# Properties

 Label 2-930-5.4-c3-0-63 Degree $2$ Conductor $930$ Sign $0.147 + 0.988i$ Analytic cond. $54.8717$ Root an. cond. $7.40754$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2i·2-s + 3i·3-s − 4·4-s + (1.65 + 11.0i)5-s + 6·6-s − 2.45i·7-s + 8i·8-s − 9·9-s + (22.1 − 3.30i)10-s − 35.2·11-s − 12i·12-s − 31.1i·13-s − 4.90·14-s + (−33.1 + 4.96i)15-s + 16·16-s − 47.1i·17-s + ⋯
 L(s)  = 1 − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.147 + 0.988i)5-s + 0.408·6-s − 0.132i·7-s + 0.353i·8-s − 0.333·9-s + (0.699 − 0.104i)10-s − 0.966·11-s − 0.288i·12-s − 0.664i·13-s − 0.0936·14-s + (−0.570 + 0.0854i)15-s + 0.250·16-s − 0.672i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.147 + 0.988i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.147 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$930$$    =    $$2 \cdot 3 \cdot 5 \cdot 31$$ Sign: $0.147 + 0.988i$ Analytic conductor: $$54.8717$$ Root analytic conductor: $$7.40754$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{930} (559, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 930,\ (\ :3/2),\ 0.147 + 0.988i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.197539569$$ $$L(\frac12)$$ $$\approx$$ $$1.197539569$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 2iT$$
3 $$1 - 3iT$$
5 $$1 + (-1.65 - 11.0i)T$$
31 $$1 + 31T$$
good7 $$1 + 2.45iT - 343T^{2}$$
11 $$1 + 35.2T + 1.33e3T^{2}$$
13 $$1 + 31.1iT - 2.19e3T^{2}$$
17 $$1 + 47.1iT - 4.91e3T^{2}$$
19 $$1 - 118.T + 6.85e3T^{2}$$
23 $$1 - 57.1iT - 1.21e4T^{2}$$
29 $$1 + 298.T + 2.43e4T^{2}$$
37 $$1 - 82.3iT - 5.06e4T^{2}$$
41 $$1 + 25.0T + 6.89e4T^{2}$$
43 $$1 + 264. iT - 7.95e4T^{2}$$
47 $$1 + 344. iT - 1.03e5T^{2}$$
53 $$1 + 422. iT - 1.48e5T^{2}$$
59 $$1 + 43.3T + 2.05e5T^{2}$$
61 $$1 - 96.2T + 2.26e5T^{2}$$
67 $$1 - 706. iT - 3.00e5T^{2}$$
71 $$1 - 1.11e3T + 3.57e5T^{2}$$
73 $$1 - 94.3iT - 3.89e5T^{2}$$
79 $$1 - 607.T + 4.93e5T^{2}$$
83 $$1 + 604. iT - 5.71e5T^{2}$$
89 $$1 - 391.T + 7.04e5T^{2}$$
97 $$1 + 1.04e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$