Dirichlet series
| L(s) = 1 | + 6.55e4·2-s + 4.29e9·4-s − 5.38e11·5-s − 3.33e13·7-s + 2.81e14·8-s − 3.53e16·10-s + 8.58e16·11-s + 1.14e18·13-s − 2.18e18·14-s + 1.84e19·16-s + 1.39e20·17-s + 8.06e19·19-s − 2.31e21·20-s + 5.62e21·22-s + 1.41e22·23-s + 1.73e23·25-s + 7.49e22·26-s − 1.43e23·28-s + 1.63e24·29-s − 1.89e24·31-s + 1.20e24·32-s + 9.11e24·34-s + 1.79e25·35-s − 9.64e25·37-s + 5.28e24·38-s − 1.51e26·40-s − 6.41e26·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.57·5-s − 0.379·7-s + 0.353·8-s − 1.11·10-s + 0.563·11-s + 0.476·13-s − 0.268·14-s + 1/4·16-s + 0.693·17-s + 0.0641·19-s − 0.789·20-s + 0.398·22-s + 0.480·23-s + 1.49·25-s + 0.337·26-s − 0.189·28-s + 1.21·29-s − 0.467·31-s + 0.176·32-s + 0.490·34-s + 0.598·35-s − 1.28·37-s + 0.0453·38-s − 0.558·40-s − 1.57·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(18\) = \(2 \cdot 3^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(124.169\) |
| Root analytic conductor: | \(11.1431\) |
| Motivic weight: | \(33\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 18,\ (\ :33/2),\ -1)\) |
Particular Values
| \(L(17)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{35}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - p^{16} T \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 + 21551965302 p^{2} T + p^{33} T^{2} \) |
| 7 | \( 1 + 4763901578824 p T + p^{33} T^{2} \) | |
| 11 | \( 1 - 7807478511398748 p T + p^{33} T^{2} \) | |
| 13 | \( 1 - 88004154528915782 p T + p^{33} T^{2} \) | |
| 17 | \( 1 - 8183157392316801294 p T + p^{33} T^{2} \) | |
| 19 | \( 1 - 4247105272322643740 p T + p^{33} T^{2} \) | |
| 23 | \( 1 - \)\(61\!\cdots\!28\)\( p T + p^{33} T^{2} \) | |
| 29 | \( 1 - \)\(56\!\cdots\!10\)\( p T + p^{33} T^{2} \) | |
| 31 | \( 1 + \)\(61\!\cdots\!88\)\( p T + p^{33} T^{2} \) | |
| 37 | \( 1 + \)\(96\!\cdots\!98\)\( T + p^{33} T^{2} \) | |
| 41 | \( 1 + \)\(64\!\cdots\!42\)\( T + p^{33} T^{2} \) | |
| 43 | \( 1 + \)\(81\!\cdots\!84\)\( T + p^{33} T^{2} \) | |
| 47 | \( 1 - \)\(62\!\cdots\!68\)\( T + p^{33} T^{2} \) | |
| 53 | \( 1 - \)\(21\!\cdots\!94\)\( T + p^{33} T^{2} \) | |
| 59 | \( 1 + \)\(29\!\cdots\!20\)\( T + p^{33} T^{2} \) | |
| 61 | \( 1 + \)\(45\!\cdots\!58\)\( T + p^{33} T^{2} \) | |
| 67 | \( 1 - \)\(11\!\cdots\!12\)\( T + p^{33} T^{2} \) | |
| 71 | \( 1 + \)\(25\!\cdots\!72\)\( T + p^{33} T^{2} \) | |
| 73 | \( 1 + \)\(28\!\cdots\!74\)\( T + p^{33} T^{2} \) | |
| 79 | \( 1 - \)\(92\!\cdots\!20\)\( T + p^{33} T^{2} \) | |
| 83 | \( 1 - \)\(16\!\cdots\!04\)\( T + p^{33} T^{2} \) | |
| 89 | \( 1 - \)\(20\!\cdots\!10\)\( T + p^{33} T^{2} \) | |
| 97 | \( 1 - \)\(22\!\cdots\!42\)\( T + p^{33} T^{2} \) | |
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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
−11.73243350868881462269913725352, −10.51706776836925879250840490578, −8.723488149304842098157463229891, −7.53270651297394893083078043988, −6.48715115074258985251923627340, −4.93944203351182696008605074011, −3.79268599961161188636273349006, −3.13076462642778461988700976318, −1.28708510680729957072325253612, 0, 1.28708510680729957072325253612, 3.13076462642778461988700976318, 3.79268599961161188636273349006, 4.93944203351182696008605074011, 6.48715115074258985251923627340, 7.53270651297394893083078043988, 8.723488149304842098157463229891, 10.51706776836925879250840490578, 11.73243350868881462269913725352