Dirichlet series
| L(s) = 1 | − 1.44e17·4-s + 2.40e24·7-s − 3.42e31·13-s + 2.07e34·16-s − 1.48e36·19-s − 6.93e39·25-s − 3.46e41·28-s − 5.52e42·31-s + 9.87e44·37-s − 2.14e46·43-s + 4.29e48·49-s + 4.93e48·52-s + 7.43e50·61-s − 2.99e51·64-s − 2.02e52·67-s + 2.23e53·73-s + 2.13e53·76-s − 8.75e53·79-s − 8.22e55·91-s + 7.40e56·97-s + 9.99e56·100-s − 4.36e57·103-s − 1.69e58·109-s + 4.99e58·112-s + ⋯ |
| L(s) = 1 | − 4-s + 1.97·7-s − 0.612·13-s + 16-s − 0.532·19-s − 25-s − 1.97·28-s − 1.73·31-s + 1.99·37-s − 0.598·43-s + 2.90·49-s + 0.612·52-s + 0.976·61-s − 64-s − 1.83·67-s + 1.75·73-s + 0.532·76-s − 0.723·79-s − 1.20·91-s + 1.76·97-s + 100-s − 1.87·103-s − 1.45·109-s + 1.97·112-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(58-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+57/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(9\) = \(3^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(185.189\) |
| Root analytic conductor: | \(13.6084\) |
| Motivic weight: | \(57\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 9,\ (\ :57/2),\ -1)\) |
Particular Values
| \(L(29)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{59}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + p^{57} T^{2} \) |
| 5 | \( 1 + p^{57} T^{2} \) | |
| 7 | \( 1 - \)\(24\!\cdots\!60\)\( T + p^{57} T^{2} \) | |
| 11 | \( 1 + p^{57} T^{2} \) | |
| 13 | \( 1 + \)\(34\!\cdots\!90\)\( T + p^{57} T^{2} \) | |
| 17 | \( 1 + p^{57} T^{2} \) | |
| 19 | \( 1 + \)\(14\!\cdots\!16\)\( T + p^{57} T^{2} \) | |
| 23 | \( 1 + p^{57} T^{2} \) | |
| 29 | \( 1 + p^{57} T^{2} \) | |
| 31 | \( 1 + \)\(55\!\cdots\!12\)\( T + p^{57} T^{2} \) | |
| 37 | \( 1 - \)\(98\!\cdots\!30\)\( T + p^{57} T^{2} \) | |
| 41 | \( 1 + p^{57} T^{2} \) | |
| 43 | \( 1 + \)\(21\!\cdots\!40\)\( T + p^{57} T^{2} \) | |
| 47 | \( 1 + p^{57} T^{2} \) | |
| 53 | \( 1 + p^{57} T^{2} \) | |
| 59 | \( 1 + p^{57} T^{2} \) | |
| 61 | \( 1 - \)\(74\!\cdots\!22\)\( T + p^{57} T^{2} \) | |
| 67 | \( 1 + \)\(20\!\cdots\!40\)\( T + p^{57} T^{2} \) | |
| 71 | \( 1 + p^{57} T^{2} \) | |
| 73 | \( 1 - \)\(22\!\cdots\!30\)\( T + p^{57} T^{2} \) | |
| 79 | \( 1 + \)\(87\!\cdots\!44\)\( T + p^{57} T^{2} \) | |
| 83 | \( 1 + p^{57} T^{2} \) | |
| 89 | \( 1 + p^{57} T^{2} \) | |
| 97 | \( 1 - \)\(74\!\cdots\!10\)\( T + p^{57} T^{2} \) | |
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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
−10.78092968396988099626671731394, −9.427923251919758812895459203918, −8.304395446215585977913758322519, −7.56442840197284527425523277924, −5.63199126031259168436197459111, −4.76282909019344974250840061534, −3.95938273250718567236128917469, −2.20467819190722058232702837249, −1.21297079885224850021979977190, 0, 1.21297079885224850021979977190, 2.20467819190722058232702837249, 3.95938273250718567236128917469, 4.76282909019344974250840061534, 5.63199126031259168436197459111, 7.56442840197284527425523277924, 8.304395446215585977913758322519, 9.427923251919758812895459203918, 10.78092968396988099626671731394