# Properties

 Degree $2$ Conductor $5376$ Sign $-0.707 + 0.707i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − i·3-s − 3.46i·5-s + 7-s − 9-s + 5.46i·11-s − 2i·13-s − 3.46·15-s + 7.46·17-s + 6.92i·19-s − i·21-s − 5.46·23-s − 6.99·25-s + i·27-s − 8.92i·29-s + 2.92·31-s + ⋯
 L(s)  = 1 − 0.577i·3-s − 1.54i·5-s + 0.377·7-s − 0.333·9-s + 1.64i·11-s − 0.554i·13-s − 0.894·15-s + 1.81·17-s + 1.58i·19-s − 0.218i·21-s − 1.13·23-s − 1.39·25-s + 0.192i·27-s − 1.65i·29-s + 0.525·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$5376$$    =    $$2^{8} \cdot 3 \cdot 7$$ Sign: $-0.707 + 0.707i$ Motivic weight: $$1$$ Character: $\chi_{5376} (2689, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 5376,\ (\ :1/2),\ -0.707 + 0.707i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.709235969$$ $$L(\frac12)$$ $$\approx$$ $$1.709235969$$ $$L(\frac{3}{2})$$ 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 + iT$$
7 $$1 - T$$
good5 $$1 + 3.46iT - 5T^{2}$$
11 $$1 - 5.46iT - 11T^{2}$$
13 $$1 + 2iT - 13T^{2}$$
17 $$1 - 7.46T + 17T^{2}$$
19 $$1 - 6.92iT - 19T^{2}$$
23 $$1 + 5.46T + 23T^{2}$$
29 $$1 + 8.92iT - 29T^{2}$$
31 $$1 - 2.92T + 31T^{2}$$
37 $$1 + 2iT - 37T^{2}$$
41 $$1 + 4.53T + 41T^{2}$$
43 $$1 + 8iT - 43T^{2}$$
47 $$1 + 2.92T + 47T^{2}$$
53 $$1 + 2iT - 53T^{2}$$
59 $$1 + 14.9iT - 59T^{2}$$
61 $$1 + 4.92iT - 61T^{2}$$
67 $$1 + 10.9iT - 67T^{2}$$
71 $$1 + 2.53T + 71T^{2}$$
73 $$1 - 0.928T + 73T^{2}$$
79 $$1 - 2.92T + 79T^{2}$$
83 $$1 - 4iT - 83T^{2}$$
89 $$1 - 3.46T + 89T^{2}$$
97 $$1 - 4.92T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$