Dirichlet series
| L(s) = 1 | − 2.06e11·2-s − 3.04e17·3-s + 2.83e22·4-s − 3.27e25·5-s + 6.26e28·6-s − 2.31e30·7-s − 3.24e33·8-s − 1.76e34·9-s + 6.74e36·10-s − 8.03e37·11-s − 8.61e39·12-s + 1.50e40·13-s + 4.77e41·14-s + 9.94e42·15-s + 3.34e44·16-s − 2.02e45·17-s + 3.64e45·18-s − 4.15e46·19-s − 9.26e47·20-s + 7.04e47·21-s + 1.65e49·22-s + 7.89e49·23-s + 9.86e50·24-s − 5.62e50·25-s − 3.09e51·26-s + 2.45e52·27-s − 6.56e52·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.16·3-s + 3·4-s − 1.00·5-s + 2.48·6-s − 0.330·7-s − 3.53·8-s − 0.261·9-s + 2.13·10-s − 0.783·11-s − 3.50·12-s + 0.329·13-s + 0.700·14-s + 1.17·15-s + 15/4·16-s − 2.47·17-s + 0.554·18-s − 0.878·19-s − 3.01·20-s + 0.386·21-s + 1.66·22-s + 1.56·23-s + 4.13·24-s − 0.531·25-s − 0.699·26-s + 1.39·27-s − 0.990·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(74-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+73/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(6\) |
| Conductor: | \(8\) = \(2^{3}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(307503.\) |
| Root analytic conductor: | \(8.21564\) |
| Motivic weight: | \(73\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(3\) |
| Selberg data: | \((6,\ 8,\ (\ :73/2, 73/2, 73/2),\ -1)\) |
Particular Values
| \(L(37)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{75}{2})\) | 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 | $C_1$ | \( ( 1 + p^{36} T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 11260711148879036 p^{3} T + \)\(20\!\cdots\!17\)\( p^{12} T^{2} + \)\(62\!\cdots\!76\)\( p^{28} T^{3} + \)\(20\!\cdots\!17\)\( p^{85} T^{4} + 11260711148879036 p^{149} T^{5} + p^{219} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + \)\(13\!\cdots\!98\)\( p^{2} T + \)\(41\!\cdots\!39\)\( p^{8} T^{2} + \)\(25\!\cdots\!08\)\( p^{19} T^{3} + \)\(41\!\cdots\!39\)\( p^{81} T^{4} + \)\(13\!\cdots\!98\)\( p^{148} T^{5} + p^{219} T^{6} \) | |
| 7 | $S_4\times C_2$ | \( 1 + \)\(23\!\cdots\!64\)\( T + \)\(10\!\cdots\!71\)\( p^{3} T^{2} + \)\(11\!\cdots\!32\)\( p^{10} T^{3} + \)\(10\!\cdots\!71\)\( p^{76} T^{4} + \)\(23\!\cdots\!64\)\( p^{146} T^{5} + p^{219} T^{6} \) | |
| 11 | $S_4\times C_2$ | \( 1 + \)\(73\!\cdots\!04\)\( p T + \)\(16\!\cdots\!05\)\( p^{4} T^{2} + \)\(66\!\cdots\!20\)\( p^{10} T^{3} + \)\(16\!\cdots\!05\)\( p^{77} T^{4} + \)\(73\!\cdots\!04\)\( p^{147} T^{5} + p^{219} T^{6} \) | |
| 13 | $S_4\times C_2$ | \( 1 - \)\(15\!\cdots\!38\)\( T + \)\(30\!\cdots\!03\)\( p^{2} T^{2} - \)\(11\!\cdots\!96\)\( p^{6} T^{3} + \)\(30\!\cdots\!03\)\( p^{75} T^{4} - \)\(15\!\cdots\!38\)\( p^{146} T^{5} + p^{219} T^{6} \) | |
| 17 | $S_4\times C_2$ | \( 1 + \)\(11\!\cdots\!02\)\( p T + \)\(35\!\cdots\!03\)\( p^{4} T^{2} + \)\(40\!\cdots\!48\)\( p^{8} T^{3} + \)\(35\!\cdots\!03\)\( p^{77} T^{4} + \)\(11\!\cdots\!02\)\( p^{147} T^{5} + p^{219} T^{6} \) | |
