# Properties

 Label 2-1152-16.5-c1-0-8 Degree $2$ Conductor $1152$ Sign $0.962 - 0.270i$ 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 + (1.74 + 1.74i)5-s − 2.55i·7-s + (0.473 + 0.473i)11-s + (−2.88 + 2.88i)13-s + 6.44·17-s + (4.55 − 4.55i)19-s + 2.82i·23-s + 1.11i·25-s + (−3.07 + 3.07i)29-s + 6.55·31-s + (4.47 − 4.47i)35-s + (2.72 + 2.72i)37-s + 0.788i·41-s + (0.389 + 0.389i)43-s − 2.82·47-s + ⋯
 L(s)  = 1 + (0.782 + 0.782i)5-s − 0.966i·7-s + (0.142 + 0.142i)11-s + (−0.800 + 0.800i)13-s + 1.56·17-s + (1.04 − 1.04i)19-s + 0.589i·23-s + 0.223i·25-s + (−0.571 + 0.571i)29-s + 1.17·31-s + (0.756 − 0.756i)35-s + (0.448 + 0.448i)37-s + 0.123i·41-s + (0.0594 + 0.0594i)43-s − 0.412·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1152$$    =    $$2^{7} \cdot 3^{2}$$ Sign: $0.962 - 0.270i$ Analytic conductor: $$9.19876$$ Root analytic conductor: $$3.03294$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1152} (865, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1152,\ (\ :1/2),\ 0.962 - 0.270i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.956173302$$ $$L(\frac12)$$ $$\approx$$ $$1.956173302$$ $$L(\frac{3}{2})$$ 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 + (-1.74 - 1.74i)T + 5iT^{2}$$
7 $$1 + 2.55iT - 7T^{2}$$
11 $$1 + (-0.473 - 0.473i)T + 11iT^{2}$$
13 $$1 + (2.88 - 2.88i)T - 13iT^{2}$$
17 $$1 - 6.44T + 17T^{2}$$
19 $$1 + (-4.55 + 4.55i)T - 19iT^{2}$$
23 $$1 - 2.82iT - 23T^{2}$$
29 $$1 + (3.07 - 3.07i)T - 29iT^{2}$$
31 $$1 - 6.55T + 31T^{2}$$
37 $$1 + (-2.72 - 2.72i)T + 37iT^{2}$$
41 $$1 - 0.788iT - 41T^{2}$$
43 $$1 + (-0.389 - 0.389i)T + 43iT^{2}$$
47 $$1 + 2.82T + 47T^{2}$$
53 $$1 + (2.57 + 2.57i)T + 53iT^{2}$$
59 $$1 + (-4 - 4i)T + 59iT^{2}$$
61 $$1 + (-4.38 + 4.38i)T - 61iT^{2}$$
67 $$1 + (-2.11 + 2.11i)T - 67iT^{2}$$
71 $$1 + 5.11iT - 71T^{2}$$
73 $$1 - 14.7iT - 73T^{2}$$
79 $$1 + 6.31T + 79T^{2}$$
83 $$1 + (-0.641 + 0.641i)T - 83iT^{2}$$
89 $$1 - 6.31iT - 89T^{2}$$
97 $$1 - 12.6T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$