# Properties

 Label 2-75-5.4-c21-0-56 Degree $2$ Conductor $75$ Sign $-0.894 - 0.447i$ Analytic cond. $209.608$ Root an. cond. $14.4778$ Motivic weight $21$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.27e3i·2-s − 5.90e4i·3-s + 4.79e5·4-s − 7.51e7·6-s + 6.32e8i·7-s − 3.27e9i·8-s − 3.48e9·9-s + 5.97e10·11-s − 2.83e10i·12-s − 7.38e11i·13-s + 8.03e11·14-s − 3.16e12·16-s − 8.35e12i·17-s + 4.43e12i·18-s − 4.19e13·19-s + ⋯
 L(s)  = 1 − 0.878i·2-s − 0.577i·3-s + 0.228·4-s − 0.507·6-s + 0.845i·7-s − 1.07i·8-s − 0.333·9-s + 0.694·11-s − 0.131i·12-s − 1.48i·13-s + 0.742·14-s − 0.719·16-s − 1.00i·17-s + 0.292i·18-s − 1.56·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(22-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$75$$    =    $$3 \cdot 5^{2}$$ Sign: $-0.894 - 0.447i$ Analytic conductor: $$209.608$$ Root analytic conductor: $$14.4778$$ Motivic weight: $$21$$ Rational: no Arithmetic: yes Character: $\chi_{75} (49, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 75,\ (\ :21/2),\ -0.894 - 0.447i)$$

## Particular Values

 $$L(11)$$ $$\approx$$ $$2.242449647$$ $$L(\frac12)$$ $$\approx$$ $$2.242449647$$ $$L(\frac{23}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + 5.90e4iT$$
5 $$1$$
good2 $$1 + 1.27e3iT - 2.09e6T^{2}$$
7 $$1 - 6.32e8iT - 5.58e17T^{2}$$
11 $$1 - 5.97e10T + 7.40e21T^{2}$$
13 $$1 + 7.38e11iT - 2.47e23T^{2}$$
17 $$1 + 8.35e12iT - 6.90e25T^{2}$$
19 $$1 + 4.19e13T + 7.14e26T^{2}$$
23 $$1 + 4.48e13iT - 3.94e28T^{2}$$
29 $$1 - 2.76e15T + 5.13e30T^{2}$$
31 $$1 - 8.36e15T + 2.08e31T^{2}$$
37 $$1 + 1.77e16iT - 8.55e32T^{2}$$
41 $$1 - 1.45e17T + 7.38e33T^{2}$$
43 $$1 + 1.24e17iT - 2.00e34T^{2}$$
47 $$1 - 4.28e17iT - 1.30e35T^{2}$$
53 $$1 - 4.77e17iT - 1.62e36T^{2}$$
59 $$1 + 1.61e18T + 1.54e37T^{2}$$
61 $$1 + 3.76e18T + 3.10e37T^{2}$$
67 $$1 + 2.81e18iT - 2.22e38T^{2}$$
71 $$1 - 1.00e19T + 7.52e38T^{2}$$
73 $$1 - 1.72e19iT - 1.34e39T^{2}$$
79 $$1 - 3.28e19T + 7.08e39T^{2}$$
83 $$1 + 3.05e17iT - 1.99e40T^{2}$$
89 $$1 + 2.34e20T + 8.65e40T^{2}$$
97 $$1 + 5.92e20iT - 5.27e41T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$