Dirichlet series
| L(s) = 1 | + 1.63e4·2-s + 4.78e6·3-s + 2.68e8·4-s − 2.10e10·5-s + 7.83e10·6-s + 1.54e12·7-s + 4.39e12·8-s + 2.28e13·9-s − 3.44e14·10-s − 1.76e15·11-s + 1.28e15·12-s − 2.81e15·13-s + 2.52e16·14-s − 1.00e17·15-s + 7.20e16·16-s − 1.21e18·17-s + 3.74e17·18-s + 2.69e18·19-s − 5.63e18·20-s + 7.36e18·21-s − 2.89e19·22-s − 8.30e19·23-s + 2.10e19·24-s + 2.54e20·25-s − 4.61e19·26-s + 1.09e20·27-s + 4.13e20·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.53·5-s + 0.408·6-s + 0.858·7-s + 0.353·8-s + 1/3·9-s − 1.08·10-s − 1.40·11-s + 0.288·12-s − 0.198·13-s + 0.607·14-s − 0.888·15-s + 1/4·16-s − 1.75·17-s + 0.235·18-s + 0.774·19-s − 0.769·20-s + 0.495·21-s − 0.990·22-s − 1.49·23-s + 0.204·24-s + 1.36·25-s − 0.140·26-s + 0.192·27-s + 0.429·28-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 | \( 1 + 168030978 p^{3} T + p^{29} T^{2} \) |
| 7 | \( 1 - 31441408568 p^{2} T + p^{29} T^{2} \) | |
| 11 | \( 1 + 160465573043868 p T + p^{29} T^{2} \) | |
| 13 | \( 1 + 216733378891498 p T + p^{29} T^{2} \) | |
| 17 | \( 1 + 1217344130484395598 T + p^{29} T^{2} \) | |
| 19 | \( 1 - 2696824417300672340 T + p^{29} T^{2} \) | |
| 23 | \( 1 + 83091607940596084104 T + p^{29} T^{2} \) | |
| 29 | \( 1 + \)\(37\!\cdots\!90\)\( T + p^{29} T^{2} \) | |
| 31 | \( 1 + \)\(25\!\cdots\!88\)\( T + p^{29} T^{2} \) | |
| 37 | \( 1 + \)\(55\!\cdots\!58\)\( T + p^{29} T^{2} \) | |
| 41 | \( 1 + \)\(39\!\cdots\!98\)\( T + p^{29} T^{2} \) | |
| 43 | \( 1 - \)\(21\!\cdots\!16\)\( T + p^{29} T^{2} \) | |
| 47 | \( 1 - \)\(25\!\cdots\!72\)\( T + p^{29} T^{2} \) | |
| 53 | \( 1 - \)\(57\!\cdots\!66\)\( T + p^{29} T^{2} \) | |
| 59 | \( 1 - \)\(64\!\cdots\!60\)\( T + p^{29} T^{2} \) | |
| 61 | \( 1 + \)\(11\!\cdots\!98\)\( T + p^{29} T^{2} \) | |
| 67 | \( 1 + \)\(24\!\cdots\!88\)\( T + p^{29} T^{2} \) | |
| 71 | \( 1 - \)\(61\!\cdots\!92\)\( T + p^{29} T^{2} \) | |
| 73 | \( 1 - \)\(14\!\cdots\!46\)\( T + p^{29} T^{2} \) | |
| 79 | \( 1 - \)\(24\!\cdots\!60\)\( T + p^{29} T^{2} \) | |
| 83 | \( 1 - \)\(52\!\cdots\!56\)\( T + p^{29} T^{2} \) | |
| 89 | \( 1 - \)\(61\!\cdots\!90\)\( T + p^{29} T^{2} \) | |
| 97 | \( 1 + \)\(62\!\cdots\!78\)\( T + p^{29} T^{2} \) | |
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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
−15.19897443619041742714408524516, −13.65120762792146067717315991152, −12.08202856367195423410634533594, −10.83800431225488915856340337597, −8.308093971112687450386561719226, −7.35499106573289120624979626834, −4.93034575380706355345921368794, −3.76261443642666072922237399897, −2.22099086297714268102338091965, 0, 2.22099086297714268102338091965, 3.76261443642666072922237399897, 4.93034575380706355345921368794, 7.35499106573289120624979626834, 8.308093971112687450386561719226, 10.83800431225488915856340337597, 12.08202856367195423410634533594, 13.65120762792146067717315991152, 15.19897443619041742714408524516