# Properties

 Label 2-1152-96.59-c1-0-6 Degree $2$ Conductor $1152$ Sign $0.739 - 0.673i$ Analytic cond. $9.19876$ Root an. cond. $3.03294$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.0963 + 0.232i)5-s + (−0.617 + 0.617i)7-s + (−0.505 + 1.21i)11-s + (3.41 − 1.41i)13-s + 2.76·17-s + (−0.189 − 0.458i)19-s + (−4.46 + 4.46i)23-s + (3.49 + 3.49i)25-s + (−0.0101 + 0.00418i)29-s − 4.03i·31-s + (−0.0841 − 0.203i)35-s + (6.30 + 2.61i)37-s + (5.34 + 5.34i)41-s + (10.1 + 4.21i)43-s + 11.5i·47-s + ⋯
 L(s)  = 1 + (−0.0430 + 0.104i)5-s + (−0.233 + 0.233i)7-s + (−0.152 + 0.367i)11-s + (0.947 − 0.392i)13-s + 0.669·17-s + (−0.0435 − 0.105i)19-s + (−0.931 + 0.931i)23-s + (0.698 + 0.698i)25-s + (−0.00187 + 0.000777i)29-s − 0.724i·31-s + (−0.0142 − 0.0343i)35-s + (1.03 + 0.429i)37-s + (0.834 + 0.834i)41-s + (1.55 + 0.642i)43-s + 1.69i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1152$$    =    $$2^{7} \cdot 3^{2}$$ Sign: $0.739 - 0.673i$ Analytic conductor: $$9.19876$$ Root analytic conductor: $$3.03294$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1152} (143, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1152,\ (\ :1/2),\ 0.739 - 0.673i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.570423736$$ $$L(\frac12)$$ $$\approx$$ $$1.570423736$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5 $$1 + (0.0963 - 0.232i)T + (-3.53 - 3.53i)T^{2}$$
7 $$1 + (0.617 - 0.617i)T - 7iT^{2}$$
11 $$1 + (0.505 - 1.21i)T + (-7.77 - 7.77i)T^{2}$$
13 $$1 + (-3.41 + 1.41i)T + (9.19 - 9.19i)T^{2}$$
17 $$1 - 2.76T + 17T^{2}$$
19 $$1 + (0.189 + 0.458i)T + (-13.4 + 13.4i)T^{2}$$
23 $$1 + (4.46 - 4.46i)T - 23iT^{2}$$
29 $$1 + (0.0101 - 0.00418i)T + (20.5 - 20.5i)T^{2}$$
31 $$1 + 4.03iT - 31T^{2}$$
37 $$1 + (-6.30 - 2.61i)T + (26.1 + 26.1i)T^{2}$$
41 $$1 + (-5.34 - 5.34i)T + 41iT^{2}$$
43 $$1 + (-10.1 - 4.21i)T + (30.4 + 30.4i)T^{2}$$
47 $$1 - 11.5iT - 47T^{2}$$
53 $$1 + (9.04 + 3.74i)T + (37.4 + 37.4i)T^{2}$$
59 $$1 + (0.939 + 0.389i)T + (41.7 + 41.7i)T^{2}$$
61 $$1 + (2.97 + 7.17i)T + (-43.1 + 43.1i)T^{2}$$
67 $$1 + (-7.40 + 3.06i)T + (47.3 - 47.3i)T^{2}$$
71 $$1 + (-1.20 - 1.20i)T + 71iT^{2}$$
73 $$1 + (-3.73 + 3.73i)T - 73iT^{2}$$
79 $$1 - 7.22T + 79T^{2}$$
83 $$1 + (-11.2 + 4.67i)T + (58.6 - 58.6i)T^{2}$$
89 $$1 + (3.70 - 3.70i)T - 89iT^{2}$$
97 $$1 + 14.0T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$