Dirichlet series
| L(s) = 1 | − 6.55e4·2-s − 1.33e8·3-s + 4.29e9·4-s + 5.38e11·5-s + 8.71e12·6-s − 3.33e13·7-s − 2.81e14·8-s + 1.21e16·9-s − 3.53e16·10-s − 8.58e16·11-s − 5.71e17·12-s + 1.14e18·13-s + 2.18e18·14-s − 7.16e19·15-s + 1.84e19·16-s − 1.39e20·17-s − 7.95e20·18-s + 8.06e19·19-s + 2.31e21·20-s + 4.43e21·21-s + 5.62e21·22-s − 1.41e22·23-s + 3.74e22·24-s + 1.73e23·25-s − 7.49e22·26-s − 8.74e23·27-s − 1.43e23·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.78·3-s + 1/2·4-s + 1.57·5-s + 1.26·6-s − 0.379·7-s − 0.353·8-s + 2.18·9-s − 1.11·10-s − 0.563·11-s − 0.891·12-s + 0.476·13-s + 0.268·14-s − 2.81·15-s + 1/4·16-s − 0.693·17-s − 1.54·18-s + 0.0641·19-s + 0.789·20-s + 0.676·21-s + 0.398·22-s − 0.480·23-s + 0.630·24-s + 1.49·25-s − 0.337·26-s − 2.10·27-s − 0.189·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(2\) |
| Sign: | $-1$ |
| Analytic conductor: | \(13.7965\) |
| Root analytic conductor: | \(3.71437\) |
| Motivic weight: | \(33\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 2,\ (\ :33/2),\ -1)\) |
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 | \( 1 + p^{16} T \) |
| good | 3 | \( 1 + 547348 p^{5} T + p^{33} T^{2} \) |
| 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
−18.29832001039939388914654156345, −17.45589220960381495964401140666, −16.19879606248037124923831154988, −12.97968492005585142239162464054, −10.97703611627622764291287564403, −9.749950439731407749482580089758, −6.55520867329295846541464440379, −5.46997840159020256031697419986, −1.67932471411918471961525408574, 0, 1.67932471411918471961525408574, 5.46997840159020256031697419986, 6.55520867329295846541464440379, 9.749950439731407749482580089758, 10.97703611627622764291287564403, 12.97968492005585142239162464054, 16.19879606248037124923831154988, 17.45589220960381495964401140666, 18.29832001039939388914654156345