# Properties

 Label 2-384-24.11-c3-0-6 Degree $2$ Conductor $384$ Sign $-0.956 + 0.290i$ Analytic cond. $22.6567$ Root an. cond. $4.75990$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−2.44 + 4.58i)3-s + 2.31·5-s + 29.6i·7-s + (−15 − 22.4i)9-s + 22.8i·11-s − 8.66i·13-s + (−5.66 + 10.6i)15-s + 93.3i·17-s + 85.4·19-s + (−135. − 72.6i)21-s − 116.·23-s − 119.·25-s + (139. − 13.7i)27-s + 103.·29-s − 10.6i·31-s + ⋯
 L(s)  = 1 + (−0.471 + 0.881i)3-s + 0.207·5-s + 1.60i·7-s + (−0.555 − 0.831i)9-s + 0.625i·11-s − 0.184i·13-s + (−0.0975 + 0.182i)15-s + 1.33i·17-s + 1.03·19-s + (−1.41 − 0.755i)21-s − 1.05·23-s − 0.957·25-s + (0.995 − 0.0979i)27-s + 0.662·29-s − 0.0614i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$384$$    =    $$2^{7} \cdot 3$$ Sign: $-0.956 + 0.290i$ Analytic conductor: $$22.6567$$ Root analytic conductor: $$4.75990$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{384} (191, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 384,\ (\ :3/2),\ -0.956 + 0.290i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.8853803139$$ $$L(\frac12)$$ $$\approx$$ $$0.8853803139$$ $$L(\frac{5}{2})$$ 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 + (2.44 - 4.58i)T$$
good5 $$1 - 2.31T + 125T^{2}$$
7 $$1 - 29.6iT - 343T^{2}$$
11 $$1 - 22.8iT - 1.33e3T^{2}$$
13 $$1 + 8.66iT - 2.19e3T^{2}$$
17 $$1 - 93.3iT - 4.91e3T^{2}$$
19 $$1 - 85.4T + 6.85e3T^{2}$$
23 $$1 + 116.T + 1.21e4T^{2}$$
29 $$1 - 103.T + 2.43e4T^{2}$$
31 $$1 + 10.6iT - 2.97e4T^{2}$$
37 $$1 + 380. iT - 5.06e4T^{2}$$
41 $$1 - 257. iT - 6.89e4T^{2}$$
43 $$1 + 359.T + 7.95e4T^{2}$$
47 $$1 - 293.T + 1.03e5T^{2}$$
53 $$1 + 679.T + 1.48e5T^{2}$$
59 $$1 - 45.8iT - 2.05e5T^{2}$$
61 $$1 + 595. iT - 2.26e5T^{2}$$
67 $$1 - 206.T + 3.00e5T^{2}$$
71 $$1 + 996.T + 3.57e5T^{2}$$
73 $$1 + 593.T + 3.89e5T^{2}$$
79 $$1 + 910. iT - 4.93e5T^{2}$$
83 $$1 - 332. iT - 5.71e5T^{2}$$
89 $$1 + 821. iT - 7.04e5T^{2}$$
97 $$1 - 420.T + 9.12e5T^{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

−11.52664078709596289540860800920, −10.35272984731578702328951424974, −9.652706181143599101126496101139, −8.875302843814684706048608340830, −7.88062022343463798556631245511, −6.19968135295221766169740177497, −5.70942827648861987749794678777, −4.65757115680071180073991879191, −3.37516432675778798063781439050, −1.98384941106184680361136379435, 0.32578687765351167420783356335, 1.40947890281369833194396881850, 3.09047918561115116620530451857, 4.51085962044791509918175844042, 5.67480354333280835379596146327, 6.75650023170814952141581424845, 7.43590767448803334340380394894, 8.266125191507075324451643458084, 9.712843727158659097259485248913, 10.47199338592195463763979408888