cdots\!76\)\( p^{34} T^{6} + 23145424510920 p^{51} T^{7} + p^{68} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 3424273979460952 T + \)\(10\!\cdots\!64\)\( T^{2} - \)\(19\!\cdots\!48\)\( T^{3} + \)\(34\!\cdots\!30\)\( T^{4} - \)\(19\!\cdots\!48\)\( p^{17} T^{5} + \)\(10\!\cdots\!64\)\( p^{34} T^{6} - 3424273979460952 p^{51} T^{7} + p^{68} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 33345190329832 p T + \)\(44\!\cdots\!04\)\( T^{2} - \)\(72\!\cdots\!64\)\( T^{3} + \)\(72\!\cdots\!50\)\( T^{4} - \)\(72\!\cdots\!64\)\( p^{17} T^{5} + \)\(44\!\cdots\!04\)\( p^{34} T^{6} - 33345190329832 p^{52} T^{7} + p^{68} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 2815776345321804 T + \)\(17\!\cdots\!64\)\( T^{2} - \)\(83\!\cdots\!76\)\( T^{3} + \)\(17\!\cdots\!70\)\( T^{4} - \)\(83\!\cdots\!76\)\( p^{17} T^{5} + \)\(17\!\cdots\!64\)\( p^{34} T^{6} - 2815776345321804 p^{51} T^{7} + p^{68} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 11330038422344784 T + \)\(93\!\cdots\!48\)\( T^{2} - \)\(43\!\cdots\!36\)\( T^{3} + \)\(26\!\cdots\!42\)\( T^{4} - \)\(43\!\cdots\!36\)\( p^{17} T^{5} + \)\(93\!\cdots\!48\)\( p^{34} T^{6} - 11330038422344784 p^{51} T^{7} + p^{68} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 12177038540666920 T + \)\(30\!\cdots\!76\)\( T^{2} + \)\(36\!\cdots\!08\)\( T^{3} + \)\(93\!\cdots\!86\)\( T^{4} + \)\(36\!\cdots\!08\)\( p^{17} T^{5} + \)\(30\!\cdots\!76\)\( p^{34} T^{6} + 12177038540666920 p^{51} T^{7} + p^{68} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 4389820874951568 T + \)\(94\!\cdots\!60\)\( T^{2} + \)\(38\!\cdots\!04\)\( T^{3} + \)\(57\!\cdots\!38\)\( T^{4} + \)\(38\!\cdots\!04\)\( p^{17} T^{5} + \)\(94\!\cdots\!60\)\( p^{34} T^{6} + 4389820874951568 p^{51} T^{7} + p^{68} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 100131520699583508 T + \)\(79\!\cdots\!44\)\( T^{2} - \)\(41\!\cdots\!24\)\( T^{3} + \)\(17\!\cdots\!50\)\( T^{4} - \)\(41\!\cdots\!24\)\( p^{17} T^{5} + \)\(79\!\cdots\!44\)\( p^{34} T^{6} - 100131520699583508 p^{51} T^{7} + p^{68} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 44947325255962464 T + \)\(19\!\cdots\!56\)\( T^{2} + \)\(70\!\cdots\!68\)\( T^{3} + \)\(16\!\cdots\!58\)\( T^{4} + \)\(70\!\cdots\!68\)\( p^{17} T^{5} + \)\(19\!\cdots\!56\)\( p^{34} T^{6} + 44947325255962464 p^{51} T^{7} + p^{68} T^{8} \) | |
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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
−10.42641771881263027874147244944, −9.897370088182859982142101051056, −9.692143425138789535744797493617, −9.634798638527570035089172752469, −9.415986201442231249603234916865, −8.237082293150908483505702558697, −8.218035956531980095727388905303, −8.112586412605847298387746225309, −7.46334315329519631459398096519, −6.95908474465170954115912332364, −6.94160379091003956903370026138, −6.25724490299920207359578789281, −6.02217542619462670575774887212, −5.56673738989229546085716359193, −5.17655224976275332039407212588, −5.06924994791029224587490035401, −4.86071467367298850383028452415, −3.71792204388266309472039066096, −3.60222866695541655438669479605, −3.58327328404935522210927624079, −2.68337861231909245995073417418, −2.25258325960527525242664499118, −1.41684220854300002647452356824, −1.21158503784375269817710950920, −1.01083736104521162868872424268, 0, 0, 0, 0, 1.01083736104521162868872424268, 1.21158503784375269817710950920, 1.41684220854300002647452356824, 2.25258325960527525242664499118, 2.68337861231909245995073417418, 3.58327328404935522210927624079, 3.60222866695541655438669479605, 3.71792204388266309472039066096, 4.86071467367298850383028452415, 5.06924994791029224587490035401, 5.17655224976275332039407212588, 5.56673738989229546085716359193, 6.02217542619462670575774887212, 6.25724490299920207359578789281, 6.94160379091003956903370026138, 6.95908474465170954115912332364, 7.46334315329519631459398096519, 8.112586412605847298387746225309, 8.218035956531980095727388905303, 8.237082293150908483505702558697, 9.415986201442231249603234916865, 9.634798638527570035089172752469, 9.692143425138789535744797493617, 9.897370088182859982142101051056, 10.42641771881263027874147244944