# Properties

 Degree $2$ Conductor $384$ Sign $0.296 + 0.955i$ Motivic weight $2$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (2.86 − 0.888i)3-s − 8.59i·5-s + 10.9·7-s + (7.41 − 5.09i)9-s + 2.75i·11-s − 4.43·13-s + (−7.63 − 24.6i)15-s + 25.4i·17-s − 17.5·19-s + (31.3 − 9.71i)21-s − 17.5i·23-s − 48.8·25-s + (16.7 − 21.1i)27-s − 19.6i·29-s + 2.58·31-s + ⋯
 L(s)  = 1 + (0.955 − 0.296i)3-s − 1.71i·5-s + 1.56·7-s + (0.824 − 0.565i)9-s + 0.250i·11-s − 0.341·13-s + (−0.509 − 1.64i)15-s + 1.49i·17-s − 0.923·19-s + (1.49 − 0.462i)21-s − 0.762i·23-s − 1.95·25-s + (0.619 − 0.784i)27-s − 0.676i·29-s + 0.0833·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$384$$    =    $$2^{7} \cdot 3$$ Sign: $0.296 + 0.955i$ Motivic weight: $$2$$ Character: $\chi_{384} (257, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 384,\ (\ :1),\ 0.296 + 0.955i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$2.15169 - 1.58538i$$ $$L(\frac12)$$ $$\approx$$ $$2.15169 - 1.58538i$$ $$L(2)$$ 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 + (-2.86 + 0.888i)T$$
good5 $$1 + 8.59iT - 25T^{2}$$
7 $$1 - 10.9T + 49T^{2}$$
11 $$1 - 2.75iT - 121T^{2}$$
13 $$1 + 4.43T + 169T^{2}$$
17 $$1 - 25.4iT - 289T^{2}$$
19 $$1 + 17.5T + 361T^{2}$$
23 $$1 + 17.5iT - 529T^{2}$$
29 $$1 + 19.6iT - 841T^{2}$$
31 $$1 - 2.58T + 961T^{2}$$
37 $$1 - 7.73T + 1.36e3T^{2}$$
41 $$1 - 58.0iT - 1.68e3T^{2}$$
43 $$1 - 42.1T + 1.84e3T^{2}$$
47 $$1 - 17.4iT - 2.20e3T^{2}$$
53 $$1 - 69.0iT - 2.80e3T^{2}$$
59 $$1 + 50.5iT - 3.48e3T^{2}$$
61 $$1 + 32.5T + 3.72e3T^{2}$$
67 $$1 - 48.0T + 4.48e3T^{2}$$
71 $$1 + 22.1iT - 5.04e3T^{2}$$
73 $$1 + 27.0T + 5.32e3T^{2}$$
79 $$1 + 97.4T + 6.24e3T^{2}$$
83 $$1 - 59.5iT - 6.88e3T^{2}$$
89 $$1 + 110. iT - 7.92e3T^{2}$$
97 $$1 - 55.1T + 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

−10.96164032584876648122143754174, −9.785080941412448886194895085865, −8.740378100530993742282106795942, −8.300693045161737115583847125912, −7.67670527712465876620856738100, −6.05069925480381234274437082962, −4.65069275089407008191282685703, −4.24713590022207372357145338015, −2.11886106713709965700863847320, −1.21971313296375126571640131338, 2.03150728254347334979019145380, 2.94714347623852754236268836460, 4.15790049549712108358714070917, 5.35919858772771544451231580974, 6.98469385856200234031243788824, 7.50437969502016079636052695909, 8.449265472493335715940035344233, 9.512621761713400865167795879734, 10.54884927630158757974316847707, 11.06244458583045468797194859986