| 19 | $S_4\times C_2$ | \( 1 + \)\(41\!\cdots\!80\)\( T + \)\(34\!\cdots\!83\)\( p T^{2} + \)\(14\!\cdots\!40\)\( p^{4} T^{3} + \)\(34\!\cdots\!83\)\( p^{74} T^{4} + \)\(41\!\cdots\!80\)\( p^{146} T^{5} + p^{219} T^{6} \) | |
| 23 | $S_4\times C_2$ | \( 1 - \)\(78\!\cdots\!88\)\( T + \)\(62\!\cdots\!93\)\( p^{2} T^{2} - \)\(25\!\cdots\!08\)\( p^{5} T^{3} + \)\(62\!\cdots\!93\)\( p^{75} T^{4} - \)\(78\!\cdots\!88\)\( p^{146} T^{5} + p^{219} T^{6} \) | |
| 29 | $S_4\times C_2$ | \( 1 - \)\(57\!\cdots\!90\)\( p T + \)\(24\!\cdots\!87\)\( p^{2} T^{2} + \)\(44\!\cdots\!80\)\( p^{5} T^{3} + \)\(24\!\cdots\!87\)\( p^{75} T^{4} - \)\(57\!\cdots\!90\)\( p^{147} T^{5} + p^{219} T^{6} \) | |
| 31 | $S_4\times C_2$ | \( 1 - \)\(95\!\cdots\!96\)\( T + \)\(54\!\cdots\!95\)\( p T^{2} - \)\(23\!\cdots\!40\)\( p^{3} T^{3} + \)\(54\!\cdots\!95\)\( p^{74} T^{4} - \)\(95\!\cdots\!96\)\( p^{146} T^{5} + p^{219} T^{6} \) | |
| 37 | $S_4\times C_2$ | \( 1 - \)\(95\!\cdots\!66\)\( T + \)\(31\!\cdots\!43\)\( T^{2} + \)\(13\!\cdots\!04\)\( p T^{3} + \)\(31\!\cdots\!43\)\( p^{73} T^{4} - \)\(95\!\cdots\!66\)\( p^{146} T^{5} + p^{219} T^{6} \) | |
| 41 | $S_4\times C_2$ | \( 1 - \)\(15\!\cdots\!46\)\( T + \)\(56\!\cdots\!35\)\( p T^{2} - \)\(10\!\cdots\!00\)\( p^{2} T^{3} + \)\(56\!\cdots\!35\)\( p^{74} T^{4} - \)\(15\!\cdots\!46\)\( p^{146} T^{5} + p^{219} T^{6} \) | |
| 43 | $S_4\times C_2$ | \( 1 + \)\(47\!\cdots\!92\)\( T + \)\(31\!\cdots\!19\)\( p T^{2} + \)\(24\!\cdots\!44\)\( p^{2} T^{3} + \)\(31\!\cdots\!19\)\( p^{74} T^{4} + \)\(47\!\cdots\!92\)\( p^{146} T^{5} + p^{219} T^{6} \) | |
| 47 | $S_4\times C_2$ | \( 1 + \)\(12\!\cdots\!64\)\( T + \)\(75\!\cdots\!79\)\( p T^{2} + \)\(12\!\cdots\!92\)\( p^{2} T^{3} + \)\(75\!\cdots\!79\)\( p^{74} T^{4} + \)\(12\!\cdots\!64\)\( p^{146} T^{5} + p^{219} T^{6} \) | |
| 53 | $S_4\times C_2$ | \( 1 - \)\(14\!\cdots\!58\)\( T + \)\(42\!\cdots\!19\)\( p T^{2} - \)\(67\!\cdots\!36\)\( p^{2} T^{3} + \)\(42\!\cdots\!19\)\( p^{74} T^{4} - \)\(14\!\cdots\!58\)\( p^{146} T^{5} + p^{219} T^{6} \) | |
| 59 | $S_4\times C_2$ | \( 1 + \)\(65\!\cdots\!20\)\( p T + \)\(17\!\cdots\!77\)\( p^{2} T^{2} + \)\(70\!\cdots\!60\)\( p^{3} T^{3} + \)\(17\!\cdots\!77\)\( p^{75} T^{4} + \)\(65\!\cdots\!20\)\( p^{147} T^{5} + p^{219} T^{6} \) | |
| 61 | $S_4\times C_2$ | \( 1 + \)\(13\!\cdots\!14\)\( T + \)\(62\!\cdots\!75\)\( p T^{2} + \)\(86\!\cdots\!40\)\( p^{2} T^{3} + \)\(62\!\cdots\!75\)\( p^{74} T^{4} + \)\(13\!\cdots\!14\)\( p^{146} T^{5} + p^{219} T^{6} \) | |
| 67 | $S_4\times C_2$ | \( 1 + \)\(15\!\cdots\!32\)\( p T + \)\(16\!\cdots\!57\)\( p^{2} T^{2} + \)\(12\!\cdots\!96\)\( p^{3} T^{3} + \)\(16\!\cdots\!57\)\( p^{75} T^{4} + \)\(15\!\cdots\!32\)\( p^{147} T^{5} + p^{219} T^{6} \) | |
| 71 | $S_4\times C_2$ | \( 1 + \)\(16\!\cdots\!44\)\( p T + \)\(16\!\cdots\!25\)\( p^{2} T^{2} + \)\(10\!\cdots\!40\)\( p^{3} T^{3} + \)\(16\!\cdots\!25\)\( p^{75} T^{4} + \)\(16\!\cdots\!44\)\( p^{147} T^{5} + p^{219} T^{6} \) | |
