Dirichlet series
| L(s) = 1 | − 1.54e26·4-s − 8.55e36·7-s + 3.61e48·13-s + 2.39e52·16-s − 4.60e55·19-s − 6.46e60·25-s + 1.32e63·28-s + 7.82e64·31-s + 2.53e68·37-s − 1.14e71·43-s + 3.97e73·49-s − 5.58e74·52-s − 5.98e77·61-s − 3.70e78·64-s − 5.19e79·67-s − 1.88e81·73-s + 7.13e81·76-s + 5.50e82·79-s − 3.08e85·91-s + 1.75e86·97-s + 9.99e86·100-s − 2.04e89·112-s + ⋯ |
| L(s) = 1 | − 4-s − 1.47·7-s + 1.26·13-s + 16-s − 1.09·19-s − 25-s + 1.47·28-s + 1.04·31-s + 1.53·37-s − 1.00·43-s + 1.18·49-s − 1.26·52-s − 1.30·61-s − 64-s − 1.91·67-s − 1.66·73-s + 1.09·76-s + 1.56·79-s − 1.86·91-s + 0.661·97-s + 100-s − 1.47·112-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(88-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+87/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(9\) = \(3^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(431.400\) |
| Root analytic conductor: | \(20.7701\) |
| Motivic weight: | \(87\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((2,\ 9,\ (\ :87/2),\ 1)\) |
Particular Values
| \(L(44)\) | \(\approx\) | \(0.6615707607\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6615707607\) |
| \(L(\frac{89}{2})\) | 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 | \( 1 + p^{87} T^{2} \) |
| 5 | \( 1 + p^{87} T^{2} \) | |
| 7 | \( 1 + \)\(85\!\cdots\!80\)\( T + p^{87} T^{2} \) | |
| 11 | \( 1 + p^{87} T^{2} \) | |
| 13 | \( 1 - \)\(36\!\cdots\!30\)\( T + p^{87} T^{2} \) | |
| 17 | \( 1 + p^{87} T^{2} \) | |
| 19 | \( 1 + \)\(46\!\cdots\!84\)\( T + p^{87} T^{2} \) | |
| 23 | \( 1 + p^{87} T^{2} \) | |
| 29 | \( 1 + p^{87} T^{2} \) | |
| 31 | \( 1 - \)\(78\!\cdots\!88\)\( T + p^{87} T^{2} \) | |
| 37 | \( 1 - \)\(25\!\cdots\!10\)\( T + p^{87} T^{2} \) | |
| 41 | \( 1 + p^{87} T^{2} \) | |
| 43 | \( 1 + \)\(11\!\cdots\!20\)\( T + p^{87} T^{2} \) | |
| 47 | \( 1 + p^{87} T^{2} \) | |
| 53 | \( 1 + p^{87} T^{2} \) | |
| 59 | \( 1 + p^{87} T^{2} \) | |
| 61 | \( 1 + \)\(59\!\cdots\!78\)\( T + p^{87} T^{2} \) | |
| 67 | \( 1 + \)\(51\!\cdots\!80\)\( T + p^{87} T^{2} \) | |
| 71 | \( 1 + p^{87} T^{2} \) | |
| 73 | \( 1 + \)\(18\!\cdots\!10\)\( T + p^{87} T^{2} \) | |
| 79 | \( 1 - \)\(55\!\cdots\!44\)\( T + p^{87} T^{2} \) | |
| 83 | \( 1 + p^{87} T^{2} \) | |
| 89 | \( 1 + p^{87} T^{2} \) | |
| 97 | \( 1 - \)\(17\!\cdots\!70\)\( T + p^{87} T^{2} \) | |
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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
−9.706181771997012289162082352501, −8.897394568782154343424385980779, −7.948692026279928096252894887559, −6.43103157594220544400993409244, −5.89057099698453189034121625179, −4.45055856541299189404893132448, −3.70944055178252040075174330835, −2.82408150831386027483323388514, −1.35631710855832651248246240068, −0.31590030436028603134398393054, 0.31590030436028603134398393054, 1.35631710855832651248246240068, 2.82408150831386027483323388514, 3.70944055178252040075174330835, 4.45055856541299189404893132448, 5.89057099698453189034121625179, 6.43103157594220544400993409244, 7.948692026279928096252894887559, 8.897394568782154343424385980779, 9.706181771997012289162082352501