Properties

 Label 2-230-1.1-c1-0-0 Degree $2$ Conductor $230$ Sign $1$ Analytic cond. $1.83655$ Root an. cond. $1.35519$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 2-s − 2.79·3-s + 4-s − 5-s + 2.79·6-s − 1.79·7-s − 8-s + 4.79·9-s + 10-s − 0.791·11-s − 2.79·12-s + 5.79·13-s + 1.79·14-s + 2.79·15-s + 16-s + 0.791·17-s − 4.79·18-s + 5.79·19-s − 20-s + 5·21-s + 0.791·22-s + 23-s + 2.79·24-s + 25-s − 5.79·26-s − 4.99·27-s − 1.79·28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1.61·3-s + 0.5·4-s − 0.447·5-s + 1.13·6-s − 0.677·7-s − 0.353·8-s + 1.59·9-s + 0.316·10-s − 0.238·11-s − 0.805·12-s + 1.60·13-s + 0.478·14-s + 0.720·15-s + 0.250·16-s + 0.191·17-s − 1.12·18-s + 1.32·19-s − 0.223·20-s + 1.09·21-s + 0.168·22-s + 0.208·23-s + 0.569·24-s + 0.200·25-s − 1.13·26-s − 0.962·27-s − 0.338·28-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$230$$    =    $$2 \cdot 5 \cdot 23$$ Sign: $1$ Analytic conductor: $$1.83655$$ Root analytic conductor: $$1.35519$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{230} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 230,\ (\ :1/2),\ 1)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.4630194007$$ $$L(\frac12)$$ $$\approx$$ $$0.4630194007$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
5 $$1 + T$$
23 $$1 - T$$
good3 $$1 + 2.79T + 3T^{2}$$
7 $$1 + 1.79T + 7T^{2}$$
11 $$1 + 0.791T + 11T^{2}$$
13 $$1 - 5.79T + 13T^{2}$$
17 $$1 - 0.791T + 17T^{2}$$
19 $$1 - 5.79T + 19T^{2}$$
29 $$1 - 7.58T + 29T^{2}$$
31 $$1 + 3.37T + 31T^{2}$$
37 $$1 + 4T + 37T^{2}$$
41 $$1 + 6.79T + 41T^{2}$$
43 $$1 - 11.1T + 43T^{2}$$
47 $$1 + 4.41T + 47T^{2}$$
53 $$1 - 6T + 53T^{2}$$
59 $$1 + 13.5T + 59T^{2}$$
61 $$1 - 10.3T + 61T^{2}$$
67 $$1 - 11.1T + 67T^{2}$$
71 $$1 - 8.37T + 71T^{2}$$
73 $$1 - 12.7T + 73T^{2}$$
79 $$1 - 8T + 79T^{2}$$
83 $$1 + 6T + 83T^{2}$$
89 $$1 - 15.1T + 89T^{2}$$
97 $$1 + 7.95T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$