Dirichlet series
| L(s) = 1 | − 8.19e3·2-s + 3.98e6·3-s + 6.71e7·4-s − 2.85e9·5-s − 3.26e10·6-s + 3.68e11·7-s − 5.49e11·8-s + 8.25e12·9-s + 2.33e13·10-s + 5.98e13·11-s + 2.67e14·12-s + 1.90e15·13-s − 3.02e15·14-s − 1.13e16·15-s + 4.50e15·16-s − 3.44e15·17-s − 6.76e16·18-s + 1.58e17·19-s − 1.91e17·20-s + 1.46e18·21-s − 4.90e17·22-s − 1.25e18·23-s − 2.19e18·24-s + 6.82e17·25-s − 1.55e19·26-s + 2.50e18·27-s + 2.47e19·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.44·3-s + 1/2·4-s − 1.04·5-s − 1.02·6-s + 1.43·7-s − 0.353·8-s + 1.08·9-s + 0.738·10-s + 0.523·11-s + 0.721·12-s + 1.74·13-s − 1.01·14-s − 1.50·15-s + 1/4·16-s − 0.0844·17-s − 0.765·18-s + 0.864·19-s − 0.522·20-s + 2.07·21-s − 0.369·22-s − 0.520·23-s − 0.510·24-s + 0.0916·25-s − 1.23·26-s + 0.118·27-s + 0.719·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(2\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.23711\) |
| Root analytic conductor: | \(3.03926\) |
| Motivic weight: | \(27\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((2,\ 2,\ (\ :27/2),\ 1)\) |
Particular Values
| \(L(14)\) | \(\approx\) | \(2.014263927\) |
| \(L(\frac12)\) | \(\approx\) | \(2.014263927\) |
| \(L(\frac{29}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + p^{13} T \) |
| good | 3 | \( 1 - 1328276 p T + p^{27} T^{2} \) |
| 5 | \( 1 + 22815114 p^{3} T + p^{27} T^{2} \) | |
| 7 | \( 1 - 52674437672 p T + p^{27} T^{2} \) | |
| 11 | \( 1 - 59896911912852 T + p^{27} T^{2} \) | |
| 13 | \( 1 - 146354702000846 p T + p^{27} T^{2} \) | |
| 17 | \( 1 + 202933185722478 p T + p^{27} T^{2} \) | |
| 19 | \( 1 - 158487413654686700 T + p^{27} T^{2} \) | |
| 23 | \( 1 + 54657494413968264 p T + p^{27} T^{2} \) | |
| 29 | \( 1 - 52884218157232223910 T + p^{27} T^{2} \) | |
| 31 | \( 1 + \)\(23\!\cdots\!28\)\( T + p^{27} T^{2} \) | |
| 37 | \( 1 + \)\(70\!\cdots\!46\)\( T + p^{27} T^{2} \) | |
| 41 | \( 1 + \)\(29\!\cdots\!78\)\( T + p^{27} T^{2} \) | |
| 43 | \( 1 - \)\(16\!\cdots\!68\)\( T + p^{27} T^{2} \) | |
| 47 | \( 1 + \)\(87\!\cdots\!36\)\( T + p^{27} T^{2} \) | |
| 53 | \( 1 + \)\(20\!\cdots\!02\)\( T + p^{27} T^{2} \) | |
| 59 | \( 1 + \)\(46\!\cdots\!80\)\( T + p^{27} T^{2} \) | |
| 61 | \( 1 - \)\(53\!\cdots\!42\)\( T + p^{27} T^{2} \) | |
| 67 | \( 1 + \)\(28\!\cdots\!36\)\( T + p^{27} T^{2} \) | |
| 71 | \( 1 - \)\(11\!\cdots\!12\)\( T + p^{27} T^{2} \) | |
| 73 | \( 1 + \)\(49\!\cdots\!02\)\( T + p^{27} T^{2} \) | |
| 79 | \( 1 - \)\(31\!\cdots\!40\)\( T + p^{27} T^{2} \) | |
| 83 | \( 1 + \)\(62\!\cdots\!92\)\( T + p^{27} T^{2} \) | |
| 89 | \( 1 + \)\(37\!\cdots\!90\)\( T + p^{27} T^{2} \) | |
| 97 | \( 1 - \)\(51\!\cdots\!54\)\( T + p^{27} T^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.83948508090625990861301403896, −19.84227335646489792549070604488, −18.28117416259955215114836326052, −15.69032251700888369811172852794, −14.22381952286637315263211680968, −11.34946122041932254648176623825, −8.723661814728507584254928121743, −7.78990091107777544051279361282, −3.68641722137474384299576645785, −1.50879160474049975885028843146, 1.50879160474049975885028843146, 3.68641722137474384299576645785, 7.78990091107777544051279361282, 8.723661814728507584254928121743, 11.34946122041932254648176623825, 14.22381952286637315263211680968, 15.69032251700888369811172852794, 18.28117416259955215114836326052, 19.84227335646489792549070604488, 20.83948508090625990861301403896