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Dirichlet series
| L(s) = 1 | + 195.·3-s + 200.·5-s + 2.40e3·7-s + 1.86e4·9-s − 6.38e4·11-s − 1.64e5·13-s + 3.93e4·15-s − 3.62e5·17-s + 4.36e5·19-s + 4.70e5·21-s − 9.18e5·23-s − 1.91e6·25-s − 2.00e5·27-s − 3.68e6·29-s − 3.47e6·31-s − 1.25e7·33-s + 4.82e5·35-s + 1.88e7·37-s − 3.22e7·39-s + 2.40e6·41-s + 1.25e7·43-s + 3.74e6·45-s + 5.54e7·47-s + 5.76e6·49-s − 7.10e7·51-s − 9.26e7·53-s − 1.28e7·55-s + ⋯ |
| L(s) = 1 | + 1.39·3-s + 0.143·5-s + 0.377·7-s + 0.948·9-s − 1.31·11-s − 1.59·13-s + 0.200·15-s − 1.05·17-s + 0.768·19-s + 0.527·21-s − 0.684·23-s − 0.979·25-s − 0.0724·27-s − 0.967·29-s − 0.676·31-s − 1.83·33-s + 0.0543·35-s + 1.65·37-s − 2.23·39-s + 0.133·41-s + 0.558·43-s + 0.136·45-s + 1.65·47-s + 0.142·49-s − 1.47·51-s − 1.61·53-s − 0.188·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(112\) = \(2^{4} \cdot 7\) |
| Sign: | $-1$ |
| Analytic conductor: | \(57.6840\) |
| Root analytic conductor: | \(7.59499\) |
| Motivic weight: | \(9\) |
| Rational: | no |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 112,\ (\ :9/2),\ -1)\) |
Particular Values
| \(L(5)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.40e3T \) | |
| good | 3 | \( 1 - 195.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 200.T + 1.95e6T^{2} \) | |
| 11 | \( 1 + 6.38e4T + 2.35e9T^{2} \) | |
| 13 | \( 1 + 1.64e5T + 1.06e10T^{2} \) | |
| 17 | \( 1 + 3.62e5T + 1.18e11T^{2} \) | |
| 19 | \( 1 - 4.36e5T + 3.22e11T^{2} \) | |
| 23 | \( 1 + 9.18e5T + 1.80e12T^{2} \) | |
| 29 | \( 1 + 3.68e6T + 1.45e13T^{2} \) | |
| 31 | \( 1 + 3.47e6T + 2.64e13T^{2} \) | |
| 37 | \( 1 - 1.88e7T + 1.29e14T^{2} \) | |
| 41 | \( 1 - 2.40e6T + 3.27e14T^{2} \) | |
| 43 | \( 1 - 1.25e7T + 5.02e14T^{2} \) | |
| 47 | \( 1 - 5.54e7T + 1.11e15T^{2} \) | |
| 53 | \( 1 + 9.26e7T + 3.29e15T^{2} \) | |
| 59 | \( 1 - 2.52e7T + 8.66e15T^{2} \) | |
| 61 | \( 1 - 6.93e7T + 1.16e16T^{2} \) | |
| 67 | \( 1 - 2.33e7T + 2.72e16T^{2} \) | |
| 71 | \( 1 - 1.06e8T + 4.58e16T^{2} \) | |
| 73 | \( 1 + 2.10e8T + 5.88e16T^{2} \) | |
| 79 | \( 1 - 1.49e5T + 1.19e17T^{2} \) | |
| 83 | \( 1 + 5.21e8T + 1.86e17T^{2} \) | |
| 89 | \( 1 - 2.98e8T + 3.50e17T^{2} \) | |
| 97 | \( 1 + 8.95e8T + 7.60e17T^{2} \) | |
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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
−11.26374988876998013529395826435, −9.941204736437172287083283808190, −9.205560221289822248806860569574, −7.910607877853971228976703146554, −7.44378078047853786658053056306, −5.51060645804642308078069050117, −4.21557918617837923510654951642, −2.70343406214420550822216851917, −2.07548256727593094130680212689, 0, 2.07548256727593094130680212689, 2.70343406214420550822216851917, 4.21557918617837923510654951642, 5.51060645804642308078069050117, 7.44378078047853786658053056306, 7.910607877853971228976703146554, 9.205560221289822248806860569574, 9.941204736437172287083283808190, 11.26374988876998013529395826435