Error: no document with id 205133349 found in table mf_hecke_traces.
Dirichlet series
| L(s) = 1 | − 2.61·2-s − 2.17·3-s + 4.85·4-s + 5.70·6-s − 4.94·7-s − 7.47·8-s + 1.74·9-s + 11-s − 10.5·12-s + 4.20·13-s + 12.9·14-s + 9.84·16-s + 6.38·17-s − 4.56·18-s + 19-s + 10.7·21-s − 2.61·22-s − 6.66·23-s + 16.2·24-s − 11.0·26-s + 2.73·27-s − 24.0·28-s − 6.65·29-s − 9.53·31-s − 10.8·32-s − 2.17·33-s − 16.7·34-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 1.25·3-s + 2.42·4-s + 2.32·6-s − 1.87·7-s − 2.64·8-s + 0.581·9-s + 0.301·11-s − 3.05·12-s + 1.16·13-s + 3.46·14-s + 2.46·16-s + 1.54·17-s − 1.07·18-s + 0.229·19-s + 2.35·21-s − 0.558·22-s − 1.38·23-s + 3.32·24-s − 2.15·26-s + 0.526·27-s − 4.53·28-s − 1.23·29-s − 1.71·31-s − 1.91·32-s − 0.379·33-s − 2.86·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(5225\) = \(5^{2} \cdot 11 \cdot 19\) |
| Sign: | $-1$ |
| Analytic conductor: | \(41.7218\) |
| Root analytic conductor: | \(6.45924\) |
| Motivic weight: | \(1\) |
| Rational: | no |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 5225,\ (\ :1/2),\ -1)\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) | |
| 19 | \( 1 - T \) | |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 + 2.17T + 3T^{2} \) | |
| 7 | \( 1 + 4.94T + 7T^{2} \) | |
| 13 | \( 1 - 4.20T + 13T^{2} \) | |
| 17 | \( 1 - 6.38T + 17T^{2} \) | |
| 23 | \( 1 + 6.66T + 23T^{2} \) | |
| 29 | \( 1 + 6.65T + 29T^{2} \) | |
| 31 | \( 1 + 9.53T + 31T^{2} \) | |
| 37 | \( 1 - 3.69T + 37T^{2} \) | |
| 41 | \( 1 + 1.08T + 41T^{2} \) | |
| 43 | \( 1 + 9.58T + 43T^{2} \) | |
| 47 | \( 1 + 2.90T + 47T^{2} \) | |
| 53 | \( 1 - 6.75T + 53T^{2} \) | |
| 59 | \( 1 + 0.613T + 59T^{2} \) | |
| 61 | \( 1 - 1.33T + 61T^{2} \) | |
| 67 | \( 1 + 0.386T + 67T^{2} \) | |
| 71 | \( 1 - 9.38T + 71T^{2} \) | |
| 73 | \( 1 - 1.97T + 73T^{2} \) | |
| 79 | \( 1 + 6.61T + 79T^{2} \) | |
| 83 | \( 1 - 0.100T + 83T^{2} \) | |
| 89 | \( 1 - 2.48T + 89T^{2} \) | |
| 97 | \( 1 - 17.3T + 97T^{2} \) | |
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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
−7.86845595177778095568443856029, −7.10364852013907742644928446902, −6.54397911164910886239194336259, −5.87343615215718459755776194521, −5.62476340948502968407864271366, −3.73649374790581478312626810831, −3.22420461124456401758565875172, −1.86104071881620734520702442742, −0.820824127909821225253441687644, 0, 0.820824127909821225253441687644, 1.86104071881620734520702442742, 3.22420461124456401758565875172, 3.73649374790581478312626810831, 5.62476340948502968407864271366, 5.87343615215718459755776194521, 6.54397911164910886239194336259, 7.10364852013907742644928446902, 7.86845595177778095568443856029