Dirichlet series
| L(s) = 1 | + 1.04e6·2-s + 1.09e12·4-s + 4.85e13·5-s − 1.19e17·7-s + 1.15e18·8-s + 5.08e19·10-s − 3.15e21·11-s − 1.14e22·13-s − 1.25e23·14-s + 1.20e24·16-s + 2.67e25·17-s + 6.79e25·19-s + 5.33e25·20-s − 3.30e27·22-s + 1.35e28·23-s − 4.31e28·25-s − 1.19e28·26-s − 1.31e29·28-s − 1.36e29·29-s + 3.06e30·31-s + 1.26e30·32-s + 2.80e31·34-s − 5.79e30·35-s − 2.21e32·37-s + 7.12e31·38-s + 5.59e31·40-s + 5.01e32·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.227·5-s − 0.565·7-s + 0.353·8-s + 0.160·10-s − 1.41·11-s − 0.166·13-s − 0.399·14-s + 1/4·16-s + 1.59·17-s + 0.414·19-s + 0.113·20-s − 0.999·22-s + 1.64·23-s − 0.948·25-s − 0.117·26-s − 0.282·28-s − 0.143·29-s + 0.818·31-s + 0.176·32-s + 1.12·34-s − 0.128·35-s − 1.57·37-s + 0.293·38-s + 0.0804·40-s + 0.435·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(42-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+41/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(18\) = \(2 \cdot 3^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(191.649\) |
| Root analytic conductor: | \(13.8437\) |
| Motivic weight: | \(41\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 18,\ (\ :41/2),\ -1)\) |
Particular Values
| \(L(21)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{43}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - p^{20} T \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 - 1940167805226 p^{2} T + p^{41} T^{2} \) |
| 7 | \( 1 + 2436580524421432 p^{2} T + p^{41} T^{2} \) | |
| 11 | \( 1 + \)\(28\!\cdots\!32\)\( p T + p^{41} T^{2} \) | |
| 13 | \( 1 + 67516566191869487746 p^{2} T + p^{41} T^{2} \) | |
| 17 | \( 1 - \)\(15\!\cdots\!74\)\( p T + p^{41} T^{2} \) | |
| 19 | \( 1 - \)\(18\!\cdots\!60\)\( p^{2} T + p^{41} T^{2} \) | |
| 23 | \( 1 - \)\(13\!\cdots\!04\)\( T + p^{41} T^{2} \) | |
| 29 | \( 1 + \)\(13\!\cdots\!10\)\( T + p^{41} T^{2} \) | |
| 31 | \( 1 - \)\(98\!\cdots\!52\)\( p T + p^{41} T^{2} \) | |
| 37 | \( 1 + \)\(59\!\cdots\!94\)\( p T + p^{41} T^{2} \) | |
| 41 | \( 1 - \)\(50\!\cdots\!38\)\( T + p^{41} T^{2} \) | |
| 43 | \( 1 + \)\(31\!\cdots\!84\)\( T + p^{41} T^{2} \) | |
| 47 | \( 1 + \)\(13\!\cdots\!92\)\( T + p^{41} T^{2} \) | |
| 53 | \( 1 - \)\(32\!\cdots\!14\)\( T + p^{41} T^{2} \) | |
| 59 | \( 1 + \)\(34\!\cdots\!20\)\( T + p^{41} T^{2} \) | |
| 61 | \( 1 + \)\(97\!\cdots\!78\)\( T + p^{41} T^{2} \) | |
| 67 | \( 1 - \)\(16\!\cdots\!52\)\( T + p^{41} T^{2} \) | |
| 71 | \( 1 + \)\(11\!\cdots\!32\)\( T + p^{41} T^{2} \) | |
| 73 | \( 1 - \)\(19\!\cdots\!66\)\( T + p^{41} T^{2} \) | |
| 79 | \( 1 + \)\(56\!\cdots\!80\)\( T + p^{41} T^{2} \) | |
| 83 | \( 1 - \)\(60\!\cdots\!84\)\( T + p^{41} T^{2} \) | |
| 89 | \( 1 + \)\(11\!\cdots\!90\)\( T + p^{41} T^{2} \) | |
| 97 | \( 1 + \)\(63\!\cdots\!98\)\( T + p^{41} T^{2} \) | |
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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
−10.59001561768409949465845246747, −9.692525788852059380819605104220, −8.045128574007036058925516849637, −7.00676614785234274848333531269, −5.67718661774929427659534630801, −4.98069800235671968110704742820, −3.40942278607602624181960276288, −2.72520307362267643608116637657, −1.34809569709815054591711406589, 0, 1.34809569709815054591711406589, 2.72520307362267643608116637657, 3.40942278607602624181960276288, 4.98069800235671968110704742820, 5.67718661774929427659534630801, 7.00676614785234274848333531269, 8.045128574007036058925516849637, 9.692525788852059380819605104220, 10.59001561768409949465845246747