# Properties

 Label 2-234-13.3-c3-0-14 Degree $2$ Conductor $234$ Sign $-0.999 + 0.0256i$ Analytic cond. $13.8064$ Root an. cond. $3.71570$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1 − 1.73i)2-s + (−1.99 + 3.46i)4-s − 2·5-s + (2.5 − 4.33i)7-s + 7.99·8-s + (2 + 3.46i)10-s + (6.5 + 11.2i)11-s + (−13 − 45.0i)13-s − 10·14-s + (−8 − 13.8i)16-s + (13.5 − 23.3i)17-s + (−37.5 + 64.9i)19-s + (3.99 − 6.92i)20-s + (12.9 − 22.5i)22-s + (−93.5 − 161. i)23-s + ⋯
 L(s)  = 1 + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.178·5-s + (0.134 − 0.233i)7-s + 0.353·8-s + (0.0632 + 0.109i)10-s + (0.178 + 0.308i)11-s + (−0.277 − 0.960i)13-s − 0.190·14-s + (−0.125 − 0.216i)16-s + (0.192 − 0.333i)17-s + (−0.452 + 0.784i)19-s + (0.0447 − 0.0774i)20-s + (0.125 − 0.218i)22-s + (−0.847 − 1.46i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0256i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0256i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$234$$    =    $$2 \cdot 3^{2} \cdot 13$$ Sign: $-0.999 + 0.0256i$ Analytic conductor: $$13.8064$$ Root analytic conductor: $$3.71570$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{234} (55, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 234,\ (\ :3/2),\ -0.999 + 0.0256i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.00678483 - 0.529057i$$ $$L(\frac12)$$ $$\approx$$ $$0.00678483 - 0.529057i$$ $$L(\frac{5}{2})$$ 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 + 1.73i)T$$
3 $$1$$
13 $$1 + (13 + 45.0i)T$$
good5 $$1 + 2T + 125T^{2}$$
7 $$1 + (-2.5 + 4.33i)T + (-171.5 - 297. i)T^{2}$$
11 $$1 + (-6.5 - 11.2i)T + (-665.5 + 1.15e3i)T^{2}$$
17 $$1 + (-13.5 + 23.3i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (37.5 - 64.9i)T + (-3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (93.5 + 161. i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + (6.5 + 11.2i)T + (-1.21e4 + 2.11e4i)T^{2}$$
31 $$1 + 104T + 2.97e4T^{2}$$
37 $$1 + (211.5 + 366. i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 + (-97.5 - 168. i)T + (-3.44e4 + 5.96e4i)T^{2}$$
43 $$1 + (99.5 - 172. i)T + (-3.97e4 - 6.88e4i)T^{2}$$
47 $$1 + 388T + 1.03e5T^{2}$$
53 $$1 + 618T + 1.48e5T^{2}$$
59 $$1 + (-245.5 + 425. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (87.5 - 151. i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (408.5 + 707. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 + (-39.5 + 68.4i)T + (-1.78e5 - 3.09e5i)T^{2}$$
73 $$1 - 230T + 3.89e5T^{2}$$
79 $$1 - 764T + 4.93e5T^{2}$$
83 $$1 - 732T + 5.71e5T^{2}$$
89 $$1 + (520.5 + 901. i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 + (-48.5 + 84.0i)T + (-4.56e5 - 7.90e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$