# Properties

 Label 2-230-115.88-c1-0-8 Degree $2$ Conductor $230$ Sign $0.648 + 0.761i$ Analytic cond. $1.83655$ Root an. cond. $1.35519$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.936 − 0.349i)2-s + (1.26 + 0.691i)3-s + (0.755 + 0.654i)4-s + (−0.584 − 2.15i)5-s + (−0.944 − 1.09i)6-s + (1.35 − 1.80i)7-s + (−0.479 − 0.877i)8-s + (−0.496 − 0.773i)9-s + (−0.206 + 2.22i)10-s + (−0.475 − 0.217i)11-s + (0.504 + 1.35i)12-s + (3.11 − 2.32i)13-s + (−1.90 + 1.22i)14-s + (0.751 − 3.13i)15-s + (0.142 + 0.989i)16-s + (−0.578 + 8.08i)17-s + ⋯
 L(s)  = 1 + (−0.662 − 0.247i)2-s + (0.730 + 0.399i)3-s + (0.377 + 0.327i)4-s + (−0.261 − 0.965i)5-s + (−0.385 − 0.445i)6-s + (0.511 − 0.683i)7-s + (−0.169 − 0.310i)8-s + (−0.165 − 0.257i)9-s + (−0.0652 + 0.704i)10-s + (−0.143 − 0.0654i)11-s + (0.145 + 0.390i)12-s + (0.863 − 0.646i)13-s + (−0.508 + 0.326i)14-s + (0.194 − 0.809i)15-s + (0.0355 + 0.247i)16-s + (−0.140 + 1.96i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$230$$    =    $$2 \cdot 5 \cdot 23$$ Sign: $0.648 + 0.761i$ Analytic conductor: $$1.83655$$ Root analytic conductor: $$1.35519$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{230} (203, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 230,\ (\ :1/2),\ 0.648 + 0.761i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.02806 - 0.474848i$$ $$L(\frac12)$$ $$\approx$$ $$1.02806 - 0.474848i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.936 + 0.349i)T$$
5 $$1 + (0.584 + 2.15i)T$$
23 $$1 + (0.430 - 4.77i)T$$
good3 $$1 + (-1.26 - 0.691i)T + (1.62 + 2.52i)T^{2}$$
7 $$1 + (-1.35 + 1.80i)T + (-1.97 - 6.71i)T^{2}$$
11 $$1 + (0.475 + 0.217i)T + (7.20 + 8.31i)T^{2}$$
13 $$1 + (-3.11 + 2.32i)T + (3.66 - 12.4i)T^{2}$$
17 $$1 + (0.578 - 8.08i)T + (-16.8 - 2.41i)T^{2}$$
19 $$1 + (-4.90 + 5.66i)T + (-2.70 - 18.8i)T^{2}$$
29 $$1 + (-1.93 + 1.67i)T + (4.12 - 28.7i)T^{2}$$
31 $$1 + (0.910 + 0.267i)T + (26.0 + 16.7i)T^{2}$$
37 $$1 + (-2.51 - 0.546i)T + (33.6 + 15.3i)T^{2}$$
41 $$1 + (5.19 + 3.34i)T + (17.0 + 37.2i)T^{2}$$
43 $$1 + (4.84 - 8.86i)T + (-23.2 - 36.1i)T^{2}$$
47 $$1 + (0.388 - 0.388i)T - 47iT^{2}$$
53 $$1 + (-7.32 - 5.48i)T + (14.9 + 50.8i)T^{2}$$
59 $$1 + (7.55 + 1.08i)T + (56.6 + 16.6i)T^{2}$$
61 $$1 + (3.43 - 11.6i)T + (-51.3 - 32.9i)T^{2}$$
67 $$1 + (3.30 - 8.86i)T + (-50.6 - 43.8i)T^{2}$$
71 $$1 + (2.35 + 5.15i)T + (-46.4 + 53.6i)T^{2}$$
73 $$1 + (-0.283 + 0.0202i)T + (72.2 - 10.3i)T^{2}$$
79 $$1 + (-0.529 + 3.68i)T + (-75.7 - 22.2i)T^{2}$$
83 $$1 + (1.46 - 6.74i)T + (-75.4 - 34.4i)T^{2}$$
89 $$1 + (-9.07 + 2.66i)T + (74.8 - 48.1i)T^{2}$$
97 $$1 + (1.22 + 5.61i)T + (-88.2 + 40.2i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$