Dirichlet series
| L(s) = 1 | + 1.96e11·3-s + 2.06e16·5-s + 5.11e19·7-s + 1.20e22·9-s − 5.29e24·11-s − 1.25e26·13-s + 4.06e27·15-s − 4.48e28·17-s + 1.11e30·19-s + 1.00e31·21-s + 1.78e32·23-s − 2.83e32·25-s − 2.85e33·27-s − 3.05e34·29-s + 1.10e35·31-s − 1.04e36·33-s + 1.05e36·35-s + 6.20e36·37-s − 2.45e37·39-s − 3.56e37·41-s − 3.38e38·43-s + 2.49e38·45-s + 2.29e39·47-s − 2.62e39·49-s − 8.81e39·51-s − 2.95e40·53-s − 1.09e41·55-s + ⋯ |
| L(s) = 1 | + 1.20·3-s + 0.775·5-s + 0.706·7-s + 0.454·9-s − 1.78·11-s − 0.830·13-s + 0.935·15-s − 0.544·17-s + 0.988·19-s + 0.852·21-s + 1.77·23-s − 0.398·25-s − 0.658·27-s − 1.31·29-s + 0.994·31-s − 2.15·33-s + 0.547·35-s + 0.870·37-s − 1.00·39-s − 0.448·41-s − 1.38·43-s + 0.352·45-s + 1.16·47-s − 0.500·49-s − 0.656·51-s − 0.892·53-s − 1.38·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(48-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+47/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(16\) = \(2^{4}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(223.852\) |
| Root analytic conductor: | \(14.9616\) |
| Motivic weight: | \(47\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 16,\ (\ :47/2),\ -1)\) |
Particular Values
| \(L(24)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{49}{2})\) | 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 | \( 1 - 809195804 p^{5} T + p^{47} T^{2} \) |
| 5 | \( 1 - 165359697351846 p^{3} T + p^{47} T^{2} \) | |
| 7 | \( 1 - 1044344527283602504 p^{2} T + p^{47} T^{2} \) | |
| 11 | \( 1 + \)\(43\!\cdots\!12\)\( p^{2} T + p^{47} T^{2} \) | |
| 13 | \( 1 + \)\(74\!\cdots\!58\)\( p^{2} T + p^{47} T^{2} \) | |
| 17 | \( 1 + \)\(26\!\cdots\!78\)\( p T + p^{47} T^{2} \) | |
| 19 | \( 1 - \)\(58\!\cdots\!00\)\( p T + p^{47} T^{2} \) | |
| 23 | \( 1 - \)\(77\!\cdots\!64\)\( p T + p^{47} T^{2} \) | |
| 29 | \( 1 + \)\(30\!\cdots\!90\)\( T + p^{47} T^{2} \) | |
| 31 | \( 1 - \)\(11\!\cdots\!28\)\( T + p^{47} T^{2} \) | |
| 37 | \( 1 - \)\(16\!\cdots\!42\)\( p T + p^{47} T^{2} \) | |
| 41 | \( 1 + \)\(86\!\cdots\!58\)\( p T + p^{47} T^{2} \) | |
| 43 | \( 1 + \)\(78\!\cdots\!76\)\( p T + p^{47} T^{2} \) | |
| 47 | \( 1 - \)\(22\!\cdots\!36\)\( T + p^{47} T^{2} \) | |
| 53 | \( 1 + \)\(29\!\cdots\!02\)\( T + p^{47} T^{2} \) | |
| 59 | \( 1 + \)\(40\!\cdots\!20\)\( T + p^{47} T^{2} \) | |
| 61 | \( 1 - \)\(57\!\cdots\!42\)\( T + p^{47} T^{2} \) | |
| 67 | \( 1 - \)\(18\!\cdots\!36\)\( T + p^{47} T^{2} \) | |
| 71 | \( 1 + \)\(55\!\cdots\!12\)\( T + p^{47} T^{2} \) | |
| 73 | \( 1 + \)\(79\!\cdots\!02\)\( T + p^{47} T^{2} \) | |
| 79 | \( 1 + \)\(88\!\cdots\!40\)\( T + p^{47} T^{2} \) | |
| 83 | \( 1 - \)\(73\!\cdots\!92\)\( T + p^{47} T^{2} \) | |
| 89 | \( 1 + \)\(82\!\cdots\!90\)\( T + p^{47} T^{2} \) | |
| 97 | \( 1 + \)\(61\!\cdots\!46\)\( T + p^{47} T^{2} \) | |
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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
−10.08617822685583289023026370439, −9.170993097398429030350047990961, −8.060165610416381045368652315138, −7.31769422791929664014118936470, −5.55123851445319859945722237933, −4.74258410470736826888595047576, −3.04962460000513013078800322229, −2.47943409185733294865204947404, −1.53498012929891415618983954991, 0, 1.53498012929891415618983954991, 2.47943409185733294865204947404, 3.04962460000513013078800322229, 4.74258410470736826888595047576, 5.55123851445319859945722237933, 7.31769422791929664014118936470, 8.060165610416381045368652315138, 9.170993097398429030350047990961, 10.08617822685583289023026370439