Dirichlet series
| L(s) = 1 | − 9.90e27·4-s + 3.64e39·7-s + 1.06e52·13-s + 9.80e55·16-s + 5.52e59·19-s − 1.00e65·25-s − 3.61e67·28-s − 1.68e69·31-s − 1.63e73·37-s − 1.75e76·43-s + 9.36e78·49-s − 1.05e80·52-s + 6.66e82·61-s − 9.71e83·64-s − 9.11e84·67-s − 8.82e86·73-s − 5.47e87·76-s − 2.68e88·79-s + 3.86e91·91-s − 4.62e92·97-s + 9.99e92·100-s + 3.57e95·112-s + ⋯ |
| L(s) = 1 | − 4-s + 1.83·7-s + 1.68·13-s + 16-s + 1.90·19-s − 25-s − 1.83·28-s − 0.755·31-s − 1.95·37-s − 1.93·43-s + 2.38·49-s − 1.68·52-s + 0.639·61-s − 64-s − 1.11·67-s − 1.99·73-s − 1.90·76-s − 1.54·79-s + 3.10·91-s − 1.90·97-s + 100-s + 1.83·112-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(94-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+93/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(9\) = \(3^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(492.952\) |
| Root analytic conductor: | \(22.2025\) |
| Motivic weight: | \(93\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 9,\ (\ :93/2),\ -1)\) |
Particular Values
| \(L(47)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{95}{2})\) | 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 | \( 1 + p^{93} T^{2} \) |
| 5 | \( 1 + p^{93} T^{2} \) | |
| 7 | \( 1 - \)\(36\!\cdots\!60\)\( T + p^{93} T^{2} \) | |
| 11 | \( 1 + p^{93} T^{2} \) | |
| 13 | \( 1 - \)\(10\!\cdots\!10\)\( T + p^{93} T^{2} \) | |
| 17 | \( 1 + p^{93} T^{2} \) | |
| 19 | \( 1 - \)\(55\!\cdots\!44\)\( T + p^{93} T^{2} \) | |
| 23 | \( 1 + p^{93} T^{2} \) | |
| 29 | \( 1 + p^{93} T^{2} \) | |
| 31 | \( 1 + \)\(16\!\cdots\!92\)\( T + p^{93} T^{2} \) | |
| 37 | \( 1 + \)\(16\!\cdots\!70\)\( T + p^{93} T^{2} \) | |
| 41 | \( 1 + p^{93} T^{2} \) | |
| 43 | \( 1 + \)\(17\!\cdots\!40\)\( T + p^{93} T^{2} \) | |
| 47 | \( 1 + p^{93} T^{2} \) | |
| 53 | \( 1 + p^{93} T^{2} \) | |
| 59 | \( 1 + p^{93} T^{2} \) | |
| 61 | \( 1 - \)\(66\!\cdots\!82\)\( T + p^{93} T^{2} \) | |
| 67 | \( 1 + \)\(91\!\cdots\!40\)\( T + p^{93} T^{2} \) | |
| 71 | \( 1 + p^{93} T^{2} \) | |
| 73 | \( 1 + \)\(88\!\cdots\!70\)\( T + p^{93} T^{2} \) | |
| 79 | \( 1 + \)\(26\!\cdots\!84\)\( T + p^{93} T^{2} \) | |
| 83 | \( 1 + p^{93} T^{2} \) | |
| 89 | \( 1 + p^{93} T^{2} \) | |
| 97 | \( 1 + \)\(46\!\cdots\!90\)\( T + p^{93} T^{2} \) | |
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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
−8.838472555343611376465934304989, −8.278611413380774856621515939461, −7.35379741881848993324065408171, −5.66160995420115274099303425534, −5.13597128662670897944260489434, −4.09890223228411094878809349850, −3.31068974410112311488355962176, −1.52840626378111158231279786929, −1.31447803313298075723192176203, 0, 1.31447803313298075723192176203, 1.52840626378111158231279786929, 3.31068974410112311488355962176, 4.09890223228411094878809349850, 5.13597128662670897944260489434, 5.66160995420115274099303425534, 7.35379741881848993324065408171, 8.278611413380774856621515939461, 8.838472555343611376465934304989