# Properties

 Label 2-930-31.5-c1-0-15 Degree $2$ Conductor $930$ Sign $0.275 + 0.961i$ Analytic cond. $7.42608$ Root an. cond. $2.72508$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s + (−0.5 + 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (0.5 − 0.866i)7-s + 8-s + (−0.499 − 0.866i)9-s + (−0.5 − 0.866i)10-s + (−2.5 − 4.33i)11-s + (−0.5 + 0.866i)12-s + (−2 − 3.46i)13-s + (0.5 − 0.866i)14-s + 0.999·15-s + 16-s + (1 − 1.73i)17-s + ⋯
 L(s)  = 1 + 0.707·2-s + (−0.288 + 0.499i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.204 + 0.353i)6-s + (0.188 − 0.327i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 − 0.273i)10-s + (−0.753 − 1.30i)11-s + (−0.144 + 0.249i)12-s + (−0.554 − 0.960i)13-s + (0.133 − 0.231i)14-s + 0.258·15-s + 0.250·16-s + (0.242 − 0.420i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$930$$    =    $$2 \cdot 3 \cdot 5 \cdot 31$$ Sign: $0.275 + 0.961i$ Analytic conductor: $$7.42608$$ Root analytic conductor: $$2.72508$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{930} (811, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 930,\ (\ :1/2),\ 0.275 + 0.961i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.38794 - 1.04654i$$ $$L(\frac12)$$ $$\approx$$ $$1.38794 - 1.04654i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - T$$
3 $$1 + (0.5 - 0.866i)T$$
5 $$1 + (0.5 + 0.866i)T$$
31 $$1 + (-3.5 - 4.33i)T$$
good7 $$1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2}$$
11 $$1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2}$$
17 $$1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + 4T + 23T^{2}$$
29 $$1 - 3T + 29T^{2}$$
37 $$1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + (2 + 3.46i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 - 2T + 47T^{2}$$
53 $$1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + 61T^{2}$$
67 $$1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + (4 + 6.92i)T + (-35.5 + 61.4i)T^{2}$$
73 $$1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 - 18T + 89T^{2}$$
97 $$1 + T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$