Dirichlet series
| L(s) = 1 | + 6.71e7·2-s + 8.89e11·3-s + 3.37e15·4-s − 5.05e17·5-s + 5.97e19·6-s − 6.43e20·7-s + 1.51e23·8-s − 3.68e24·9-s − 3.39e25·10-s + 8.63e24·11-s + 3.00e27·12-s − 3.08e28·13-s − 4.31e28·14-s − 4.49e29·15-s + 6.33e30·16-s − 2.87e31·17-s − 2.47e32·18-s − 7.33e32·19-s − 1.70e33·20-s − 5.72e32·21-s + 5.79e32·22-s + 3.72e34·23-s + 1.34e35·24-s + 1.36e34·25-s − 2.07e36·26-s − 5.34e36·27-s − 2.17e36·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.606·3-s + 3/2·4-s − 0.758·5-s + 0.857·6-s − 0.181·7-s + 1.41·8-s − 1.71·9-s − 1.07·10-s + 0.0240·11-s + 0.909·12-s − 1.21·13-s − 0.256·14-s − 0.459·15-s + 5/4·16-s − 1.20·17-s − 2.41·18-s − 1.80·19-s − 1.13·20-s − 0.109·21-s + 0.0339·22-s + 0.704·23-s + 0.857·24-s + 0.0307·25-s − 1.71·26-s − 1.69·27-s − 0.271·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(52-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+51/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(4\) = \(2^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1085.45\) |
| Root analytic conductor: | \(5.73988\) |
| Motivic weight: | \(51\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 4,\ (\ :51/2, 51/2),\ 1)\) |
Particular Values
| \(L(26)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{53}{2})\) | 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 | $C_1$ | \( ( 1 - p^{25} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 3660986728 p^{5} T + 935600343575284342 p^{14} T^{2} - 3660986728 p^{56} T^{3} + p^{102} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 20219353608241956 p^{2} T + \)\(30\!\cdots\!54\)\( p^{7} T^{2} + 20219353608241956 p^{53} T^{3} + p^{102} T^{4} \) | |
| 7 | $D_{4}$ | \( 1 + 91907810512415242544 p T - \)\(36\!\cdots\!14\)\( p^{5} T^{2} + 91907810512415242544 p^{52} T^{3} + p^{102} T^{4} \) | |
| 11 | $D_{4}$ | \( 1 - \)\(78\!\cdots\!44\)\( p T + \)\(11\!\cdots\!86\)\( p^{5} T^{2} - \)\(78\!\cdots\!44\)\( p^{52} T^{3} + p^{102} T^{4} \) | |
| 13 | $D_{4}$ | \( 1 + \)\(23\!\cdots\!52\)\( p T + \)\(38\!\cdots\!22\)\( p^{2} T^{2} + \)\(23\!\cdots\!52\)\( p^{52} T^{3} + p^{102} T^{4} \) | |
| 17 | $D_{4}$ | \( 1 + \)\(16\!\cdots\!84\)\( p T + \)\(46\!\cdots\!58\)\( p^{2} T^{2} + \)\(16\!\cdots\!84\)\( p^{52} T^{3} + p^{102} T^{4} \) | |
| 19 | $D_{4}$ | \( 1 + \)\(38\!\cdots\!00\)\( p T + \)\(67\!\cdots\!82\)\( p^{3} T^{2} + \)\(38\!\cdots\!00\)\( p^{52} T^{3} + p^{102} T^{4} \) | |
| 23 | $D_{4}$ | \( 1 - \)\(16\!\cdots\!88\)\( p T + \)\(28\!\cdots\!94\)\( p^{3} T^{2} - \)\(16\!\cdots\!88\)\( p^{52} T^{3} + p^{102} T^{4} \) | |
| 29 | $D_{4}$ | \( 1 + \)\(23\!\cdots\!80\)\( T + \)\(29\!\cdots\!02\)\( p T^{2} + \)\(23\!\cdots\!80\)\( p^{51} T^{3} + p^{102} T^{4} \) | |
| 31 | $D_{4}$ | \( 1 + \)\(12\!\cdots\!16\)\( T + \)\(86\!\cdots\!46\)\( p T^{2} + \)\(12\!\cdots\!16\)\( p^{51} T^{3} + p^{102} T^{4} \) | |
| 37 | $D_{4}$ | \( 1 + \)\(25\!\cdots\!48\)\( T + \)\(91\!\cdots\!46\)\( p T^{2} + \)\(25\!\cdots\!48\)\( p^{51} T^{3} + p^{102} T^{4} \) | |
| 41 | $D_{4}$ | \( 1 + \)\(49\!\cdots\!96\)\( p T + \)\(17\!\cdots\!26\)\( p^{2} T^{2} + \)\(49\!\cdots\!96\)\( p^{52} T^{3} + p^{102} T^{4} \) | |
