Properties

 Degree $2$ Conductor $3$ Sign $-1$ Motivic weight $35$ Primitive yes Self-dual yes Analytic rank $1$

Related objects

Dirichlet series

 L(s)  = 1 + 2.18e5·2-s + 1.29e8·3-s + 1.34e10·4-s − 9.68e11·5-s + 2.82e13·6-s − 5.65e14·7-s − 4.57e15·8-s + 1.66e16·9-s − 2.11e17·10-s − 1.62e18·11-s + 1.73e18·12-s − 2.52e19·13-s − 1.23e20·14-s − 1.25e20·15-s − 1.46e21·16-s + 4.37e21·17-s + 3.64e21·18-s + 1.72e22·19-s − 1.29e22·20-s − 7.30e22·21-s − 3.54e23·22-s − 5.13e23·23-s − 5.91e23·24-s − 1.97e24·25-s − 5.51e24·26-s + 2.15e24·27-s − 7.57e24·28-s + ⋯
 L(s)  = 1 + 1.17·2-s + 0.577·3-s + 0.390·4-s − 0.567·5-s + 0.680·6-s − 0.918·7-s − 0.719·8-s + 0.333·9-s − 0.669·10-s − 0.968·11-s + 0.225·12-s − 0.809·13-s − 1.08·14-s − 0.327·15-s − 1.23·16-s + 1.28·17-s + 0.393·18-s + 0.720·19-s − 0.221·20-s − 0.530·21-s − 1.14·22-s − 0.758·23-s − 0.415·24-s − 0.677·25-s − 0.954·26-s + 0.192·27-s − 0.358·28-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(36-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+35/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$3$$ Sign: $-1$ Motivic weight: $$35$$ Character: $\chi_{3} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 3,\ (\ :35/2),\ -1)$$

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$$F_p(T)$
bad3 $$1 - 1.29e8T$$
good2 $$1 - 2.18e5T + 3.43e10T^{2}$$
5 $$1 + 9.68e11T + 2.91e24T^{2}$$
7 $$1 + 5.65e14T + 3.78e29T^{2}$$
11 $$1 + 1.62e18T + 2.81e36T^{2}$$
13 $$1 + 2.52e19T + 9.72e38T^{2}$$
17 $$1 - 4.37e21T + 1.16e43T^{2}$$
19 $$1 - 1.72e22T + 5.70e44T^{2}$$
23 $$1 + 5.13e23T + 4.57e47T^{2}$$
29 $$1 + 3.39e25T + 1.52e51T^{2}$$
31 $$1 + 9.89e25T + 1.57e52T^{2}$$
37 $$1 - 4.58e27T + 7.71e54T^{2}$$
41 $$1 + 2.87e28T + 2.80e56T^{2}$$
43 $$1 - 6.85e28T + 1.48e57T^{2}$$
47 $$1 + 4.79e26T + 3.33e58T^{2}$$
53 $$1 - 2.32e30T + 2.23e60T^{2}$$
59 $$1 - 5.75e30T + 9.54e61T^{2}$$
61 $$1 + 2.67e31T + 3.06e62T^{2}$$
67 $$1 - 7.66e31T + 8.17e63T^{2}$$
71 $$1 + 5.08e31T + 6.22e64T^{2}$$
73 $$1 + 2.74e32T + 1.64e65T^{2}$$
79 $$1 + 2.13e33T + 2.61e66T^{2}$$
83 $$1 + 2.22e33T + 1.47e67T^{2}$$
89 $$1 - 2.07e34T + 1.69e68T^{2}$$
97 $$1 - 1.55e34T + 3.44e69T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$