Dirichlet series
| L(s) = 1 | − 1.63e4·2-s − 4.78e6·3-s + 2.68e8·4-s + 1.95e9·5-s + 7.83e10·6-s − 1.58e12·7-s − 4.39e12·8-s + 2.28e13·9-s − 3.19e13·10-s + 3.03e14·11-s − 1.28e15·12-s + 1.74e16·13-s + 2.60e16·14-s − 9.33e15·15-s + 7.20e16·16-s − 5.28e16·17-s − 3.74e17·18-s + 3.26e18·19-s + 5.23e17·20-s + 7.60e18·21-s − 4.97e18·22-s + 4.42e19·23-s + 2.10e19·24-s − 1.82e20·25-s − 2.86e20·26-s − 1.09e20·27-s − 4.26e20·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.142·5-s + 0.408·6-s − 0.885·7-s − 0.353·8-s + 1/3·9-s − 0.101·10-s + 0.241·11-s − 0.288·12-s + 1.23·13-s + 0.626·14-s − 0.0825·15-s + 1/4·16-s − 0.0761·17-s − 0.235·18-s + 0.937·19-s + 0.0714·20-s + 0.511·21-s − 0.170·22-s + 0.795·23-s + 0.204·24-s − 0.979·25-s − 0.871·26-s − 0.192·27-s − 0.442·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(6\) = \(2 \cdot 3\) |
| Sign: | $-1$ |
| Analytic conductor: | \(31.9668\) |
| Root analytic conductor: | \(5.65392\) |
| Motivic weight: | \(29\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 6,\ (\ :29/2),\ -1)\) |
Particular Values
| \(L(15)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{31}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + p^{14} T \) |
| 3 | \( 1 + p^{14} T \) | |
| good | 5 | \( 1 - 3122526 p^{4} T + p^{29} T^{2} \) |
| 7 | \( 1 + 227110141456 p T + p^{29} T^{2} \) | |
| 11 | \( 1 - 2510502369252 p^{2} T + p^{29} T^{2} \) | |
| 13 | \( 1 - 1345161226540358 p T + p^{29} T^{2} \) | |
| 17 | \( 1 + 3109691254400766 p T + p^{29} T^{2} \) | |
| 19 | \( 1 - 3266460704183826620 T + p^{29} T^{2} \) | |
| 23 | \( 1 - 1922083515762359208 p T + p^{29} T^{2} \) | |
| 29 | \( 1 + \)\(46\!\cdots\!70\)\( T + p^{29} T^{2} \) | |
| 31 | \( 1 + \)\(33\!\cdots\!28\)\( T + p^{29} T^{2} \) | |
| 37 | \( 1 + \)\(90\!\cdots\!82\)\( T + p^{29} T^{2} \) | |
| 41 | \( 1 + \)\(39\!\cdots\!38\)\( T + p^{29} T^{2} \) | |
| 43 | \( 1 + \)\(73\!\cdots\!56\)\( T + p^{29} T^{2} \) | |
| 47 | \( 1 - \)\(14\!\cdots\!88\)\( T + p^{29} T^{2} \) | |
| 53 | \( 1 - \)\(13\!\cdots\!74\)\( T + p^{29} T^{2} \) | |
| 59 | \( 1 - \)\(18\!\cdots\!60\)\( T + p^{29} T^{2} \) | |
| 61 | \( 1 + \)\(40\!\cdots\!58\)\( T + p^{29} T^{2} \) | |
| 67 | \( 1 + \)\(21\!\cdots\!72\)\( T + p^{29} T^{2} \) | |
| 71 | \( 1 + \)\(13\!\cdots\!68\)\( T + p^{29} T^{2} \) | |
| 73 | \( 1 + \)\(18\!\cdots\!66\)\( T + p^{29} T^{2} \) | |
| 79 | \( 1 - \)\(22\!\cdots\!80\)\( T + p^{29} T^{2} \) | |
| 83 | \( 1 - \)\(23\!\cdots\!64\)\( T + p^{29} T^{2} \) | |
| 89 | \( 1 + \)\(10\!\cdots\!10\)\( T + p^{29} T^{2} \) | |
| 97 | \( 1 + \)\(95\!\cdots\!62\)\( T + p^{29} T^{2} \) | |
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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
−15.59697602723723591163978568193, −13.38801401482606141407558336960, −11.75251055520723675201892362446, −10.33472409030779432633663564596, −8.963551090442122626539520420494, −7.03772569243247688704878544707, −5.74028766118540309092711523259, −3.45411402976144106785361109829, −1.42818850567111383524969159462, 0, 1.42818850567111383524969159462, 3.45411402976144106785361109829, 5.74028766118540309092711523259, 7.03772569243247688704878544707, 8.963551090442122626539520420494, 10.33472409030779432633663564596, 11.75251055520723675201892362446, 13.38801401482606141407558336960, 15.59697602723723591163978568193