# Properties

 Label 2-69-69.68-c5-0-6 Degree $2$ Conductor $69$ Sign $-0.678 + 0.734i$ Analytic cond. $11.0664$ Root an. cond. $3.32663$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5.41i·2-s + (3.03 + 15.2i)3-s + 2.64·4-s − 41.1·5-s + (−82.8 + 16.4i)6-s − 67.8i·7-s + 187. i·8-s + (−224. + 92.8i)9-s − 223. i·10-s − 718.·11-s + (8.01 + 40.3i)12-s − 17.0·13-s + 367.·14-s + (−125. − 629. i)15-s − 932.·16-s + 1.85e3·17-s + ⋯
 L(s)  = 1 + 0.957i·2-s + (0.194 + 0.980i)3-s + 0.0825·4-s − 0.736·5-s + (−0.939 + 0.186i)6-s − 0.523i·7-s + 1.03i·8-s + (−0.924 + 0.381i)9-s − 0.705i·10-s − 1.79·11-s + (0.0160 + 0.0809i)12-s − 0.0279·13-s + 0.501·14-s + (−0.143 − 0.722i)15-s − 0.910·16-s + 1.55·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$69$$    =    $$3 \cdot 23$$ Sign: $-0.678 + 0.734i$ Analytic conductor: $$11.0664$$ Root analytic conductor: $$3.32663$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{69} (68, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 69,\ (\ :5/2),\ -0.678 + 0.734i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.382206 - 0.873004i$$ $$L(\frac12)$$ $$\approx$$ $$0.382206 - 0.873004i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-3.03 - 15.2i)T$$
23 $$1 + (1.49e3 + 2.05e3i)T$$
good2 $$1 - 5.41iT - 32T^{2}$$
5 $$1 + 41.1T + 3.12e3T^{2}$$
7 $$1 + 67.8iT - 1.68e4T^{2}$$
11 $$1 + 718.T + 1.61e5T^{2}$$
13 $$1 + 17.0T + 3.71e5T^{2}$$
17 $$1 - 1.85e3T + 1.41e6T^{2}$$
19 $$1 - 1.13e3iT - 2.47e6T^{2}$$
29 $$1 - 1.74e3iT - 2.05e7T^{2}$$
31 $$1 - 3.91e3T + 2.86e7T^{2}$$
37 $$1 - 9.22e3iT - 6.93e7T^{2}$$
41 $$1 - 5.01e3iT - 1.15e8T^{2}$$
43 $$1 - 5.50e3iT - 1.47e8T^{2}$$
47 $$1 - 2.05e4iT - 2.29e8T^{2}$$
53 $$1 + 3.66e4T + 4.18e8T^{2}$$
59 $$1 - 3.00e4iT - 7.14e8T^{2}$$
61 $$1 - 3.03e4iT - 8.44e8T^{2}$$
67 $$1 + 4.26e4iT - 1.35e9T^{2}$$
71 $$1 + 4.28e4iT - 1.80e9T^{2}$$
73 $$1 - 1.04e4T + 2.07e9T^{2}$$
79 $$1 - 3.71e4iT - 3.07e9T^{2}$$
83 $$1 - 6.57e4T + 3.93e9T^{2}$$
89 $$1 + 1.07e5T + 5.58e9T^{2}$$
97 $$1 + 6.92e4iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$