# Properties

 Label 2-69-23.22-c6-0-11 Degree $2$ Conductor $69$ Sign $0.990 - 0.139i$ Analytic cond. $15.8737$ Root an. cond. $3.98418$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.368·2-s + 15.5·3-s − 63.8·4-s + 113. i·5-s − 5.74·6-s − 569. i·7-s + 47.1·8-s + 243·9-s − 41.9i·10-s + 814. i·11-s − 995.·12-s + 3.51e3·13-s + 210. i·14-s + 1.77e3i·15-s + 4.06e3·16-s + 7.48e3i·17-s + ⋯
 L(s)  = 1 − 0.0460·2-s + 0.577·3-s − 0.997·4-s + 0.911i·5-s − 0.0265·6-s − 1.66i·7-s + 0.0920·8-s + 0.333·9-s − 0.0419i·10-s + 0.611i·11-s − 0.576·12-s + 1.59·13-s + 0.0765i·14-s + 0.526i·15-s + 0.993·16-s + 1.52i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$69$$    =    $$3 \cdot 23$$ Sign: $0.990 - 0.139i$ Analytic conductor: $$15.8737$$ Root analytic conductor: $$3.98418$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{69} (22, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 69,\ (\ :3),\ 0.990 - 0.139i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$1.86608 + 0.130762i$$ $$L(\frac12)$$ $$\approx$$ $$1.86608 + 0.130762i$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - 15.5T$$
23 $$1 + (-1.20e4 + 1.69e3i)T$$
good2 $$1 + 0.368T + 64T^{2}$$
5 $$1 - 113. iT - 1.56e4T^{2}$$
7 $$1 + 569. iT - 1.17e5T^{2}$$
11 $$1 - 814. iT - 1.77e6T^{2}$$
13 $$1 - 3.51e3T + 4.82e6T^{2}$$
17 $$1 - 7.48e3iT - 2.41e7T^{2}$$
19 $$1 + 4.28e3iT - 4.70e7T^{2}$$
29 $$1 + 3.31e3T + 5.94e8T^{2}$$
31 $$1 - 3.44e4T + 8.87e8T^{2}$$
37 $$1 - 7.16e3iT - 2.56e9T^{2}$$
41 $$1 - 7.79e4T + 4.75e9T^{2}$$
43 $$1 + 1.13e5iT - 6.32e9T^{2}$$
47 $$1 + 6.28e4T + 1.07e10T^{2}$$
53 $$1 - 1.35e5iT - 2.21e10T^{2}$$
59 $$1 + 1.01e5T + 4.21e10T^{2}$$
61 $$1 - 3.82e4iT - 5.15e10T^{2}$$
67 $$1 + 2.29e5iT - 9.04e10T^{2}$$
71 $$1 + 6.45e4T + 1.28e11T^{2}$$
73 $$1 + 5.36e4T + 1.51e11T^{2}$$
79 $$1 - 4.58e5iT - 2.43e11T^{2}$$
83 $$1 + 1.13e6iT - 3.26e11T^{2}$$
89 $$1 - 8.29e5iT - 4.96e11T^{2}$$
97 $$1 - 5.25e5iT - 8.32e11T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$