Properties

 Degree $2$ Conductor $37$ Sign $1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + 3-s − 2·4-s − 7-s − 2·9-s + 3·11-s − 2·12-s − 4·13-s + 4·16-s + 6·17-s + 2·19-s − 21-s + 6·23-s − 5·25-s − 5·27-s + 2·28-s − 6·29-s − 4·31-s + 3·33-s + 4·36-s + 37-s − 4·39-s − 9·41-s + 8·43-s − 6·44-s + 3·47-s + 4·48-s − 6·49-s + ⋯
 L(s)  = 1 + 0.577·3-s − 4-s − 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.577·12-s − 1.10·13-s + 16-s + 1.45·17-s + 0.458·19-s − 0.218·21-s + 1.25·23-s − 25-s − 0.962·27-s + 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.522·33-s + 2/3·36-s + 0.164·37-s − 0.640·39-s − 1.40·41-s + 1.21·43-s − 0.904·44-s + 0.437·47-s + 0.577·48-s − 6/7·49-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$37$$ Sign: $1$ Motivic weight: $$1$$ Character: $\chi_{37} (1, \cdot )$ Sato-Tate group: $\mathrm{SU}(2)$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 37,\ (\ :1/2),\ 1)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.7256810619$$ $$L(\frac12)$$ $$\approx$$ $$0.7256810619$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad37 $$1 - T$$
good2 $$1 + p T^{2}$$
3 $$1 - T + p T^{2}$$
5 $$1 + p T^{2}$$
7 $$1 + T + p T^{2}$$
11 $$1 - 3 T + p T^{2}$$
13 $$1 + 4 T + p T^{2}$$
17 $$1 - 6 T + p T^{2}$$
19 $$1 - 2 T + p T^{2}$$
23 $$1 - 6 T + p T^{2}$$
29 $$1 + 6 T + p T^{2}$$
31 $$1 + 4 T + p T^{2}$$
41 $$1 + 9 T + p T^{2}$$
43 $$1 - 8 T + p T^{2}$$
47 $$1 - 3 T + p T^{2}$$
53 $$1 + 3 T + p T^{2}$$
59 $$1 - 12 T + p T^{2}$$
61 $$1 - 8 T + p T^{2}$$
67 $$1 + 4 T + p T^{2}$$
71 $$1 + 15 T + p T^{2}$$
73 $$1 - 11 T + p T^{2}$$
79 $$1 + 10 T + p T^{2}$$
83 $$1 - 9 T + p T^{2}$$
89 $$1 - 6 T + p T^{2}$$
97 $$1 - 8 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$