# Properties

 Label 2-69-69.68-c5-0-34 Degree $2$ Conductor $69$ Sign $-0.700 - 0.713i$ 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 − 8.75i·2-s + (10.6 + 11.3i)3-s − 44.7·4-s − 43.1·5-s + (99.7 − 93.2i)6-s − 151. i·7-s + 111. i·8-s + (−16.4 + 242. i)9-s + 378. i·10-s − 610.·11-s + (−476. − 509. i)12-s + 132.·13-s − 1.32e3·14-s + (−459. − 491. i)15-s − 454.·16-s − 1.72e3·17-s + ⋯
 L(s)  = 1 − 1.54i·2-s + (0.682 + 0.730i)3-s − 1.39·4-s − 0.772·5-s + (1.13 − 1.05i)6-s − 1.16i·7-s + 0.615i·8-s + (−0.0676 + 0.997i)9-s + 1.19i·10-s − 1.52·11-s + (−0.954 − 1.02i)12-s + 0.217·13-s − 1.80·14-s + (−0.527 − 0.564i)15-s − 0.444·16-s − 1.44·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$69$$    =    $$3 \cdot 23$$ Sign: $-0.700 - 0.713i$ 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: $$(2,\ 69,\ (\ :5/2),\ -0.700 - 0.713i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.246682 + 0.587979i$$ $$L(\frac12)$$ $$\approx$$ $$0.246682 + 0.587979i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-10.6 - 11.3i)T$$
23 $$1 + (-2.53e3 + 62.8i)T$$
good2 $$1 + 8.75iT - 32T^{2}$$
5 $$1 + 43.1T + 3.12e3T^{2}$$
7 $$1 + 151. iT - 1.68e4T^{2}$$
11 $$1 + 610.T + 1.61e5T^{2}$$
13 $$1 - 132.T + 3.71e5T^{2}$$
17 $$1 + 1.72e3T + 1.41e6T^{2}$$
19 $$1 - 554. iT - 2.47e6T^{2}$$
29 $$1 + 7.00e3iT - 2.05e7T^{2}$$
31 $$1 - 4.81e3T + 2.86e7T^{2}$$
37 $$1 + 1.32e3iT - 6.93e7T^{2}$$
41 $$1 - 231. iT - 1.15e8T^{2}$$
43 $$1 + 1.84e4iT - 1.47e8T^{2}$$
47 $$1 + 1.17e4iT - 2.29e8T^{2}$$
53 $$1 - 1.89e3T + 4.18e8T^{2}$$
59 $$1 - 7.71e3iT - 7.14e8T^{2}$$
61 $$1 + 1.20e3iT - 8.44e8T^{2}$$
67 $$1 - 4.10e4iT - 1.35e9T^{2}$$
71 $$1 + 8.19e3iT - 1.80e9T^{2}$$
73 $$1 + 7.82e4T + 2.07e9T^{2}$$
79 $$1 + 7.74e4iT - 3.07e9T^{2}$$
83 $$1 - 1.70e4T + 3.93e9T^{2}$$
89 $$1 - 1.35e5T + 5.58e9T^{2}$$
97 $$1 - 7.51e4iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$