# Properties

 Label 2-350-5.4-c9-0-20 Degree $2$ Conductor $350$ Sign $-0.447 - 0.894i$ Analytic cond. $180.262$ Root an. cond. $13.4261$ Motivic weight $9$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 16i·2-s − 6i·3-s − 256·4-s + 96·6-s + 2.40e3i·7-s − 4.09e3i·8-s + 1.96e4·9-s − 5.41e4·11-s + 1.53e3i·12-s − 1.13e5i·13-s − 3.84e4·14-s + 6.55e4·16-s − 6.26e3i·17-s + 3.14e5i·18-s − 2.57e5·19-s + ⋯
 L(s)  = 1 + 0.707i·2-s − 0.0427i·3-s − 0.5·4-s + 0.0302·6-s + 0.377i·7-s − 0.353i·8-s + 0.998·9-s − 1.11·11-s + 0.0213i·12-s − 1.09i·13-s − 0.267·14-s + 0.250·16-s − 0.0181i·17-s + 0.705i·18-s − 0.452·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(10-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$350$$    =    $$2 \cdot 5^{2} \cdot 7$$ Sign: $-0.447 - 0.894i$ Analytic conductor: $$180.262$$ Root analytic conductor: $$13.4261$$ Motivic weight: $$9$$ Rational: no Arithmetic: yes Character: $\chi_{350} (99, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 350,\ (\ :9/2),\ -0.447 - 0.894i)$$

## Particular Values

 $$L(5)$$ $$\approx$$ $$1.399983985$$ $$L(\frac12)$$ $$\approx$$ $$1.399983985$$ $$L(\frac{11}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 16iT$$
5 $$1$$
7 $$1 - 2.40e3iT$$
good3 $$1 + 6iT - 1.96e4T^{2}$$
11 $$1 + 5.41e4T + 2.35e9T^{2}$$
13 $$1 + 1.13e5iT - 1.06e10T^{2}$$
17 $$1 + 6.26e3iT - 1.18e11T^{2}$$
19 $$1 + 2.57e5T + 3.22e11T^{2}$$
23 $$1 + 2.66e5iT - 1.80e12T^{2}$$
29 $$1 + 1.57e6T + 1.45e13T^{2}$$
31 $$1 + 4.63e6T + 2.64e13T^{2}$$
37 $$1 - 1.19e7iT - 1.29e14T^{2}$$
41 $$1 - 2.19e7T + 3.27e14T^{2}$$
43 $$1 - 2.75e7iT - 5.02e14T^{2}$$
47 $$1 + 5.29e7iT - 1.11e15T^{2}$$
53 $$1 - 1.62e7iT - 3.29e15T^{2}$$
59 $$1 - 1.40e8T + 8.66e15T^{2}$$
61 $$1 + 2.02e8T + 1.16e16T^{2}$$
67 $$1 + 1.53e8iT - 2.72e16T^{2}$$
71 $$1 - 2.79e8T + 4.58e16T^{2}$$
73 $$1 + 4.04e8iT - 5.88e16T^{2}$$
79 $$1 - 1.30e8T + 1.19e17T^{2}$$
83 $$1 - 4.20e8iT - 1.86e17T^{2}$$
89 $$1 - 4.69e8T + 3.50e17T^{2}$$
97 $$1 - 8.72e8iT - 7.60e17T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$