| 73 | $S_4\times C_2$ | \( 1 + \)\(15\!\cdots\!82\)\( T + \)\(34\!\cdots\!07\)\( T^{2} + \)\(32\!\cdots\!96\)\( T^{3} + \)\(34\!\cdots\!07\)\( p^{73} T^{4} + \)\(15\!\cdots\!82\)\( p^{146} T^{5} + p^{219} T^{6} \) | |
| 79 | $S_4\times C_2$ | \( 1 + \)\(40\!\cdots\!00\)\( T + \)\(12\!\cdots\!17\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!17\)\( p^{73} T^{4} + \)\(40\!\cdots\!00\)\( p^{146} T^{5} + p^{219} T^{6} \) | |
| 83 | $S_4\times C_2$ | \( 1 - \)\(48\!\cdots\!88\)\( T + \)\(83\!\cdots\!37\)\( T^{2} - \)\(24\!\cdots\!24\)\( T^{3} + \)\(83\!\cdots\!37\)\( p^{73} T^{4} - \)\(48\!\cdots\!88\)\( p^{146} T^{5} + p^{219} T^{6} \) | |
| 89 | $S_4\times C_2$ | \( 1 - \)\(45\!\cdots\!70\)\( T + \)\(11\!\cdots\!07\)\( T^{2} - \)\(20\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!07\)\( p^{73} T^{4} - \)\(45\!\cdots\!70\)\( p^{146} T^{5} + p^{219} T^{6} \) | |
| 97 | $S_4\times C_2$ | \( 1 - \)\(76\!\cdots\!26\)\( T + \)\(30\!\cdots\!23\)\( T^{2} - \)\(42\!\cdots\!92\)\( T^{3} + \)\(30\!\cdots\!23\)\( p^{73} T^{4} - \)\(76\!\cdots\!26\)\( p^{146} T^{5} + p^{219} T^{6} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.58805341947394253725274877344, −11.69924772708517400643708234512, −11.64634428293895705985415551673, −11.30193706570880635085387165995, −10.62888852381961838494519931772, −10.59704288706777926072819972010, −10.07020783405444104817877758402, −8.986834764273305817308949168806, −8.977723660216591352630663012940, −8.631111554541227607800746801140, −7.988691924381378440584050687617, −7.47141372178201152864596624219, −7.14249160472746145017407124556, −6.39119662768870588145050258849, −6.31929144656032340805686992663, −5.82454343625433112359562989855, −5.02155636073421785653005595409, −4.48555450003534516601196077681, −4.08633000240039831521118812534, −3.05185698823116177633704619048, −2.78887457854937774929367771242, −2.47442393605397948668074291459, −1.60866412184828572701398358947, −1.32949285260893221464985841136, −0.63713644765416817125420834111, 0, 0, 0, 0.63713644765416817125420834111, 1.32949285260893221464985841136, 1.60866412184828572701398358947, 2.47442393605397948668074291459, 2.78887457854937774929367771242, 3.05185698823116177633704619048, 4.08633000240039831521118812534, 4.48555450003534516601196077681, 5.02155636073421785653005595409, 5.82454343625433112359562989855, 6.31929144656032340805686992663, 6.39119662768870588145050258849, 7.14249160472746145017407124556, 7.47141372178201152864596624219, 7.988691924381378440584050687617, 8.631111554541227607800746801140, 8.977723660216591352630663012940, 8.986834764273305817308949168806, 10.07020783405444104817877758402, 10.59704288706777926072819972010, 10.62888852381961838494519931772, 11.30193706570880635085387165995, 11.64634428293895705985415551673, 11.69924772708517400643708234512, 12.58805341947394253725274877344