Dirichlet series
| L(s) = 1 | − 2·4-s − 4·7-s − 7·13-s + 4·16-s − 19-s − 5·25-s + 8·28-s + 11·31-s − 10·37-s + 5·43-s + 9·49-s + 14·52-s − 61-s − 8·64-s + 5·67-s − 7·73-s + 2·76-s − 13·79-s + 28·91-s + 5·97-s + 10·100-s − 13·103-s − 19·109-s − 16·112-s + ⋯ |
| L(s) = 1 | − 4-s − 1.51·7-s − 1.94·13-s + 16-s − 0.229·19-s − 25-s + 1.51·28-s + 1.97·31-s − 1.64·37-s + 0.762·43-s + 9/7·49-s + 1.94·52-s − 0.128·61-s − 64-s + 0.610·67-s − 0.819·73-s + 0.229·76-s − 1.46·79-s + 2.93·91-s + 0.507·97-s + 100-s − 1.28·103-s − 1.81·109-s − 1.51·112-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(243\) = \(3^{5}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.94036\) |
| Root analytic conductor: | \(1.39296\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 243,\ (\ :1/2),\ -1)\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ | |
|---|---|---|---|
| bad | 3 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a | |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e | |
| 11 | \( 1 + p T^{2} \) | 1.11.a | |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h | |
| 17 | \( 1 + p T^{2} \) | 1.17.a | |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b | |
| 23 | \( 1 + p T^{2} \) | 1.23.a | |
| 29 | \( 1 + p T^{2} \) | 1.29.a | |
| 31 | \( 1 - 11 T + p T^{2} \) | 1.31.al | |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k | |
| 41 | \( 1 + p T^{2} \) | 1.41.a | |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af | |
| 47 | \( 1 + p T^{2} \) | 1.47.a | |
| 53 | \( 1 + p T^{2} \) | 1.53.a | |
| 59 | \( 1 + p T^{2} \) | 1.59.a | |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b | |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af | |
| 71 | \( 1 + p T^{2} \) | 1.71.a | |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h | |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n | |
| 83 | \( 1 + p T^{2} \) | 1.83.a | |
| 89 | \( 1 + p T^{2} \) | 1.89.a | |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af | |
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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
−12.00563141299999196241656264872, −10.16325612920902098193571359997, −9.839140997218714811663358114347, −8.910401959588073671653022038986, −7.67478550580751206163059313011, −6.55678077622873572890302306211, −5.32179199644498256876875394013, −4.15706479427404740125135154289, −2.83018486415551887293164017672, 0, 2.83018486415551887293164017672, 4.15706479427404740125135154289, 5.32179199644498256876875394013, 6.55678077622873572890302306211, 7.67478550580751206163059313011, 8.910401959588073671653022038986, 9.839140997218714811663358114347, 10.16325612920902098193571359997, 12.00563141299999196241656264872