# Properties

 Label 2-930-93.26-c1-0-16 Degree $2$ Conductor $930$ Sign $0.292 + 0.956i$ 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 − i·2-s + (−1.23 + 1.21i)3-s − 4-s + (−0.866 + 0.5i)5-s + (1.21 + 1.23i)6-s + (−1.74 + 3.01i)7-s + i·8-s + (0.0529 − 2.99i)9-s + (0.5 + 0.866i)10-s + (−2.28 − 3.95i)11-s + (1.23 − 1.21i)12-s + (−6.16 + 3.55i)13-s + (3.01 + 1.74i)14-s + (0.463 − 1.66i)15-s + 16-s + (1.01 − 1.75i)17-s + ⋯
 L(s)  = 1 − 0.707i·2-s + (−0.713 + 0.700i)3-s − 0.5·4-s + (−0.387 + 0.223i)5-s + (0.495 + 0.504i)6-s + (−0.658 + 1.14i)7-s + 0.353i·8-s + (0.0176 − 0.999i)9-s + (0.158 + 0.273i)10-s + (−0.688 − 1.19i)11-s + (0.356 − 0.350i)12-s + (−1.70 + 0.986i)13-s + (0.806 + 0.465i)14-s + (0.119 − 0.430i)15-s + 0.250·16-s + (0.245 − 0.424i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$930$$    =    $$2 \cdot 3 \cdot 5 \cdot 31$$ Sign: $0.292 + 0.956i$ Analytic conductor: $$7.42608$$ Root analytic conductor: $$2.72508$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{930} (491, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 930,\ (\ :1/2),\ 0.292 + 0.956i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.491363 - 0.363349i$$ $$L(\frac12)$$ $$\approx$$ $$0.491363 - 0.363349i$$ $$L(\frac{3}{2})$$ 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 + (1.23 - 1.21i)T$$
5 $$1 + (0.866 - 0.5i)T$$
31 $$1 + (1.29 - 5.41i)T$$
good7 $$1 + (1.74 - 3.01i)T + (-3.5 - 6.06i)T^{2}$$
11 $$1 + (2.28 + 3.95i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + (6.16 - 3.55i)T + (6.5 - 11.2i)T^{2}$$
17 $$1 + (-1.01 + 1.75i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-3.87 + 6.71i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 - 7.42T + 23T^{2}$$
29 $$1 - 6.35T + 29T^{2}$$
37 $$1 + (-7.71 - 4.45i)T + (18.5 + 32.0i)T^{2}$$
41 $$1 + (-2.17 + 1.25i)T + (20.5 - 35.5i)T^{2}$$
43 $$1 + (3.21 + 1.85i)T + (21.5 + 37.2i)T^{2}$$
47 $$1 + 5.93iT - 47T^{2}$$
53 $$1 + (5.09 + 8.83i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (7.20 + 4.16i)T + (29.5 + 51.0i)T^{2}$$
61 $$1 - 2.03iT - 61T^{2}$$
67 $$1 + (3.08 + 5.35i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + (-2.26 + 1.30i)T + (35.5 - 61.4i)T^{2}$$
73 $$1 + (-7.81 + 4.51i)T + (36.5 - 63.2i)T^{2}$$
79 $$1 + (7.33 + 4.23i)T + (39.5 + 68.4i)T^{2}$$
83 $$1 + (4.54 + 7.87i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 - 3.74T + 89T^{2}$$
97 $$1 - 5.16T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$