Dirichlet series
| L(s) = 1 | − 8.58e9·2-s + 1.14e15·3-s + 5.53e19·4-s − 7.47e22·5-s − 9.87e24·6-s + 2.70e27·7-s − 3.16e29·8-s − 1.17e31·9-s + 6.42e32·10-s + 7.81e33·11-s + 6.35e34·12-s + 2.87e35·13-s − 2.32e37·14-s − 8.59e37·15-s + 1.70e39·16-s + 3.84e39·17-s + 1.00e41·18-s − 1.39e41·19-s − 4.13e42·20-s + 3.11e42·21-s − 6.71e43·22-s + 1.42e44·23-s − 3.64e44·24-s + 2.59e45·25-s − 2.47e45·26-s − 1.66e46·27-s + 1.49e47·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.358·3-s + 3/2·4-s − 1.43·5-s − 0.506·6-s + 0.926·7-s − 1.41·8-s − 1.14·9-s + 2.03·10-s + 1.11·11-s + 0.537·12-s + 0.180·13-s − 1.31·14-s − 0.514·15-s + 5/4·16-s + 0.393·17-s + 1.61·18-s − 0.383·19-s − 2.15·20-s + 0.331·21-s − 1.57·22-s + 0.787·23-s − 0.506·24-s + 0.958·25-s − 0.255·26-s − 0.504·27-s + 1.39·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(66-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+65/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(4\) = \(2^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2863.79\) |
| Root analytic conductor: | \(7.31535\) |
| Motivic weight: | \(65\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 4,\ (\ :65/2, 65/2),\ 1)\) |
Particular Values
| \(L(33)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{67}{2})\) | 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 | $C_1$ | \( ( 1 + p^{32} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 4728641811544 p^{5} T + \)\(10\!\cdots\!54\)\( p^{17} T^{2} - 4728641811544 p^{70} T^{3} + p^{130} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + \)\(29\!\cdots\!56\)\( p^{2} T + \)\(76\!\cdots\!74\)\( p^{8} T^{2} + \)\(29\!\cdots\!56\)\( p^{67} T^{3} + p^{130} T^{4} \) | |
| 7 | $D_{4}$ | \( 1 - \)\(55\!\cdots\!76\)\( p^{2} T + \)\(32\!\cdots\!58\)\( p^{8} T^{2} - \)\(55\!\cdots\!76\)\( p^{67} T^{3} + p^{130} T^{4} \) | |
| 11 | $D_{4}$ | \( 1 - \)\(64\!\cdots\!64\)\( p^{2} T + \)\(53\!\cdots\!46\)\( p^{4} T^{2} - \)\(64\!\cdots\!64\)\( p^{67} T^{3} + p^{130} T^{4} \) | |
| 13 | $D_{4}$ | \( 1 - \)\(13\!\cdots\!76\)\( p^{3} T + \)\(17\!\cdots\!62\)\( p^{4} T^{2} - \)\(13\!\cdots\!76\)\( p^{68} T^{3} + p^{130} T^{4} \) | |
| 17 | $D_{4}$ | \( 1 - \)\(22\!\cdots\!92\)\( p T + \)\(38\!\cdots\!26\)\( p^{3} T^{2} - \)\(22\!\cdots\!92\)\( p^{66} T^{3} + p^{130} T^{4} \) | |
| 19 | $D_{4}$ | \( 1 + \)\(73\!\cdots\!80\)\( p T - \)\(35\!\cdots\!62\)\( p^{4} T^{2} + \)\(73\!\cdots\!80\)\( p^{66} T^{3} + p^{130} T^{4} \) | |
| 23 | $D_{4}$ | \( 1 - \)\(14\!\cdots\!12\)\( T + \)\(29\!\cdots\!14\)\( p T^{2} - \)\(14\!\cdots\!12\)\( p^{65} T^{3} + p^{130} T^{4} \) | |
| 29 | $D_{4}$ | \( 1 - \)\(22\!\cdots\!60\)\( p T + \)\(35\!\cdots\!78\)\( p^{2} T^{2} - \)\(22\!\cdots\!60\)\( p^{66} T^{3} + p^{130} T^{4} \) | |
| 31 | $D_{4}$ | \( 1 - \)\(97\!\cdots\!64\)\( p T + \)\(58\!\cdots\!26\)\( p^{3} T^{2} - \)\(97\!\cdots\!64\)\( p^{66} T^{3} + p^{130} T^{4} \) | |
| 37 | $D_{4}$ | \( 1 + \)\(21\!\cdots\!96\)\( T + \)\(65\!\cdots\!14\)\( p T^{2} + \)\(21\!\cdots\!96\)\( p^{65} T^{3} + p^{130} T^{4} \) | |
