# Properties

 Degree $2$ Conductor $4800$ Sign $-0.894 - 0.447i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·3-s − 4i·7-s − 9-s + 2i·13-s + 6i·17-s − 4·19-s + 4·21-s − i·27-s − 6·29-s + 8·31-s − 2i·37-s − 2·39-s − 6·41-s − 4i·43-s − 9·49-s + ⋯
 L(s)  = 1 + 0.577i·3-s − 1.51i·7-s − 0.333·9-s + 0.554i·13-s + 1.45i·17-s − 0.917·19-s + 0.872·21-s − 0.192i·27-s − 1.11·29-s + 1.43·31-s − 0.328i·37-s − 0.320·39-s − 0.937·41-s − 0.609i·43-s − 1.28·49-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$4800$$    =    $$2^{6} \cdot 3 \cdot 5^{2}$$ Sign: $-0.894 - 0.447i$ Motivic weight: $$1$$ Character: $\chi_{4800} (3649, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 4800,\ (\ :1/2),\ -0.894 - 0.447i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.5180487010$$ $$L(\frac12)$$ $$\approx$$ $$0.5180487010$$ $$L(\frac{3}{2})$$ 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$$
5 $$1$$
good7 $$1 + 4iT - 7T^{2}$$
11 $$1 + 11T^{2}$$
13 $$1 - 2iT - 13T^{2}$$
17 $$1 - 6iT - 17T^{2}$$
19 $$1 + 4T + 19T^{2}$$
23 $$1 - 23T^{2}$$
29 $$1 + 6T + 29T^{2}$$
31 $$1 - 8T + 31T^{2}$$
37 $$1 + 2iT - 37T^{2}$$
41 $$1 + 6T + 41T^{2}$$
43 $$1 + 4iT - 43T^{2}$$
47 $$1 - 47T^{2}$$
53 $$1 + 6iT - 53T^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 - 10T + 61T^{2}$$
67 $$1 - 4iT - 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + 2iT - 73T^{2}$$
79 $$1 + 8T + 79T^{2}$$
83 $$1 - 12iT - 83T^{2}$$
89 $$1 + 18T + 89T^{2}$$
97 $$1 - 2iT - 97T^{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

−8.455233135705387212861052314482, −8.029451383413718376665135192763, −7.00973490429193110249063852313, −6.56133756697981748068188510870, −5.65708101710557455717847389203, −4.70127456620026194380752292530, −3.94566661095528265927170750141, −3.70510808288950872780027081159, −2.31470118259016022338550526527, −1.24523399701690396111495928999, 0.14036159239926709240687008141, 1.56798595818064913692494655373, 2.59785736426635415508276587317, 2.97933740423861091418010559601, 4.33627945846281192038255049054, 5.25132457113131706421315713230, 5.74118921287138040355002067898, 6.53632454707921671253133712783, 7.19546948894517705814261930376, 8.158297965893395875536536566238