Dirichlet series
| L(s) = 1 | − 3.27e4·2-s − 1.43e7·3-s + 1.07e9·4-s + 2.14e10·5-s + 4.70e11·6-s + 1.46e13·7-s − 3.51e13·8-s + 2.05e14·9-s − 7.02e14·10-s + 1.02e16·11-s − 1.54e16·12-s − 7.33e16·13-s − 4.78e17·14-s − 3.07e17·15-s + 1.15e18·16-s − 1.45e19·17-s − 6.74e18·18-s + 4.71e19·19-s + 2.30e19·20-s − 2.09e20·21-s − 3.37e20·22-s − 7.27e20·23-s + 5.04e20·24-s − 4.19e21·25-s + 2.40e21·26-s − 2.95e21·27-s + 1.56e22·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.314·5-s + 0.408·6-s + 1.16·7-s − 0.353·8-s + 1/3·9-s − 0.222·10-s + 0.742·11-s − 0.288·12-s − 0.397·13-s − 0.822·14-s − 0.181·15-s + 1/4·16-s − 1.23·17-s − 0.235·18-s + 0.712·19-s + 0.157·20-s − 0.671·21-s − 0.525·22-s − 0.568·23-s + 0.204·24-s − 0.901·25-s + 0.281·26-s − 0.192·27-s + 0.581·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+31/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(6\) = \(2 \cdot 3\) |
| Sign: | $1$ |
| Analytic conductor: | \(36.5262\) |
| Root analytic conductor: | \(6.04369\) |
| Motivic weight: | \(31\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((2,\ 6,\ (\ :31/2),\ 1)\) |
Particular Values
| \(L(16)\) | \(\approx\) | \(1.435869661\) |
| \(L(\frac12)\) | \(\approx\) | \(1.435869661\) |
| \(L(\frac{33}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + p^{15} T \) |
| 3 | \( 1 + p^{15} T \) | |
| good | 5 | \( 1 - 4287222606 p T + p^{31} T^{2} \) |
| 7 | \( 1 - 298050133664 p^{2} T + p^{31} T^{2} \) | |
| 11 | \( 1 - 10293018436866252 T + p^{31} T^{2} \) | |
| 13 | \( 1 + 434037979962682 p^{2} T + p^{31} T^{2} \) | |
| 17 | \( 1 + 858779793185279742 p T + p^{31} T^{2} \) | |
| 19 | \( 1 - 2481387923053559900 p T + p^{31} T^{2} \) | |
| 23 | \( 1 + 31615975999979931336 p T + p^{31} T^{2} \) | |
| 29 | \( 1 - \)\(10\!\cdots\!50\)\( p T + p^{31} T^{2} \) | |
| 31 | \( 1 - \)\(15\!\cdots\!92\)\( T + p^{31} T^{2} \) | |
| 37 | \( 1 - \)\(36\!\cdots\!86\)\( T + p^{31} T^{2} \) | |
| 41 | \( 1 - \)\(16\!\cdots\!02\)\( T + p^{31} T^{2} \) | |
| 43 | \( 1 - \)\(33\!\cdots\!72\)\( T + p^{31} T^{2} \) | |
| 47 | \( 1 - \)\(61\!\cdots\!56\)\( T + p^{31} T^{2} \) | |
| 53 | \( 1 + \)\(75\!\cdots\!58\)\( T + p^{31} T^{2} \) | |
| 59 | \( 1 + \)\(31\!\cdots\!60\)\( T + p^{31} T^{2} \) | |
| 61 | \( 1 - \)\(24\!\cdots\!02\)\( T + p^{31} T^{2} \) | |
| 67 | \( 1 + \)\(22\!\cdots\!44\)\( T + p^{31} T^{2} \) | |
| 71 | \( 1 + \)\(12\!\cdots\!88\)\( T + p^{31} T^{2} \) | |
| 73 | \( 1 - \)\(13\!\cdots\!22\)\( T + p^{31} T^{2} \) | |
| 79 | \( 1 - \)\(40\!\cdots\!00\)\( T + p^{31} T^{2} \) | |
| 83 | \( 1 - \)\(40\!\cdots\!72\)\( T + p^{31} T^{2} \) | |
| 89 | \( 1 - \)\(58\!\cdots\!10\)\( T + p^{31} T^{2} \) | |
| 97 | \( 1 - \)\(67\!\cdots\!06\)\( T + p^{31} T^{2} \) | |
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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
−15.72278996468397063413051982543, −14.09773768942591882000515482196, −11.99408155678497279539439092777, −10.91342772855950508595428500189, −9.358432608583296080175133967499, −7.75958144209362613377949309367, −6.18456754625275668298494001953, −4.52637352131602005376717619612, −2.11874037283258946076700286642, −0.854587465772234981849564940917, 0.854587465772234981849564940917, 2.11874037283258946076700286642, 4.52637352131602005376717619612, 6.18456754625275668298494001953, 7.75958144209362613377949309367, 9.358432608583296080175133967499, 10.91342772855950508595428500189, 11.99408155678497279539439092777, 14.09773768942591882000515482196, 15.72278996468397063413051982543