# Properties

 Label 2-1148-41.9-c1-0-6 Degree $2$ Conductor $1148$ Sign $0.989 + 0.144i$ Analytic cond. $9.16682$ Root an. cond. $3.02767$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.350 − 0.350i)3-s + 3.08i·5-s + (−0.707 − 0.707i)7-s − 2.75i·9-s + (1.14 + 1.14i)11-s + (−2.88 − 2.88i)13-s + (1.08 − 1.08i)15-s + (3.58 − 3.58i)17-s + (2.29 − 2.29i)19-s + 0.495i·21-s + 3.46·23-s − 4.51·25-s + (−2.01 + 2.01i)27-s + (4.60 + 4.60i)29-s + 2.08·31-s + ⋯
 L(s)  = 1 + (−0.202 − 0.202i)3-s + 1.37i·5-s + (−0.267 − 0.267i)7-s − 0.917i·9-s + (0.344 + 0.344i)11-s + (−0.799 − 0.799i)13-s + (0.279 − 0.279i)15-s + (0.869 − 0.869i)17-s + (0.526 − 0.526i)19-s + 0.108i·21-s + 0.723·23-s − 0.902·25-s + (−0.388 + 0.388i)27-s + (0.855 + 0.855i)29-s + 0.374·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1148$$    =    $$2^{2} \cdot 7 \cdot 41$$ Sign: $0.989 + 0.144i$ Analytic conductor: $$9.16682$$ Root analytic conductor: $$3.02767$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1148} (337, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1148,\ (\ :1/2),\ 0.989 + 0.144i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.481845347$$ $$L(\frac12)$$ $$\approx$$ $$1.481845347$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
7 $$1 + (0.707 + 0.707i)T$$
41 $$1 + (-6.11 + 1.90i)T$$
good3 $$1 + (0.350 + 0.350i)T + 3iT^{2}$$
5 $$1 - 3.08iT - 5T^{2}$$
11 $$1 + (-1.14 - 1.14i)T + 11iT^{2}$$
13 $$1 + (2.88 + 2.88i)T + 13iT^{2}$$
17 $$1 + (-3.58 + 3.58i)T - 17iT^{2}$$
19 $$1 + (-2.29 + 2.29i)T - 19iT^{2}$$
23 $$1 - 3.46T + 23T^{2}$$
29 $$1 + (-4.60 - 4.60i)T + 29iT^{2}$$
31 $$1 - 2.08T + 31T^{2}$$
37 $$1 - 6.46T + 37T^{2}$$
43 $$1 - 6.95iT - 43T^{2}$$
47 $$1 + (3.66 - 3.66i)T - 47iT^{2}$$
53 $$1 + (-5.69 - 5.69i)T + 53iT^{2}$$
59 $$1 - 2.42T + 59T^{2}$$
61 $$1 + 8.24iT - 61T^{2}$$
67 $$1 + (5.59 - 5.59i)T - 67iT^{2}$$
71 $$1 + (3.14 + 3.14i)T + 71iT^{2}$$
73 $$1 + 8.57iT - 73T^{2}$$
79 $$1 + (-5.97 - 5.97i)T + 79iT^{2}$$
83 $$1 - 13.2T + 83T^{2}$$
89 $$1 + (3.25 + 3.25i)T + 89iT^{2}$$
97 $$1 + (3.75 - 3.75i)T - 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$