# Properties

 Label 2-690-5.2-c2-0-26 Degree $2$ Conductor $690$ Sign $0.908 + 0.417i$ Analytic cond. $18.8011$ Root an. cond. $4.33602$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1 + i)2-s + (−1.22 + 1.22i)3-s + 2i·4-s + (−4.90 − 0.988i)5-s − 2.44·6-s + (2.14 + 2.14i)7-s + (−2 + 2i)8-s − 2.99i·9-s + (−3.91 − 5.89i)10-s − 12.9·11-s + (−2.44 − 2.44i)12-s + (10.2 − 10.2i)13-s + 4.29i·14-s + (7.21 − 4.79i)15-s − 4·16-s + (−2.98 − 2.98i)17-s + ⋯
 L(s)  = 1 + (0.5 + 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s + (−0.980 − 0.197i)5-s − 0.408·6-s + (0.307 + 0.307i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (−0.391 − 0.589i)10-s − 1.18·11-s + (−0.204 − 0.204i)12-s + (0.788 − 0.788i)13-s + 0.307i·14-s + (0.480 − 0.319i)15-s − 0.250·16-s + (−0.175 − 0.175i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$690$$    =    $$2 \cdot 3 \cdot 5 \cdot 23$$ Sign: $0.908 + 0.417i$ Analytic conductor: $$18.8011$$ Root analytic conductor: $$4.33602$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{690} (277, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 690,\ (\ :1),\ 0.908 + 0.417i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.181742001$$ $$L(\frac12)$$ $$\approx$$ $$1.181742001$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1 - i)T$$
3 $$1 + (1.22 - 1.22i)T$$
5 $$1 + (4.90 + 0.988i)T$$
23 $$1 + (-3.39 + 3.39i)T$$
good7 $$1 + (-2.14 - 2.14i)T + 49iT^{2}$$
11 $$1 + 12.9T + 121T^{2}$$
13 $$1 + (-10.2 + 10.2i)T - 169iT^{2}$$
17 $$1 + (2.98 + 2.98i)T + 289iT^{2}$$
19 $$1 + 26.4iT - 361T^{2}$$
29 $$1 - 45.6iT - 841T^{2}$$
31 $$1 - 37.4T + 961T^{2}$$
37 $$1 + (44.9 + 44.9i)T + 1.36e3iT^{2}$$
41 $$1 - 39.3T + 1.68e3T^{2}$$
43 $$1 + (-29.7 + 29.7i)T - 1.84e3iT^{2}$$
47 $$1 + (10.2 + 10.2i)T + 2.20e3iT^{2}$$
53 $$1 + (-34.6 + 34.6i)T - 2.80e3iT^{2}$$
59 $$1 + 22.8iT - 3.48e3T^{2}$$
61 $$1 + 70.3T + 3.72e3T^{2}$$
67 $$1 + (-19.6 - 19.6i)T + 4.48e3iT^{2}$$
71 $$1 - 131.T + 5.04e3T^{2}$$
73 $$1 + (-49.0 + 49.0i)T - 5.32e3iT^{2}$$
79 $$1 + 59.2iT - 6.24e3T^{2}$$
83 $$1 + (31.0 - 31.0i)T - 6.88e3iT^{2}$$
89 $$1 + 169. iT - 7.92e3T^{2}$$
97 $$1 + (-8.17 - 8.17i)T + 9.40e3iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$