Dirichlet series
| L(s) = 1 | − 9.22e18·4-s − 4.63e26·7-s + 2.23e35·13-s + 8.50e37·16-s + 3.12e40·19-s − 1.08e44·25-s + 4.27e45·28-s − 1.73e47·31-s − 4.45e49·37-s + 2.68e51·43-s + 4.05e52·49-s − 2.06e54·52-s − 2.69e56·61-s − 7.84e56·64-s − 4.30e57·67-s + 9.83e58·73-s − 2.88e59·76-s + 1.17e60·79-s − 1.03e62·91-s + 3.40e62·97-s + 9.99e62·100-s − 9.62e62·103-s + 7.16e63·109-s − 3.94e64·112-s + ⋯ |
| L(s) = 1 | − 4-s − 1.11·7-s + 1.82·13-s + 16-s + 1.63·19-s − 25-s + 1.11·28-s − 1.83·31-s − 1.78·37-s + 0.942·43-s + 0.232·49-s − 1.82·52-s − 1.55·61-s − 64-s − 1.29·67-s + 1.98·73-s − 1.63·76-s + 1.97·79-s − 2.02·91-s + 0.889·97-s + 100-s − 0.379·103-s + 0.474·109-s − 1.11·112-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(64-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+63/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(9\) = \(3^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(226.224\) |
| Root analytic conductor: | \(15.0407\) |
| Motivic weight: | \(63\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((2,\ 9,\ (\ :63/2),\ 1)\) |
Particular Values
| \(L(32)\) | \(\approx\) | \(1.220188413\) |
| \(L(\frac12)\) | \(\approx\) | \(1.220188413\) |
| \(L(\frac{65}{2})\) | 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 | \( 1 + p^{63} T^{2} \) |
| 5 | \( 1 + p^{63} T^{2} \) | |
| 7 | \( 1 + \)\(46\!\cdots\!20\)\( T + p^{63} T^{2} \) | |
| 11 | \( 1 + p^{63} T^{2} \) | |
| 13 | \( 1 - \)\(22\!\cdots\!70\)\( T + p^{63} T^{2} \) | |
| 17 | \( 1 + p^{63} T^{2} \) | |
| 19 | \( 1 - \)\(31\!\cdots\!56\)\( T + p^{63} T^{2} \) | |
| 23 | \( 1 + p^{63} T^{2} \) | |
| 29 | \( 1 + p^{63} T^{2} \) | |
| 31 | \( 1 + \)\(17\!\cdots\!92\)\( T + p^{63} T^{2} \) | |
| 37 | \( 1 + \)\(44\!\cdots\!10\)\( T + p^{63} T^{2} \) | |
| 41 | \( 1 + p^{63} T^{2} \) | |
| 43 | \( 1 - \)\(26\!\cdots\!20\)\( T + p^{63} T^{2} \) | |
| 47 | \( 1 + p^{63} T^{2} \) | |
| 53 | \( 1 + p^{63} T^{2} \) | |
| 59 | \( 1 + p^{63} T^{2} \) | |
| 61 | \( 1 + \)\(26\!\cdots\!18\)\( T + p^{63} T^{2} \) | |
| 67 | \( 1 + \)\(43\!\cdots\!20\)\( T + p^{63} T^{2} \) | |
| 71 | \( 1 + p^{63} T^{2} \) | |
| 73 | \( 1 - \)\(98\!\cdots\!10\)\( T + p^{63} T^{2} \) | |
| 79 | \( 1 - \)\(11\!\cdots\!84\)\( T + p^{63} T^{2} \) | |
| 83 | \( 1 + p^{63} T^{2} \) | |
| 89 | \( 1 + p^{63} T^{2} \) | |
| 97 | \( 1 - \)\(34\!\cdots\!30\)\( T + p^{63} T^{2} \) | |
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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
−10.79540007831739090753598484802, −9.569798015850832448105640650404, −8.888348634462466384563285431240, −7.59131101955290344939154999307, −6.17145859891056162872034459508, −5.30765299722665848427127689674, −3.74997066241995771308632144525, −3.36010656066226554933496863253, −1.51735414119408947359156931096, −0.48043863845065493474090482748, 0.48043863845065493474090482748, 1.51735414119408947359156931096, 3.36010656066226554933496863253, 3.74997066241995771308632144525, 5.30765299722665848427127689674, 6.17145859891056162872034459508, 7.59131101955290344939154999307, 8.888348634462466384563285431240, 9.569798015850832448105640650404, 10.79540007831739090753598484802