# Properties

 Label 2-75-15.14-c8-0-18 Degree $2$ Conductor $75$ Sign $0.868 + 0.495i$ Analytic cond. $30.5533$ Root an. cond. $5.52751$ Motivic weight $8$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 22.4·2-s + (−67.3 + 45i)3-s + 248·4-s + (1.51e3 − 1.01e3i)6-s + 1.75e3i·7-s + 179.·8-s + (2.51e3 − 6.06e3i)9-s − 6.95e3i·11-s + (−1.67e4 + 1.11e4i)12-s + 2.57e4i·13-s − 3.92e4i·14-s − 6.75e4·16-s − 7.48e4·17-s + (−5.63e4 + 1.36e5i)18-s − 1.89e4·19-s + ⋯
 L(s)  = 1 − 1.40·2-s + (−0.831 + 0.555i)3-s + 0.968·4-s + (1.16 − 0.779i)6-s + 0.728i·7-s + 0.0438·8-s + (0.382 − 0.923i)9-s − 0.475i·11-s + (−0.805 + 0.538i)12-s + 0.900i·13-s − 1.02i·14-s − 1.03·16-s − 0.896·17-s + (−0.536 + 1.29i)18-s − 0.145·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$75$$    =    $$3 \cdot 5^{2}$$ Sign: $0.868 + 0.495i$ Analytic conductor: $$30.5533$$ Root analytic conductor: $$5.52751$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{75} (74, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 75,\ (\ :4),\ 0.868 + 0.495i)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$0.307347 - 0.0814514i$$ $$L(\frac12)$$ $$\approx$$ $$0.307347 - 0.0814514i$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (67.3 - 45i)T$$
5 $$1$$
good2 $$1 + 22.4T + 256T^{2}$$
7 $$1 - 1.75e3iT - 5.76e6T^{2}$$
11 $$1 + 6.95e3iT - 2.14e8T^{2}$$
13 $$1 - 2.57e4iT - 8.15e8T^{2}$$
17 $$1 + 7.48e4T + 6.97e9T^{2}$$
19 $$1 + 1.89e4T + 1.69e10T^{2}$$
23 $$1 + 4.70e5T + 7.83e10T^{2}$$
29 $$1 - 4.60e5iT - 5.00e11T^{2}$$
31 $$1 + 3.51e5T + 8.52e11T^{2}$$
37 $$1 + 1.33e6iT - 3.51e12T^{2}$$
41 $$1 + 1.87e6iT - 7.98e12T^{2}$$
43 $$1 + 3.52e6iT - 1.16e13T^{2}$$
47 $$1 - 4.08e6T + 2.38e13T^{2}$$
53 $$1 + 6.60e6T + 6.22e13T^{2}$$
59 $$1 - 1.37e7iT - 1.46e14T^{2}$$
61 $$1 - 7.53e5T + 1.91e14T^{2}$$
67 $$1 + 2.26e6iT - 4.06e14T^{2}$$
71 $$1 + 1.70e7iT - 6.45e14T^{2}$$
73 $$1 - 2.76e7iT - 8.06e14T^{2}$$
79 $$1 - 2.29e7T + 1.51e15T^{2}$$
83 $$1 + 4.63e7T + 2.25e15T^{2}$$
89 $$1 - 7.26e7iT - 3.93e15T^{2}$$
97 $$1 + 1.47e8iT - 7.83e15T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$