Dirichlet series
| L(s) = 1 | − 1.63e4·2-s − 2.79e6·3-s + 2.68e8·4-s + 6.65e9·5-s + 4.57e10·6-s + 1.43e12·7-s − 4.39e12·8-s − 6.08e13·9-s − 1.08e14·10-s − 7.77e14·11-s − 7.49e14·12-s − 2.70e16·13-s − 2.34e16·14-s − 1.85e16·15-s + 7.20e16·16-s + 6.23e17·17-s + 9.96e17·18-s + 2.33e16·19-s + 1.78e18·20-s − 4.00e18·21-s + 1.27e19·22-s − 1.01e20·23-s + 1.22e19·24-s − 1.42e20·25-s + 4.43e20·26-s + 3.61e20·27-s + 3.84e20·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.337·3-s + 1/2·4-s + 0.487·5-s + 0.238·6-s + 0.798·7-s − 0.353·8-s − 0.886·9-s − 0.344·10-s − 0.616·11-s − 0.168·12-s − 1.90·13-s − 0.564·14-s − 0.164·15-s + 1/4·16-s + 0.898·17-s + 0.626·18-s + 0.00671·19-s + 0.243·20-s − 0.269·21-s + 0.436·22-s − 1.83·23-s + 0.119·24-s − 0.762·25-s + 1.34·26-s + 0.635·27-s + 0.399·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(2\) |
| Sign: | $-1$ |
| Analytic conductor: | \(10.6556\) |
| Root analytic conductor: | \(3.26429\) |
| Motivic weight: | \(29\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 2,\ (\ :29/2),\ -1)\) |
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 | \( 1 + p^{14} T \) |
| good | 3 | \( 1 + 11492 p^{5} T + p^{29} T^{2} \) |
| 5 | \( 1 - 1330371294 p T + p^{29} T^{2} \) | |
| 7 | \( 1 - 29235070952 p^{2} T + p^{29} T^{2} \) | |
| 11 | \( 1 + 70638420377868 p T + p^{29} T^{2} \) | |
| 13 | \( 1 + 2084288506988362 p T + p^{29} T^{2} \) | |
| 17 | \( 1 - 623720384075229138 T + p^{29} T^{2} \) | |
| 19 | \( 1 - 1231450551444620 p T + p^{29} T^{2} \) | |
| 23 | \( 1 + \)\(10\!\cdots\!96\)\( T + p^{29} T^{2} \) | |
| 29 | \( 1 + \)\(19\!\cdots\!50\)\( T + p^{29} T^{2} \) | |
| 31 | \( 1 + \)\(44\!\cdots\!08\)\( T + p^{29} T^{2} \) | |
| 37 | \( 1 - \)\(68\!\cdots\!78\)\( T + p^{29} T^{2} \) | |
| 41 | \( 1 + \)\(13\!\cdots\!58\)\( T + p^{29} T^{2} \) | |
| 43 | \( 1 + \)\(65\!\cdots\!96\)\( T + p^{29} T^{2} \) | |
| 47 | \( 1 - \)\(18\!\cdots\!28\)\( T + p^{29} T^{2} \) | |
| 53 | \( 1 - \)\(65\!\cdots\!14\)\( T + p^{29} T^{2} \) | |
| 59 | \( 1 - \)\(44\!\cdots\!00\)\( T + p^{29} T^{2} \) | |
| 61 | \( 1 - \)\(35\!\cdots\!82\)\( T + p^{29} T^{2} \) | |
| 67 | \( 1 + \)\(38\!\cdots\!52\)\( T + p^{29} T^{2} \) | |
| 71 | \( 1 - \)\(72\!\cdots\!72\)\( T + p^{29} T^{2} \) | |
| 73 | \( 1 - \)\(72\!\cdots\!74\)\( T + p^{29} T^{2} \) | |
| 79 | \( 1 + \)\(20\!\cdots\!60\)\( T + p^{29} T^{2} \) | |
| 83 | \( 1 + \)\(72\!\cdots\!36\)\( T + p^{29} T^{2} \) | |
| 89 | \( 1 - \)\(13\!\cdots\!90\)\( T + p^{29} T^{2} \) | |
| 97 | \( 1 - \)\(22\!\cdots\!38\)\( T + p^{29} T^{2} \) | |
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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
−20.02376809385089576048093839028, −18.01306522708052658857765247446, −16.84330969952440568675630218017, −14.52286368909969979431555115523, −11.85050229904196924382496261333, −9.987983134515026750411946322362, −7.84888592463133211101043410300, −5.47416387335663328419677168437, −2.20837635375416625120828932042, 0, 2.20837635375416625120828932042, 5.47416387335663328419677168437, 7.84888592463133211101043410300, 9.987983134515026750411946322362, 11.85050229904196924382496261333, 14.52286368909969979431555115523, 16.84330969952440568675630218017, 18.01306522708052658857765247446, 20.02376809385089576048093839028