Dirichlet series
| L(s) = 1 | + 8.38e6·2-s − 1.96e11·3-s + 7.03e13·4-s + 2.06e16·5-s − 1.64e18·6-s − 5.11e19·7-s + 5.90e20·8-s + 1.20e22·9-s + 1.73e23·10-s + 5.29e24·11-s − 1.38e25·12-s − 1.25e26·13-s − 4.29e26·14-s − 4.06e27·15-s + 4.95e27·16-s − 4.48e28·17-s + 1.01e29·18-s − 1.11e30·19-s + 1.45e30·20-s + 1.00e31·21-s + 4.44e31·22-s − 1.78e32·23-s − 1.16e32·24-s − 2.83e32·25-s − 1.04e33·26-s + 2.85e33·27-s − 3.60e33·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.20·3-s + 1/2·4-s + 0.775·5-s − 0.852·6-s − 0.706·7-s + 0.353·8-s + 0.454·9-s + 0.548·10-s + 1.78·11-s − 0.602·12-s − 0.830·13-s − 0.499·14-s − 0.935·15-s + 1/4·16-s − 0.544·17-s + 0.321·18-s − 0.988·19-s + 0.387·20-s + 0.852·21-s + 1.26·22-s − 1.77·23-s − 0.426·24-s − 0.398·25-s − 0.587·26-s + 0.658·27-s − 0.353·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(48-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+47/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(2\) |
| Sign: | $-1$ |
| Analytic conductor: | \(27.9815\) |
| Root analytic conductor: | \(5.28975\) |
| Motivic weight: | \(47\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 2,\ (\ :47/2),\ -1)\) |
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 | \( 1 - p^{23} T \) |
| 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
−16.50690140691374186946979193440, −14.41934631887018138093812925902, −12.67467475024587745538068871040, −11.41167712682993798568045806326, −9.670791487927185637443775027015, −6.60017203139532905940607846544, −5.80879569785079717513056325231, −4.09167792003467702657852794553, −1.91234233200451036802198779504, 0, 1.91234233200451036802198779504, 4.09167792003467702657852794553, 5.80879569785079717513056325231, 6.60017203139532905940607846544, 9.670791487927185637443775027015, 11.41167712682993798568045806326, 12.67467475024587745538068871040, 14.41934631887018138093812925902, 16.50690140691374186946979193440