Dirichlet series
| L(s) = 1 | + 1.33e8·3-s + 5.38e11·5-s + 3.33e13·7-s + 1.21e16·9-s + 8.58e16·11-s + 1.14e18·13-s + 7.16e19·15-s − 1.39e20·17-s − 8.06e19·19-s + 4.43e21·21-s + 1.41e22·23-s + 1.73e23·25-s + 8.74e23·27-s − 1.63e24·29-s + 1.89e24·31-s + 1.14e25·33-s + 1.79e25·35-s − 9.64e25·37-s + 1.52e26·39-s + 6.41e26·41-s + 8.17e26·43-s + 6.53e27·45-s + 6.22e27·47-s − 6.61e27·49-s − 1.85e28·51-s − 2.13e28·53-s + 4.62e28·55-s + ⋯ |
| L(s) = 1 | + 1.78·3-s + 1.57·5-s + 0.379·7-s + 2.18·9-s + 0.563·11-s + 0.476·13-s + 2.81·15-s − 0.693·17-s − 0.0641·19-s + 0.676·21-s + 0.480·23-s + 1.49·25-s + 2.10·27-s − 1.21·29-s + 0.467·31-s + 1.00·33-s + 0.598·35-s − 1.28·37-s + 0.850·39-s + 1.57·41-s + 0.913·43-s + 3.44·45-s + 1.60·47-s − 0.856·49-s − 1.23·51-s − 0.755·53-s + 0.889·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(16\) = \(2^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(110.372\) |
| Root analytic conductor: | \(10.5058\) |
| Motivic weight: | \(33\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((2,\ 16,\ (\ :33/2),\ 1)\) |
Particular Values
| \(L(17)\) | \(\approx\) | \(7.113268194\) |
| \(L(\frac12)\) | \(\approx\) | \(7.113268194\) |
| \(L(\frac{35}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 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
−12.84089346670686978739838696752, −10.65226886031872023349321635786, −9.322198537589821445715864418176, −8.909332740952746506525573876788, −7.44502574436192333668272294878, −6.06585849536248334458552640254, −4.40722137056938184846369479598, −3.05394694313291105963626214645, −2.03706819371030997466337730786, −1.37319871623788687131781904559, 1.37319871623788687131781904559, 2.03706819371030997466337730786, 3.05394694313291105963626214645, 4.40722137056938184846369479598, 6.06585849536248334458552640254, 7.44502574436192333668272294878, 8.909332740952746506525573876788, 9.322198537589821445715864418176, 10.65226886031872023349321635786, 12.84089346670686978739838696752