# Properties

 Label 2-75-5.3-c8-0-17 Degree $2$ Conductor $75$ Sign $0.991 - 0.130i$ 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 + (7.34 − 7.34i)2-s + (33.0 + 33.0i)3-s + 148i·4-s + 486·6-s + (2.87e3 − 2.87e3i)7-s + (2.96e3 + 2.96e3i)8-s + 2.18e3i·9-s − 234·11-s + (−4.89e3 + 4.89e3i)12-s + (−1.19e4 − 1.19e4i)13-s − 4.22e4i·14-s + 5.74e3·16-s + (8.90e4 − 8.90e4i)17-s + (1.60e4 + 1.60e4i)18-s + 1.81e5i·19-s + ⋯
 L(s)  = 1 + (0.459 − 0.459i)2-s + (0.408 + 0.408i)3-s + 0.578i·4-s + 0.375·6-s + (1.19 − 1.19i)7-s + (0.724 + 0.724i)8-s + 0.333i·9-s − 0.0159·11-s + (−0.236 + 0.236i)12-s + (−0.417 − 0.417i)13-s − 1.09i·14-s + 0.0876·16-s + (1.06 − 1.06i)17-s + (0.153 + 0.153i)18-s + 1.39i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$75$$    =    $$3 \cdot 5^{2}$$ Sign: $0.991 - 0.130i$ Analytic conductor: $$30.5533$$ Root analytic conductor: $$5.52751$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{75} (43, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 75,\ (\ :4),\ 0.991 - 0.130i)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$3.58949 + 0.235710i$$ $$L(\frac12)$$ $$\approx$$ $$3.58949 + 0.235710i$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-33.0 - 33.0i)T$$
5 $$1$$
good2 $$1 + (-7.34 + 7.34i)T - 256iT^{2}$$
7 $$1 + (-2.87e3 + 2.87e3i)T - 5.76e6iT^{2}$$
11 $$1 + 234T + 2.14e8T^{2}$$
13 $$1 + (1.19e4 + 1.19e4i)T + 8.15e8iT^{2}$$
17 $$1 + (-8.90e4 + 8.90e4i)T - 6.97e9iT^{2}$$
19 $$1 - 1.81e5iT - 1.69e10T^{2}$$
23 $$1 + (-2.69e5 - 2.69e5i)T + 7.83e10iT^{2}$$
29 $$1 + 2.40e5iT - 5.00e11T^{2}$$
31 $$1 - 8.36e5T + 8.52e11T^{2}$$
37 $$1 + (-6.08e5 + 6.08e5i)T - 3.51e12iT^{2}$$
41 $$1 - 2.82e6T + 7.98e12T^{2}$$
43 $$1 + (-2.80e6 - 2.80e6i)T + 1.16e13iT^{2}$$
47 $$1 + (5.39e6 - 5.39e6i)T - 2.38e13iT^{2}$$
53 $$1 + (1.19e6 + 1.19e6i)T + 6.22e13iT^{2}$$
59 $$1 + 1.27e7iT - 1.46e14T^{2}$$
61 $$1 - 5.17e5T + 1.91e14T^{2}$$
67 $$1 + (-2.06e6 + 2.06e6i)T - 4.06e14iT^{2}$$
71 $$1 + 2.08e7T + 6.45e14T^{2}$$
73 $$1 + (-2.88e7 - 2.88e7i)T + 8.06e14iT^{2}$$
79 $$1 + 4.21e7iT - 1.51e15T^{2}$$
83 $$1 + (6.66e7 + 6.66e7i)T + 2.25e15iT^{2}$$
89 $$1 - 9.51e7iT - 3.93e15T^{2}$$
97 $$1 + (-7.08e7 + 7.08e7i)T - 7.83e15iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$