# Properties

 Label 2-690-69.68-c1-0-13 Degree $2$ Conductor $690$ Sign $0.989 - 0.144i$ Analytic cond. $5.50967$ Root an. cond. $2.34727$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·2-s + (−1.30 + 1.14i)3-s − 4-s − 5-s + (−1.14 − 1.30i)6-s − 1.02i·7-s − i·8-s + (0.384 − 2.97i)9-s − i·10-s + 0.336·11-s + (1.30 − 1.14i)12-s − 4.92·13-s + 1.02·14-s + (1.30 − 1.14i)15-s + 16-s + 7.57·17-s + ⋯
 L(s)  = 1 + 0.707i·2-s + (−0.751 + 0.660i)3-s − 0.5·4-s − 0.447·5-s + (−0.466 − 0.531i)6-s − 0.385i·7-s − 0.353i·8-s + (0.128 − 0.991i)9-s − 0.316i·10-s + 0.101·11-s + (0.375 − 0.330i)12-s − 1.36·13-s + 0.272·14-s + (0.335 − 0.295i)15-s + 0.250·16-s + 1.83·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$690$$    =    $$2 \cdot 3 \cdot 5 \cdot 23$$ Sign: $0.989 - 0.144i$ Analytic conductor: $$5.50967$$ Root analytic conductor: $$2.34727$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{690} (551, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 690,\ (\ :1/2),\ 0.989 - 0.144i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.821502 + 0.0598259i$$ $$L(\frac12)$$ $$\approx$$ $$0.821502 + 0.0598259i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - iT$$
3 $$1 + (1.30 - 1.14i)T$$
5 $$1 + T$$
23 $$1 + (-2.61 + 4.02i)T$$
good7 $$1 + 1.02iT - 7T^{2}$$
11 $$1 - 0.336T + 11T^{2}$$
13 $$1 + 4.92T + 13T^{2}$$
17 $$1 - 7.57T + 17T^{2}$$
19 $$1 + 3.95iT - 19T^{2}$$
29 $$1 + 2.20iT - 29T^{2}$$
31 $$1 + 1.27T + 31T^{2}$$
37 $$1 - 0.378iT - 37T^{2}$$
41 $$1 + 7.82iT - 41T^{2}$$
43 $$1 - 10.0iT - 43T^{2}$$
47 $$1 - 5.34iT - 47T^{2}$$
53 $$1 - 13.2T + 53T^{2}$$
59 $$1 - 2.53iT - 59T^{2}$$
61 $$1 + 12.9iT - 61T^{2}$$
67 $$1 + 3.67iT - 67T^{2}$$
71 $$1 + 10.8iT - 71T^{2}$$
73 $$1 + 7.22T + 73T^{2}$$
79 $$1 + 4.78iT - 79T^{2}$$
83 $$1 + 16.0T + 83T^{2}$$
89 $$1 - 14.1T + 89T^{2}$$
97 $$1 + 11.6iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$