Dirichlet series
| L(s) = 1 | − 1.04e6·2-s + 5.04e9·3-s + 1.09e12·4-s − 4.85e13·5-s − 5.28e15·6-s − 1.19e17·7-s − 1.15e18·8-s − 1.10e19·9-s + 5.08e19·10-s + 3.15e21·11-s + 5.54e21·12-s − 1.14e22·13-s + 1.25e23·14-s − 2.44e23·15-s + 1.20e24·16-s − 2.67e25·17-s + 1.15e25·18-s + 6.79e25·19-s − 5.33e25·20-s − 6.02e26·21-s − 3.30e27·22-s − 1.35e28·23-s − 5.81e27·24-s − 4.31e28·25-s + 1.19e28·26-s − 2.39e29·27-s − 1.31e29·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.835·3-s + 1/2·4-s − 0.227·5-s − 0.590·6-s − 0.565·7-s − 0.353·8-s − 0.302·9-s + 0.160·10-s + 1.41·11-s + 0.417·12-s − 0.166·13-s + 0.399·14-s − 0.189·15-s + 1/4·16-s − 1.59·17-s + 0.213·18-s + 0.414·19-s − 0.113·20-s − 0.472·21-s − 0.999·22-s − 1.64·23-s − 0.295·24-s − 0.948·25-s + 0.117·26-s − 1.08·27-s − 0.282·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(42-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+41/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(2\) |
| Sign: | $-1$ |
| Analytic conductor: | \(21.2943\) |
| Root analytic conductor: | \(4.61457\) |
| Motivic weight: | \(41\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 2,\ (\ :41/2),\ -1)\) |
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 | \( 1 + p^{20} T \) |
| good | 3 | \( 1 - 6918404 p^{6} T + p^{41} T^{2} \) |
| 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
−17.42635804582235831545073259863, −15.65383792604942111326636438046, −13.96921270041742277130727864878, −11.72817598372064561374838745052, −9.598517821897693977014552375650, −8.377379889930806691296240666831, −6.55461737338123172920406101733, −3.67838765534465907917156277613, −2.01911258865289250076112753959, 0, 2.01911258865289250076112753959, 3.67838765534465907917156277613, 6.55461737338123172920406101733, 8.377379889930806691296240666831, 9.598517821897693977014552375650, 11.72817598372064561374838745052, 13.96921270041742277130727864878, 15.65383792604942111326636438046, 17.42635804582235831545073259863