# Properties

 Label 2-230-23.12-c3-0-9 Degree $2$ Conductor $230$ Sign $0.997 - 0.0661i$ Analytic cond. $13.5704$ Root an. cond. $3.68380$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.68 + 1.08i)2-s + (−1.10 + 7.65i)3-s + (1.66 − 3.63i)4-s + (−4.79 + 1.40i)5-s + (−6.42 − 14.0i)6-s + (−11.9 − 13.7i)7-s + (1.13 + 7.91i)8-s + (−31.4 − 9.22i)9-s + (6.54 − 7.55i)10-s + (10.1 + 6.52i)11-s + (26.0 + 16.7i)12-s + (59.7 − 68.9i)13-s + (34.9 + 10.2i)14-s + (−5.50 − 38.2i)15-s + (−10.4 − 12.0i)16-s + (−23.5 − 51.4i)17-s + ⋯
 L(s)  = 1 + (−0.594 + 0.382i)2-s + (−0.211 + 1.47i)3-s + (0.207 − 0.454i)4-s + (−0.429 + 0.125i)5-s + (−0.436 − 0.956i)6-s + (−0.644 − 0.743i)7-s + (0.0503 + 0.349i)8-s + (−1.16 − 0.341i)9-s + (0.207 − 0.238i)10-s + (0.278 + 0.178i)11-s + (0.625 + 0.402i)12-s + (1.27 − 1.47i)13-s + (0.667 + 0.195i)14-s + (−0.0946 − 0.658i)15-s + (−0.163 − 0.188i)16-s + (−0.335 − 0.734i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0661i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0661i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$230$$    =    $$2 \cdot 5 \cdot 23$$ Sign: $0.997 - 0.0661i$ Analytic conductor: $$13.5704$$ Root analytic conductor: $$3.68380$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{230} (81, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 230,\ (\ :3/2),\ 0.997 - 0.0661i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.799828 + 0.0265028i$$ $$L(\frac12)$$ $$\approx$$ $$0.799828 + 0.0265028i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.68 - 1.08i)T$$
5 $$1 + (4.79 - 1.40i)T$$
23 $$1 + (80.7 + 75.1i)T$$
good3 $$1 + (1.10 - 7.65i)T + (-25.9 - 7.60i)T^{2}$$
7 $$1 + (11.9 + 13.7i)T + (-48.8 + 339. i)T^{2}$$
11 $$1 + (-10.1 - 6.52i)T + (552. + 1.21e3i)T^{2}$$
13 $$1 + (-59.7 + 68.9i)T + (-312. - 2.17e3i)T^{2}$$
17 $$1 + (23.5 + 51.4i)T + (-3.21e3 + 3.71e3i)T^{2}$$
19 $$1 + (18.4 - 40.3i)T + (-4.49e3 - 5.18e3i)T^{2}$$
29 $$1 + (-109. - 240. i)T + (-1.59e4 + 1.84e4i)T^{2}$$
31 $$1 + (20.9 + 145. i)T + (-2.85e4 + 8.39e3i)T^{2}$$
37 $$1 + (-184. - 54.2i)T + (4.26e4 + 2.73e4i)T^{2}$$
41 $$1 + (-217. + 63.8i)T + (5.79e4 - 3.72e4i)T^{2}$$
43 $$1 + (-55.5 + 386. i)T + (-7.62e4 - 2.23e4i)T^{2}$$
47 $$1 + 352.T + 1.03e5T^{2}$$
53 $$1 + (-101. - 117. i)T + (-2.11e4 + 1.47e5i)T^{2}$$
59 $$1 + (-51.7 + 59.7i)T + (-2.92e4 - 2.03e5i)T^{2}$$
61 $$1 + (-48.2 - 335. i)T + (-2.17e5 + 6.39e4i)T^{2}$$
67 $$1 + (-614. + 394. i)T + (1.24e5 - 2.73e5i)T^{2}$$
71 $$1 + (-383. + 246. i)T + (1.48e5 - 3.25e5i)T^{2}$$
73 $$1 + (-443. + 970. i)T + (-2.54e5 - 2.93e5i)T^{2}$$
79 $$1 + (696. - 803. i)T + (-7.01e4 - 4.88e5i)T^{2}$$
83 $$1 + (-220. - 64.6i)T + (4.81e5 + 3.09e5i)T^{2}$$
89 $$1 + (-182. + 1.26e3i)T + (-6.76e5 - 1.98e5i)T^{2}$$
97 $$1 + (640. - 188. i)T + (7.67e5 - 4.93e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$