| 43 | $D_{4}$ | \( 1 + \)\(77\!\cdots\!52\)\( p T - \)\(21\!\cdots\!38\)\( p^{2} T^{2} + \)\(77\!\cdots\!52\)\( p^{52} T^{3} + p^{102} T^{4} \) | |
| 47 | $D_{4}$ | \( 1 - \)\(18\!\cdots\!16\)\( p T + \)\(25\!\cdots\!98\)\( p^{2} T^{2} - \)\(18\!\cdots\!16\)\( p^{52} T^{3} + p^{102} T^{4} \) | |
| 53 | $D_{4}$ | \( 1 + \)\(99\!\cdots\!36\)\( T + \)\(18\!\cdots\!18\)\( T^{2} + \)\(99\!\cdots\!36\)\( p^{51} T^{3} + p^{102} T^{4} \) | |
| 59 | $D_{4}$ | \( 1 - \)\(71\!\cdots\!40\)\( T + \)\(41\!\cdots\!18\)\( T^{2} - \)\(71\!\cdots\!40\)\( p^{51} T^{3} + p^{102} T^{4} \) | |
| 61 | $D_{4}$ | \( 1 - \)\(24\!\cdots\!64\)\( T + \)\(12\!\cdots\!46\)\( T^{2} - \)\(24\!\cdots\!64\)\( p^{51} T^{3} + p^{102} T^{4} \) | |
| 67 | $D_{4}$ | \( 1 - \)\(58\!\cdots\!92\)\( T + \)\(28\!\cdots\!82\)\( T^{2} - \)\(58\!\cdots\!92\)\( p^{51} T^{3} + p^{102} T^{4} \) | |
| 71 | $D_{4}$ | \( 1 - \)\(46\!\cdots\!84\)\( T + \)\(64\!\cdots\!06\)\( T^{2} - \)\(46\!\cdots\!84\)\( p^{51} T^{3} + p^{102} T^{4} \) | |
| 73 | $D_{4}$ | \( 1 + \)\(37\!\cdots\!16\)\( T + \)\(65\!\cdots\!18\)\( T^{2} + \)\(37\!\cdots\!16\)\( p^{51} T^{3} + p^{102} T^{4} \) | |
| 79 | $D_{4}$ | \( 1 - \)\(45\!\cdots\!80\)\( T + \)\(16\!\cdots\!58\)\( T^{2} - \)\(45\!\cdots\!80\)\( p^{51} T^{3} + p^{102} T^{4} \) | |
| 83 | $D_{4}$ | \( 1 - \)\(65\!\cdots\!24\)\( T + \)\(12\!\cdots\!78\)\( T^{2} - \)\(65\!\cdots\!24\)\( p^{51} T^{3} + p^{102} T^{4} \) | |
| 89 | $D_{4}$ | \( 1 + \)\(74\!\cdots\!80\)\( T + \)\(62\!\cdots\!78\)\( T^{2} + \)\(74\!\cdots\!80\)\( p^{51} T^{3} + p^{102} T^{4} \) | |
| 97 | $D_{4}$ | \( 1 - \)\(78\!\cdots\!72\)\( T + \)\(53\!\cdots\!02\)\( T^{2} - \)\(78\!\cdots\!72\)\( p^{51} T^{3} + p^{102} T^{4} \) | |
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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
−15.52369735453227332253056440728, −15.05924423262710940542830961172, −14.45656681159848802607764427989, −13.71055463666524664183868612287, −12.80994733459928500610855036445, −12.07642422990345218882556770676, −11.26240971601040779110528440340, −10.65435403092182391104974280463, −9.023029809735260653592434266038, −8.434900975367784749415899426337, −7.33465811909553970725330874401, −6.62929253419421850704778541718, −5.54797490704522574490087122499, −4.91721398347124056158221824287, −3.82218584475551430314354067677, −3.41384541241399954135432201846, −2.29129101385189169421575281666, −2.14487875219918345519777024489, 0, 0, 2.14487875219918345519777024489, 2.29129101385189169421575281666, 3.41384541241399954135432201846, 3.82218584475551430314354067677, 4.91721398347124056158221824287, 5.54797490704522574490087122499, 6.62929253419421850704778541718, 7.33465811909553970725330874401, 8.434900975367784749415899426337, 9.023029809735260653592434266038, 10.65435403092182391104974280463, 11.26240971601040779110528440340, 12.07642422990345218882556770676, 12.80994733459928500610855036445, 13.71055463666524664183868612287, 14.45656681159848802607764427989, 15.05924423262710940542830961172, 15.52369735453227332253056440728