Dirichlet series
| L(s) = 1 | + 3.27e4·2-s − 1.99e7·3-s + 1.07e9·4-s + 4.29e10·5-s − 6.54e11·6-s − 1.68e13·7-s + 3.51e13·8-s − 2.18e14·9-s + 1.40e15·10-s − 7.20e15·11-s − 2.14e16·12-s − 2.70e17·13-s − 5.51e17·14-s − 8.58e17·15-s + 1.15e18·16-s − 1.62e19·17-s − 7.15e18·18-s + 1.09e20·19-s + 4.61e19·20-s + 3.36e20·21-s − 2.36e20·22-s − 3.05e20·23-s − 7.03e20·24-s − 2.81e21·25-s − 8.84e21·26-s + 1.67e22·27-s − 1.80e22·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.804·3-s + 1/2·4-s + 0.629·5-s − 0.568·6-s − 1.34·7-s + 0.353·8-s − 0.353·9-s + 0.445·10-s − 0.520·11-s − 0.402·12-s − 1.46·13-s − 0.947·14-s − 0.506·15-s + 1/4·16-s − 1.37·17-s − 0.249·18-s + 1.64·19-s + 0.314·20-s + 1.07·21-s − 0.367·22-s − 0.239·23-s − 0.284·24-s − 0.603·25-s − 1.03·26-s + 1.08·27-s − 0.670·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+31/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(2\) |
| Sign: | $-1$ |
| Analytic conductor: | \(12.1754\) |
| Root analytic conductor: | \(3.48933\) |
| Motivic weight: | \(31\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 2,\ (\ :31/2),\ -1)\) |
Particular Values
| \(L(16)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{33}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - p^{15} T \) |
| good | 3 | \( 1 + 740156 p^{3} T + p^{31} T^{2} \) |
| 5 | \( 1 - 68722734 p^{4} T + p^{31} T^{2} \) | |
| 7 | \( 1 + 2405051285368 p T + p^{31} T^{2} \) | |
| 11 | \( 1 + 655257518635668 p T + p^{31} T^{2} \) | |
| 13 | \( 1 + 20773356529370914 p T + p^{31} T^{2} \) | |
| 17 | \( 1 + 957381350642703918 p T + p^{31} T^{2} \) | |
| 19 | \( 1 - \)\(10\!\cdots\!40\)\( T + p^{31} T^{2} \) | |
| 23 | \( 1 + \)\(30\!\cdots\!72\)\( T + p^{31} T^{2} \) | |
| 29 | \( 1 - \)\(11\!\cdots\!10\)\( p T + p^{31} T^{2} \) | |
| 31 | \( 1 - \)\(16\!\cdots\!12\)\( T + p^{31} T^{2} \) | |
| 37 | \( 1 + \)\(25\!\cdots\!46\)\( T + p^{31} T^{2} \) | |
| 41 | \( 1 + \)\(92\!\cdots\!38\)\( T + p^{31} T^{2} \) | |
| 43 | \( 1 - \)\(58\!\cdots\!88\)\( T + p^{31} T^{2} \) | |
| 47 | \( 1 + \)\(91\!\cdots\!56\)\( T + p^{31} T^{2} \) | |
| 53 | \( 1 - \)\(58\!\cdots\!98\)\( T + p^{31} T^{2} \) | |
| 59 | \( 1 + \)\(32\!\cdots\!20\)\( T + p^{31} T^{2} \) | |
| 61 | \( 1 - \)\(40\!\cdots\!22\)\( T + p^{31} T^{2} \) | |
| 67 | \( 1 - \)\(16\!\cdots\!64\)\( T + p^{31} T^{2} \) | |
| 71 | \( 1 + \)\(68\!\cdots\!68\)\( T + p^{31} T^{2} \) | |
| 73 | \( 1 + \)\(23\!\cdots\!62\)\( T + p^{31} T^{2} \) | |
| 79 | \( 1 + \)\(63\!\cdots\!20\)\( T + p^{31} T^{2} \) | |
| 83 | \( 1 + \)\(28\!\cdots\!12\)\( T + p^{31} T^{2} \) | |
| 89 | \( 1 - \)\(19\!\cdots\!10\)\( T + p^{31} T^{2} \) | |
| 97 | \( 1 + \)\(77\!\cdots\!86\)\( T + p^{31} T^{2} \) | |
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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
−19.82036241963506430028412251567, −17.47603438438533824530166581327, −15.89237566213601395673959016277, −13.61730904869683664435349480057, −12.02321796978139000163474493390, −9.973738672045923656707532446673, −6.63451214515703080832035004205, −5.21046160744584569408378307685, −2.73719898346923336203718419371, 0, 2.73719898346923336203718419371, 5.21046160744584569408378307685, 6.63451214515703080832035004205, 9.973738672045923656707532446673, 12.02321796978139000163474493390, 13.61730904869683664435349480057, 15.89237566213601395673959016277, 17.47603438438533824530166581327, 19.82036241963506430028412251567