# Properties

 Label 2-384-8.5-c3-0-13 Degree $2$ Conductor $384$ Sign $0.707 + 0.707i$ Analytic cond. $22.6567$ Root an. cond. $4.75990$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3i·3-s + 8i·5-s − 12·7-s − 9·9-s − 12i·11-s + 20i·13-s + 24·15-s + 62·17-s − 108i·19-s + 36i·21-s + 72·23-s + 61·25-s + 27i·27-s − 128i·29-s + 204·31-s + ⋯
 L(s)  = 1 − 0.577i·3-s + 0.715i·5-s − 0.647·7-s − 0.333·9-s − 0.328i·11-s + 0.426i·13-s + 0.413·15-s + 0.884·17-s − 1.30i·19-s + 0.374i·21-s + 0.652·23-s + 0.487·25-s + 0.192i·27-s − 0.819i·29-s + 1.18·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$384$$    =    $$2^{7} \cdot 3$$ Sign: $0.707 + 0.707i$ Analytic conductor: $$22.6567$$ Root analytic conductor: $$4.75990$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{384} (193, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 384,\ (\ :3/2),\ 0.707 + 0.707i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.662048924$$ $$L(\frac12)$$ $$\approx$$ $$1.662048924$$ $$L(\frac{5}{2})$$ 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 + 3iT$$
good5 $$1 - 8iT - 125T^{2}$$
7 $$1 + 12T + 343T^{2}$$
11 $$1 + 12iT - 1.33e3T^{2}$$
13 $$1 - 20iT - 2.19e3T^{2}$$
17 $$1 - 62T + 4.91e3T^{2}$$
19 $$1 + 108iT - 6.85e3T^{2}$$
23 $$1 - 72T + 1.21e4T^{2}$$
29 $$1 + 128iT - 2.43e4T^{2}$$
31 $$1 - 204T + 2.97e4T^{2}$$
37 $$1 - 228iT - 5.06e4T^{2}$$
41 $$1 + 22T + 6.89e4T^{2}$$
43 $$1 + 204iT - 7.95e4T^{2}$$
47 $$1 - 600T + 1.03e5T^{2}$$
53 $$1 + 256iT - 1.48e5T^{2}$$
59 $$1 + 828iT - 2.05e5T^{2}$$
61 $$1 + 84iT - 2.26e5T^{2}$$
67 $$1 + 348iT - 3.00e5T^{2}$$
71 $$1 + 456T + 3.57e5T^{2}$$
73 $$1 - 822T + 3.89e5T^{2}$$
79 $$1 - 1.35e3T + 4.93e5T^{2}$$
83 $$1 + 108iT - 5.71e5T^{2}$$
89 $$1 + 938T + 7.04e5T^{2}$$
97 $$1 - 1.27e3T + 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

−10.88758045829862777463725896634, −9.908757253579809690057150085985, −8.970381761779671909165505777745, −7.88366026597079824288528619576, −6.84660943388273585944637826379, −6.33506803100283482008421661977, −4.99321205682029885702958347931, −3.41710824678293383153661424220, −2.50033972193491986500237740914, −0.72028353492479047861093909844, 1.04797873822628727908765330350, 2.92198359829609070233142221255, 4.04303773165259767684355044715, 5.16738416771866755775375752886, 6.03029862734700909799207496381, 7.36200899198744230967914767662, 8.405451713842020603646094967365, 9.278962047227124056166872836397, 10.07581722560803102455752514939, 10.80794673550282631160108035630