# Properties

 Label 2-234-13.12-c5-0-5 Degree $2$ Conductor $234$ Sign $-0.234 - 0.971i$ Analytic cond. $37.5298$ Root an. cond. $6.12615$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 4i·2-s − 16·4-s + 86.8i·5-s + 98.7i·7-s + 64i·8-s + 347.·10-s − 610. i·11-s + (592. − 143. i)13-s + 395.·14-s + 256·16-s + 1.14e3·17-s + 2.26e3i·19-s − 1.38e3i·20-s − 2.44e3·22-s − 433.·23-s + ⋯
 L(s)  = 1 − 0.707i·2-s − 0.5·4-s + 1.55i·5-s + 0.761i·7-s + 0.353i·8-s + 1.09·10-s − 1.52i·11-s + (0.971 − 0.234i)13-s + 0.538·14-s + 0.250·16-s + 0.963·17-s + 1.44i·19-s − 0.776i·20-s − 1.07·22-s − 0.170·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.234 - 0.971i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.234 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$234$$    =    $$2 \cdot 3^{2} \cdot 13$$ Sign: $-0.234 - 0.971i$ Analytic conductor: $$37.5298$$ Root analytic conductor: $$6.12615$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{234} (181, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 234,\ (\ :5/2),\ -0.234 - 0.971i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.261933762$$ $$L(\frac12)$$ $$\approx$$ $$1.261933762$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 4iT$$
3 $$1$$
13 $$1 + (-592. + 143. i)T$$
good5 $$1 - 86.8iT - 3.12e3T^{2}$$
7 $$1 - 98.7iT - 1.68e4T^{2}$$
11 $$1 + 610. iT - 1.61e5T^{2}$$
17 $$1 - 1.14e3T + 1.41e6T^{2}$$
19 $$1 - 2.26e3iT - 2.47e6T^{2}$$
23 $$1 + 433.T + 6.43e6T^{2}$$
29 $$1 + 7.66e3T + 2.05e7T^{2}$$
31 $$1 - 7.36e3iT - 2.86e7T^{2}$$
37 $$1 - 1.05e4iT - 6.93e7T^{2}$$
41 $$1 - 3.69e3iT - 1.15e8T^{2}$$
43 $$1 + 6.06e3T + 1.47e8T^{2}$$
47 $$1 + 8.74e3iT - 2.29e8T^{2}$$
53 $$1 + 3.47e4T + 4.18e8T^{2}$$
59 $$1 - 1.19e4iT - 7.14e8T^{2}$$
61 $$1 + 4.54e4T + 8.44e8T^{2}$$
67 $$1 + 4.56e4iT - 1.35e9T^{2}$$
71 $$1 - 2.38e4iT - 1.80e9T^{2}$$
73 $$1 - 3.77e4iT - 2.07e9T^{2}$$
79 $$1 + 3.53e4T + 3.07e9T^{2}$$
83 $$1 + 3.14e4iT - 3.93e9T^{2}$$
89 $$1 + 5.90e4iT - 5.58e9T^{2}$$
97 $$1 - 4.96e3iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$