# Properties

 Label 2-690-23.18-c1-0-8 Degree $2$ Conductor $690$ Sign $0.906 - 0.421i$ 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 + (0.959 + 0.281i)2-s + (−0.654 + 0.755i)3-s + (0.841 + 0.540i)4-s + (0.142 − 0.989i)5-s + (−0.841 + 0.540i)6-s + (0.0700 + 0.153i)7-s + (0.654 + 0.755i)8-s + (−0.142 − 0.989i)9-s + (0.415 − 0.909i)10-s + (5.28 − 1.55i)11-s + (−0.959 + 0.281i)12-s + (1.37 − 3.01i)13-s + (0.0239 + 0.166i)14-s + (0.654 + 0.755i)15-s + (0.415 + 0.909i)16-s + (−3.71 + 2.38i)17-s + ⋯
 L(s)  = 1 + (0.678 + 0.199i)2-s + (−0.378 + 0.436i)3-s + (0.420 + 0.270i)4-s + (0.0636 − 0.442i)5-s + (−0.343 + 0.220i)6-s + (0.0264 + 0.0579i)7-s + (0.231 + 0.267i)8-s + (−0.0474 − 0.329i)9-s + (0.131 − 0.287i)10-s + (1.59 − 0.467i)11-s + (−0.276 + 0.0813i)12-s + (0.381 − 0.835i)13-s + (0.00641 + 0.0445i)14-s + (0.169 + 0.195i)15-s + (0.103 + 0.227i)16-s + (−0.901 + 0.579i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 - 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$690$$    =    $$2 \cdot 3 \cdot 5 \cdot 23$$ Sign: $0.906 - 0.421i$ Analytic conductor: $$5.50967$$ Root analytic conductor: $$2.34727$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{690} (271, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 690,\ (\ :1/2),\ 0.906 - 0.421i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.16067 + 0.477264i$$ $$L(\frac12)$$ $$\approx$$ $$2.16067 + 0.477264i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.959 - 0.281i)T$$
3 $$1 + (0.654 - 0.755i)T$$
5 $$1 + (-0.142 + 0.989i)T$$
23 $$1 + (-3.07 - 3.67i)T$$
good7 $$1 + (-0.0700 - 0.153i)T + (-4.58 + 5.29i)T^{2}$$
11 $$1 + (-5.28 + 1.55i)T + (9.25 - 5.94i)T^{2}$$
13 $$1 + (-1.37 + 3.01i)T + (-8.51 - 9.82i)T^{2}$$
17 $$1 + (3.71 - 2.38i)T + (7.06 - 15.4i)T^{2}$$
19 $$1 + (-3.75 - 2.41i)T + (7.89 + 17.2i)T^{2}$$
29 $$1 + (-3.33 + 2.14i)T + (12.0 - 26.3i)T^{2}$$
31 $$1 + (2.47 + 2.85i)T + (-4.41 + 30.6i)T^{2}$$
37 $$1 + (-1.34 - 9.36i)T + (-35.5 + 10.4i)T^{2}$$
41 $$1 + (1.37 - 9.53i)T + (-39.3 - 11.5i)T^{2}$$
43 $$1 + (-7.12 + 8.21i)T + (-6.11 - 42.5i)T^{2}$$
47 $$1 + 8.61T + 47T^{2}$$
53 $$1 + (4.59 + 10.0i)T + (-34.7 + 40.0i)T^{2}$$
59 $$1 + (1.51 - 3.32i)T + (-38.6 - 44.5i)T^{2}$$
61 $$1 + (-3.28 - 3.79i)T + (-8.68 + 60.3i)T^{2}$$
67 $$1 + (13.1 + 3.84i)T + (56.3 + 36.2i)T^{2}$$
71 $$1 + (9.05 + 2.65i)T + (59.7 + 38.3i)T^{2}$$
73 $$1 + (-8.80 - 5.65i)T + (30.3 + 66.4i)T^{2}$$
79 $$1 + (-5.83 + 12.7i)T + (-51.7 - 59.7i)T^{2}$$
83 $$1 + (1.07 + 7.46i)T + (-79.6 + 23.3i)T^{2}$$
89 $$1 + (0.489 - 0.564i)T + (-12.6 - 88.0i)T^{2}$$
97 $$1 + (1.17 - 8.17i)T + (-93.0 - 27.3i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$