# Properties

 Label 2-1152-12.11-c1-0-15 Degree $2$ Conductor $1152$ Sign $-0.816 + 0.577i$ 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 − 3.41i·5-s − 4.82i·7-s + 2.82·11-s − 2.82·13-s − 5.41i·17-s + 5.65i·19-s + 1.17·23-s − 6.65·25-s − 0.585i·29-s + 3.17i·31-s − 16.4·35-s − 3.65·37-s − 2.58i·41-s + 9.65i·43-s + 12.4·47-s + ⋯
 L(s)  = 1 − 1.52i·5-s − 1.82i·7-s + 0.852·11-s − 0.784·13-s − 1.31i·17-s + 1.29i·19-s + 0.244·23-s − 1.33·25-s − 0.108i·29-s + 0.569i·31-s − 2.78·35-s − 0.601·37-s − 0.403i·41-s + 1.47i·43-s + 1.82·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1152$$    =    $$2^{7} \cdot 3^{2}$$ Sign: $-0.816 + 0.577i$ Analytic conductor: $$9.19876$$ Root analytic conductor: $$3.03294$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1152} (1151, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1152,\ (\ :1/2),\ -0.816 + 0.577i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.394543776$$ $$L(\frac12)$$ $$\approx$$ $$1.394543776$$ $$L(\frac{3}{2})$$ 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 + 3.41iT - 5T^{2}$$
7 $$1 + 4.82iT - 7T^{2}$$
11 $$1 - 2.82T + 11T^{2}$$
13 $$1 + 2.82T + 13T^{2}$$
17 $$1 + 5.41iT - 17T^{2}$$
19 $$1 - 5.65iT - 19T^{2}$$
23 $$1 - 1.17T + 23T^{2}$$
29 $$1 + 0.585iT - 29T^{2}$$
31 $$1 - 3.17iT - 31T^{2}$$
37 $$1 + 3.65T + 37T^{2}$$
41 $$1 + 2.58iT - 41T^{2}$$
43 $$1 - 9.65iT - 43T^{2}$$
47 $$1 - 12.4T + 47T^{2}$$
53 $$1 - 5.07iT - 53T^{2}$$
59 $$1 + 2.34T + 59T^{2}$$
61 $$1 + 7.65T + 61T^{2}$$
67 $$1 + 12iT - 67T^{2}$$
71 $$1 + 4.48T + 71T^{2}$$
73 $$1 - 4T + 73T^{2}$$
79 $$1 + 6.48iT - 79T^{2}$$
83 $$1 - 5.17T + 83T^{2}$$
89 $$1 + 12.2iT - 89T^{2}$$
97 $$1 - 13.6T + 97T^{2}$$
show less
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−9.476622045970790126001086190014, −8.754222283827963497321952429461, −7.66361027103555869249621635624, −7.24804426401360082030751942465, −6.07035896559468798857056440758, −4.84483636622478370861229163232, −4.42600481305550002691736340030, −3.42690940081122356402879640653, −1.53446204890742761300930115359, −0.62423180138408618841870644975, 2.12154522017822879867372762996, 2.74492907252988811700255468447, 3.81416085765955287388015694261, 5.17545816460098451451907089048, 6.08189120771413023156901515609, 6.68289168531714400911084600491, 7.52833824145856802900753599008, 8.699465912945540436526496732826, 9.195604143171364641328255906468, 10.17239221224634724450516424780