| 41 | $D_{4}$ | \( 1 + \)\(93\!\cdots\!76\)\( T + \)\(18\!\cdots\!06\)\( p T^{2} + \)\(93\!\cdots\!76\)\( p^{65} T^{3} + p^{130} T^{4} \) | |
| 43 | $D_{4}$ | \( 1 - \)\(35\!\cdots\!04\)\( p T + \)\(19\!\cdots\!18\)\( p^{2} T^{2} - \)\(35\!\cdots\!04\)\( p^{66} T^{3} + p^{130} T^{4} \) | |
| 47 | $D_{4}$ | \( 1 + \)\(30\!\cdots\!96\)\( T + \)\(25\!\cdots\!94\)\( p T^{2} + \)\(30\!\cdots\!96\)\( p^{65} T^{3} + p^{130} T^{4} \) | |
| 53 | $D_{4}$ | \( 1 + \)\(52\!\cdots\!88\)\( T + \)\(33\!\cdots\!74\)\( p T^{2} + \)\(52\!\cdots\!88\)\( p^{65} T^{3} + p^{130} T^{4} \) | |
| 59 | $D_{4}$ | \( 1 + \)\(11\!\cdots\!80\)\( p T + \)\(80\!\cdots\!58\)\( p^{2} T^{2} + \)\(11\!\cdots\!80\)\( p^{66} T^{3} + p^{130} T^{4} \) | |
| 61 | $D_{4}$ | \( 1 + \)\(48\!\cdots\!76\)\( p T + \)\(11\!\cdots\!06\)\( p^{2} T^{2} + \)\(48\!\cdots\!76\)\( p^{66} T^{3} + p^{130} T^{4} \) | |
| 67 | $D_{4}$ | \( 1 + \)\(16\!\cdots\!76\)\( T + \)\(95\!\cdots\!58\)\( T^{2} + \)\(16\!\cdots\!76\)\( p^{65} T^{3} + p^{130} T^{4} \) | |
| 71 | $D_{4}$ | \( 1 - \)\(24\!\cdots\!64\)\( T + \)\(40\!\cdots\!26\)\( T^{2} - \)\(24\!\cdots\!64\)\( p^{65} T^{3} + p^{130} T^{4} \) | |
| 73 | $D_{4}$ | \( 1 - \)\(29\!\cdots\!32\)\( T + \)\(20\!\cdots\!42\)\( T^{2} - \)\(29\!\cdots\!32\)\( p^{65} T^{3} + p^{130} T^{4} \) | |
| 79 | $D_{4}$ | \( 1 + \)\(72\!\cdots\!00\)\( T + \)\(55\!\cdots\!98\)\( T^{2} + \)\(72\!\cdots\!00\)\( p^{65} T^{3} + p^{130} T^{4} \) | |
| 83 | $D_{4}$ | \( 1 + \)\(21\!\cdots\!48\)\( T + \)\(12\!\cdots\!62\)\( T^{2} + \)\(21\!\cdots\!48\)\( p^{65} T^{3} + p^{130} T^{4} \) | |
| 89 | $D_{4}$ | \( 1 + \)\(26\!\cdots\!20\)\( T + \)\(10\!\cdots\!98\)\( T^{2} + \)\(26\!\cdots\!20\)\( p^{65} T^{3} + p^{130} T^{4} \) | |
| 97 | $D_{4}$ | \( 1 + \)\(59\!\cdots\!36\)\( T - \)\(13\!\cdots\!62\)\( T^{2} + \)\(59\!\cdots\!36\)\( p^{65} T^{3} + p^{130} T^{4} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.50585826568165695420702706957, −14.05038424983186729223448934002, −12.11516262784532923352961072083, −12.07418719110149735378284632565, −11.00650203861176988462755067699, −10.87509831698148236449499847472, −9.504235129703417686099242247066, −8.773650496389907882926035363201, −8.253556057623910300942773880441, −7.919141235857927421284308733405, −6.95591752821259754277221512982, −6.30210515639798906454265839639, −5.08416716072312763036874894869, −4.20552825022655629107139508238, −3.21109506436071256892383807905, −2.83751588423496613046798587259, −1.47240413047536480591812892851, −1.30009721069984967462026459068, 0, 0, 1.30009721069984967462026459068, 1.47240413047536480591812892851, 2.83751588423496613046798587259, 3.21109506436071256892383807905, 4.20552825022655629107139508238, 5.08416716072312763036874894869, 6.30210515639798906454265839639, 6.95591752821259754277221512982, 7.919141235857927421284308733405, 8.253556057623910300942773880441, 8.773650496389907882926035363201, 9.504235129703417686099242247066, 10.87509831698148236449499847472, 11.00650203861176988462755067699, 12.07418719110149735378284632565, 12.11516262784532923352961072083, 14.05038424983186729223448934002, 14.50585826568165695420702706957