# Properties

 Degree $2$ Conductor $1152$ Sign $0.959 - 0.282i$ Motivic weight $2$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.909 − 0.909i)5-s + 0.654·7-s + (13.3 + 13.3i)11-s + (−8.32 − 8.32i)13-s + 3.93·17-s + (16.8 − 16.8i)19-s − 23.1·23-s + 23.3i·25-s + (35.6 + 35.6i)29-s − 45.5i·31-s + (0.595 − 0.595i)35-s + (−10.1 + 10.1i)37-s + 28.4i·41-s + (22.7 + 22.7i)43-s + 10.7i·47-s + ⋯
 L(s)  = 1 + (0.181 − 0.181i)5-s + 0.0935·7-s + (1.21 + 1.21i)11-s + (−0.640 − 0.640i)13-s + 0.231·17-s + (0.889 − 0.889i)19-s − 1.00·23-s + 0.933i·25-s + (1.22 + 1.22i)29-s − 1.46i·31-s + (0.0170 − 0.0170i)35-s + (−0.274 + 0.274i)37-s + 0.694i·41-s + (0.528 + 0.528i)43-s + 0.229i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1152$$    =    $$2^{7} \cdot 3^{2}$$ Sign: $0.959 - 0.282i$ Motivic weight: $$2$$ Character: $\chi_{1152} (991, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1152,\ (\ :1),\ 0.959 - 0.282i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$2.152188835$$ $$L(\frac12)$$ $$\approx$$ $$2.152188835$$ $$L(2)$$ 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 + (-0.909 + 0.909i)T - 25iT^{2}$$
7 $$1 - 0.654T + 49T^{2}$$
11 $$1 + (-13.3 - 13.3i)T + 121iT^{2}$$
13 $$1 + (8.32 + 8.32i)T + 169iT^{2}$$
17 $$1 - 3.93T + 289T^{2}$$
19 $$1 + (-16.8 + 16.8i)T - 361iT^{2}$$
23 $$1 + 23.1T + 529T^{2}$$
29 $$1 + (-35.6 - 35.6i)T + 841iT^{2}$$
31 $$1 + 45.5iT - 961T^{2}$$
37 $$1 + (10.1 - 10.1i)T - 1.36e3iT^{2}$$
41 $$1 - 28.4iT - 1.68e3T^{2}$$
43 $$1 + (-22.7 - 22.7i)T + 1.84e3iT^{2}$$
47 $$1 - 10.7iT - 2.20e3T^{2}$$
53 $$1 + (-41.5 + 41.5i)T - 2.80e3iT^{2}$$
59 $$1 + (-21.0 - 21.0i)T + 3.48e3iT^{2}$$
61 $$1 + (-68.7 - 68.7i)T + 3.72e3iT^{2}$$
67 $$1 + (-67.8 + 67.8i)T - 4.48e3iT^{2}$$
71 $$1 - 33.3T + 5.04e3T^{2}$$
73 $$1 + 18.6iT - 5.32e3T^{2}$$
79 $$1 + 6.29iT - 6.24e3T^{2}$$
83 $$1 + (-72.0 + 72.0i)T - 6.88e3iT^{2}$$
89 $$1 + 10.6iT - 7.92e3T^{2}$$
97 $$1 - 143.T + 9.40e3T^{2}$$
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$$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

−9.666201886484629595359810688013, −9.004505267193249454321848372307, −7.895864965251895691539721132264, −7.18456394696865100254930102706, −6.37890918883922519192669181387, −5.24896707322265160618030971235, −4.55335868491476053465524911727, −3.43570861360936629331968189453, −2.20477640284435797901819954534, −1.01236126514455232198926141294, 0.835647616734344109998864347392, 2.13829315130620250707441961140, 3.40963412201781933671141841840, 4.21292965413262698394301701683, 5.42368475363009764966934423432, 6.25911280151017487462682187579, 6.93293885022585827487598551046, 8.077507342984178875448761761749, 8.681158961221507835566199647004, 9.672461569401080757928407598861