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Dirichlet series
| L(s) = 1 | − 24.2·3-s − 51.2·5-s + 25.3·7-s + 346.·9-s + 228.·13-s + 1.24e3·15-s + 1.04e3·17-s + 395.·19-s − 614.·21-s − 437.·23-s − 500.·25-s − 2.51e3·27-s + 2.05e3·29-s + 7.46e3·31-s − 1.29e3·35-s + 9.58e3·37-s − 5.55e3·39-s − 7.72e3·41-s + 1.96e4·43-s − 1.77e4·45-s − 1.21e4·47-s − 1.61e4·49-s − 2.52e4·51-s − 3.18e4·53-s − 9.61e3·57-s − 5.12e4·59-s + 4.40e4·61-s + ⋯ |
| L(s) = 1 | − 1.55·3-s − 0.916·5-s + 0.195·7-s + 1.42·9-s + 0.375·13-s + 1.42·15-s + 0.872·17-s + 0.251·19-s − 0.304·21-s − 0.172·23-s − 0.160·25-s − 0.662·27-s + 0.452·29-s + 1.39·31-s − 0.178·35-s + 1.15·37-s − 0.584·39-s − 0.717·41-s + 1.62·43-s − 1.30·45-s − 0.801·47-s − 0.961·49-s − 1.35·51-s − 1.55·53-s − 0.391·57-s − 1.91·59-s + 1.51·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(968\) = \(2^{3} \cdot 11^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(155.251\) |
| Root analytic conductor: | \(12.4599\) |
| Motivic weight: | \(5\) |
| Rational: | no |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((2,\ 968,\ (\ :5/2),\ 1)\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(0.8538505762\) |
| \(L(\frac12)\) | \(\approx\) | \(0.8538505762\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + 24.2T + 243T^{2} \) |
| 5 | \( 1 + 51.2T + 3.12e3T^{2} \) | |
| 7 | \( 1 - 25.3T + 1.68e4T^{2} \) | |
| 13 | \( 1 - 228.T + 3.71e5T^{2} \) | |
| 17 | \( 1 - 1.04e3T + 1.41e6T^{2} \) | |
| 19 | \( 1 - 395.T + 2.47e6T^{2} \) | |
| 23 | \( 1 + 437.T + 6.43e6T^{2} \) | |
| 29 | \( 1 - 2.05e3T + 2.05e7T^{2} \) | |
| 31 | \( 1 - 7.46e3T + 2.86e7T^{2} \) | |
| 37 | \( 1 - 9.58e3T + 6.93e7T^{2} \) | |
| 41 | \( 1 + 7.72e3T + 1.15e8T^{2} \) | |
| 43 | \( 1 - 1.96e4T + 1.47e8T^{2} \) | |
| 47 | \( 1 + 1.21e4T + 2.29e8T^{2} \) | |
| 53 | \( 1 + 3.18e4T + 4.18e8T^{2} \) | |
| 59 | \( 1 + 5.12e4T + 7.14e8T^{2} \) | |
| 61 | \( 1 - 4.40e4T + 8.44e8T^{2} \) | |
| 67 | \( 1 + 1.56e4T + 1.35e9T^{2} \) | |
| 71 | \( 1 + 5.13e4T + 1.80e9T^{2} \) | |
| 73 | \( 1 + 6.98e4T + 2.07e9T^{2} \) | |
| 79 | \( 1 - 1.04e5T + 3.07e9T^{2} \) | |
| 83 | \( 1 - 7.24e4T + 3.93e9T^{2} \) | |
| 89 | \( 1 + 2.57e4T + 5.58e9T^{2} \) | |
| 97 | \( 1 + 2.87e3T + 8.58e9T^{2} \) | |
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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
−9.484397106889693021567783235151, −8.160653917398298645354818093582, −7.62157102399003796521042585949, −6.51690775811701629674343047726, −5.91857275897058834020070115023, −4.91483807473727336785970820266, −4.25039940479837116871256948402, −3.09121642297887435961786365018, −1.35150802726891775275131412103, −0.49175320550226277528115136222, 0.49175320550226277528115136222, 1.35150802726891775275131412103, 3.09121642297887435961786365018, 4.25039940479837116871256948402, 4.91483807473727336785970820266, 5.91857275897058834020070115023, 6.51690775811701629674343047726, 7.62157102399003796521042585949, 8.160653917398298645354818093582, 9.484397106889693021567783235151