Dirichlet series
| L(s) = 1 | + 1.31e5·2-s + 1.59e8·3-s + 1.71e10·4-s − 2.83e12·5-s + 2.09e13·6-s − 7.82e14·7-s + 2.25e15·8-s − 2.44e16·9-s − 3.72e17·10-s + 7.38e17·11-s + 2.74e18·12-s + 1.22e19·13-s − 1.02e20·14-s − 4.54e20·15-s + 2.95e20·16-s − 5.84e21·17-s − 3.20e21·18-s − 1.00e22·19-s − 4.87e22·20-s − 1.25e23·21-s + 9.67e22·22-s + 4.89e23·23-s + 3.60e23·24-s + 5.14e24·25-s + 1.60e24·26-s − 1.19e25·27-s − 1.34e25·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.715·3-s + 1/2·4-s − 1.66·5-s + 0.505·6-s − 1.27·7-s + 0.353·8-s − 0.488·9-s − 1.17·10-s + 0.440·11-s + 0.357·12-s + 0.392·13-s − 0.898·14-s − 1.18·15-s + 1/4·16-s − 1.71·17-s − 0.345·18-s − 0.422·19-s − 0.831·20-s − 0.908·21-s + 0.311·22-s + 0.722·23-s + 0.252·24-s + 1.76·25-s + 0.277·26-s − 1.06·27-s − 0.635·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(36-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+35/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(2\) |
| Sign: | $-1$ |
| Analytic conductor: | \(15.5190\) |
| Root analytic conductor: | \(3.93941\) |
| Motivic weight: | \(35\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 2,\ (\ :35/2),\ -1)\) |
Particular Values
| \(L(18)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{37}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - p^{17} T \) |
| good | 3 | \( 1 - 219388 p^{6} T + p^{35} T^{2} \) |
| 5 | \( 1 + 567748515738 p T + p^{35} T^{2} \) | |
| 7 | \( 1 + 15964936060456 p^{2} T + p^{35} T^{2} \) | |
| 11 | \( 1 - 6103322984829012 p^{2} T + p^{35} T^{2} \) | |
| 13 | \( 1 - 942303923082785774 p T + p^{35} T^{2} \) | |
| 17 | \( 1 + \)\(34\!\cdots\!42\)\( p T + p^{35} T^{2} \) | |
| 19 | \( 1 + \)\(53\!\cdots\!80\)\( p T + p^{35} T^{2} \) | |
| 23 | \( 1 - \)\(48\!\cdots\!72\)\( T + p^{35} T^{2} \) | |
| 29 | \( 1 - \)\(57\!\cdots\!70\)\( T + p^{35} T^{2} \) | |
| 31 | \( 1 + \)\(61\!\cdots\!08\)\( p T + p^{35} T^{2} \) | |
| 37 | \( 1 - \)\(30\!\cdots\!46\)\( T + p^{35} T^{2} \) | |
| 41 | \( 1 + \)\(11\!\cdots\!98\)\( T + p^{35} T^{2} \) | |
| 43 | \( 1 + \)\(15\!\cdots\!08\)\( T + p^{35} T^{2} \) | |
| 47 | \( 1 + \)\(13\!\cdots\!24\)\( T + p^{35} T^{2} \) | |
| 53 | \( 1 + \)\(16\!\cdots\!98\)\( T + p^{35} T^{2} \) | |
| 59 | \( 1 - \)\(18\!\cdots\!40\)\( T + p^{35} T^{2} \) | |
| 61 | \( 1 - \)\(91\!\cdots\!02\)\( T + p^{35} T^{2} \) | |
| 67 | \( 1 + \)\(10\!\cdots\!64\)\( T + p^{35} T^{2} \) | |
| 71 | \( 1 + \)\(82\!\cdots\!48\)\( T + p^{35} T^{2} \) | |
| 73 | \( 1 + \)\(24\!\cdots\!78\)\( T + p^{35} T^{2} \) | |
| 79 | \( 1 + \)\(51\!\cdots\!80\)\( T + p^{35} T^{2} \) | |
| 83 | \( 1 + \)\(48\!\cdots\!68\)\( T + p^{35} T^{2} \) | |
| 89 | \( 1 + \)\(12\!\cdots\!90\)\( T + p^{35} T^{2} \) | |
| 97 | \( 1 + \)\(65\!\cdots\!74\)\( T + p^{35} T^{2} \) | |
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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
−19.42357754450956700323629345564, −16.14422364142802242180978570506, −14.97177213660149549818970643613, −13.05020417893333870047803216694, −11.37938432030516438591164614878, −8.646450023499857407243191919963, −6.77828570256558341258414517265, −4.06588325037257894361366761584, −2.96289097694313536139586547016, 0, 2.96289097694313536139586547016, 4.06588325037257894361366761584, 6.77828570256558341258414517265, 8.646450023499857407243191919963, 11.37938432030516438591164614878, 13.05020417893333870047803216694, 14.97177213660149549818970643613, 16.14422364142802242180978570506, 19.42357754450956700323629345564