# Properties

 Label 2-75-3.2-c8-0-32 Degree $2$ Conductor $75$ Sign $0.555 + 0.831i$ 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 + 22.4i·2-s + (−45 − 67.3i)3-s − 248·4-s + (1.51e3 − 1.01e3i)6-s + 1.75e3·7-s + 179. i·8-s + (−2.51e3 + 6.06e3i)9-s − 6.95e3i·11-s + (1.11e4 + 1.67e4i)12-s − 2.57e4·13-s + 3.92e4i·14-s − 6.75e4·16-s + 7.48e4i·17-s + (−1.36e5 − 5.63e4i)18-s + 1.89e4·19-s + ⋯
 L(s)  = 1 + 1.40i·2-s + (−0.555 − 0.831i)3-s − 0.968·4-s + (1.16 − 0.779i)6-s + 0.728·7-s + 0.0438i·8-s + (−0.382 + 0.923i)9-s − 0.475i·11-s + (0.538 + 0.805i)12-s − 0.900·13-s + 1.02i·14-s − 1.03·16-s + 0.896i·17-s + (−1.29 − 0.536i)18-s + 0.145·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$75$$    =    $$3 \cdot 5^{2}$$ Sign: $0.555 + 0.831i$ Analytic conductor: $$30.5533$$ Root analytic conductor: $$5.52751$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{75} (26, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 75,\ (\ :4),\ 0.555 + 0.831i)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$0.621602 - 0.332260i$$ $$L(\frac12)$$ $$\approx$$ $$0.621602 - 0.332260i$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (45 + 67.3i)T$$
5 $$1$$
good2 $$1 - 22.4iT - 256T^{2}$$
7 $$1 - 1.75e3T + 5.76e6T^{2}$$
11 $$1 + 6.95e3iT - 2.14e8T^{2}$$
13 $$1 + 2.57e4T + 8.15e8T^{2}$$
17 $$1 - 7.48e4iT - 6.97e9T^{2}$$
19 $$1 - 1.89e4T + 1.69e10T^{2}$$
23 $$1 + 4.70e5iT - 7.83e10T^{2}$$
29 $$1 + 4.60e5iT - 5.00e11T^{2}$$
31 $$1 + 3.51e5T + 8.52e11T^{2}$$
37 $$1 + 1.33e6T + 3.51e12T^{2}$$
41 $$1 + 1.87e6iT - 7.98e12T^{2}$$
43 $$1 - 3.52e6T + 1.16e13T^{2}$$
47 $$1 + 4.08e6iT - 2.38e13T^{2}$$
53 $$1 + 6.60e6iT - 6.22e13T^{2}$$
59 $$1 + 1.37e7iT - 1.46e14T^{2}$$
61 $$1 - 7.53e5T + 1.91e14T^{2}$$
67 $$1 + 2.26e6T + 4.06e14T^{2}$$
71 $$1 + 1.70e7iT - 6.45e14T^{2}$$
73 $$1 + 2.76e7T + 8.06e14T^{2}$$
79 $$1 + 2.29e7T + 1.51e15T^{2}$$
83 $$1 + 4.63e7iT - 2.25e15T^{2}$$
89 $$1 + 7.26e7iT - 3.93e15T^{2}$$
97 $$1 + 1.47e8T + 7.83e15T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$