# Properties

 Label 2-3150-5.4-c1-0-10 Degree $2$ Conductor $3150$ Sign $-0.447 - 0.894i$ Analytic cond. $25.1528$ Root an. cond. $5.01526$ 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 − 4-s + i·7-s − i·8-s − 4·11-s − 6i·13-s − 14-s + 16-s + 2i·17-s − 4i·22-s + 6·26-s − i·28-s + 6·29-s + 8·31-s + i·32-s + ⋯
 L(s)  = 1 + 0.707i·2-s − 0.5·4-s + 0.377i·7-s − 0.353i·8-s − 1.20·11-s − 1.66i·13-s − 0.267·14-s + 0.250·16-s + 0.485i·17-s − 0.852i·22-s + 1.17·26-s − 0.188i·28-s + 1.11·29-s + 1.43·31-s + 0.176i·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3150$$    =    $$2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Sign: $-0.447 - 0.894i$ Analytic conductor: $$25.1528$$ Root analytic conductor: $$5.01526$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{3150} (2899, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3150,\ (\ :1/2),\ -0.447 - 0.894i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.219013087$$ $$L(\frac12)$$ $$\approx$$ $$1.219013087$$ $$L(\frac{3}{2})$$ 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$$
5 $$1$$
7 $$1 - iT$$
good11 $$1 + 4T + 11T^{2}$$
13 $$1 + 6iT - 13T^{2}$$
17 $$1 - 2iT - 17T^{2}$$
19 $$1 + 19T^{2}$$
23 $$1 - 23T^{2}$$
29 $$1 - 6T + 29T^{2}$$
31 $$1 - 8T + 31T^{2}$$
37 $$1 - 10iT - 37T^{2}$$
41 $$1 + 2T + 41T^{2}$$
43 $$1 - 4iT - 43T^{2}$$
47 $$1 - 8iT - 47T^{2}$$
53 $$1 - 2iT - 53T^{2}$$
59 $$1 + 8T + 59T^{2}$$
61 $$1 + 14T + 61T^{2}$$
67 $$1 - 12iT - 67T^{2}$$
71 $$1 - 16T + 71T^{2}$$
73 $$1 - 2iT - 73T^{2}$$
79 $$1 - 8T + 79T^{2}$$
83 $$1 + 8iT - 83T^{2}$$
89 $$1 - 10T + 89T^{2}$$
97 $$1 + 2iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$