# Properties

 Label 2-95-95.94-c10-0-19 Degree $2$ Conductor $95$ Sign $0.632 - 0.774i$ Analytic cond. $60.3589$ Root an. cond. $7.76910$ Motivic weight $10$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.02e3·4-s + (1.97e3 − 2.42e3i)5-s − 8.48e3i·7-s − 5.90e4·9-s − 2.03e5·11-s + 1.04e6·16-s + 1.85e6i·17-s − 2.47e6·19-s + (−2.02e6 + 2.47e6i)20-s − 1.18e7i·23-s + (−1.96e6 − 9.56e6i)25-s + 8.69e6i·28-s + (−2.05e7 − 1.67e7i)35-s + 6.04e7·36-s + 2.03e8i·43-s + 2.08e8·44-s + ⋯
 L(s)  = 1 − 4-s + (0.632 − 0.774i)5-s − 0.504i·7-s − 0.999·9-s − 1.26·11-s + 16-s + 1.30i·17-s − 19-s + (−0.632 + 0.774i)20-s − 1.83i·23-s + (−0.200 − 0.979i)25-s + 0.504i·28-s + (−0.391 − 0.319i)35-s + 0.999·36-s + 1.38i·43-s + 1.26·44-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(11-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$95$$    =    $$5 \cdot 19$$ Sign: $0.632 - 0.774i$ Analytic conductor: $$60.3589$$ Root analytic conductor: $$7.76910$$ Motivic weight: $$10$$ Rational: no Arithmetic: yes Character: $\chi_{95} (94, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 95,\ (\ :5),\ 0.632 - 0.774i)$$

## Particular Values

 $$L(\frac{11}{2})$$ $$\approx$$ $$0.612044 + 0.290556i$$ $$L(\frac12)$$ $$\approx$$ $$0.612044 + 0.290556i$$ $$L(6)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + (-1.97e3 + 2.42e3i)T$$
19 $$1 + 2.47e6T$$
good2 $$1 + 1.02e3T^{2}$$
3 $$1 + 5.90e4T^{2}$$
7 $$1 + 8.48e3iT - 2.82e8T^{2}$$
11 $$1 + 2.03e5T + 2.59e10T^{2}$$
13 $$1 + 1.37e11T^{2}$$
17 $$1 - 1.85e6iT - 2.01e12T^{2}$$
23 $$1 + 1.18e7iT - 4.14e13T^{2}$$
29 $$1 - 4.20e14T^{2}$$
31 $$1 - 8.19e14T^{2}$$
37 $$1 + 4.80e15T^{2}$$
41 $$1 - 1.34e16T^{2}$$
43 $$1 - 2.03e8iT - 2.16e16T^{2}$$
47 $$1 + 4.28e7iT - 5.25e16T^{2}$$
53 $$1 + 1.74e17T^{2}$$
59 $$1 - 5.11e17T^{2}$$
61 $$1 - 1.60e9T + 7.13e17T^{2}$$
67 $$1 + 1.82e18T^{2}$$
71 $$1 - 3.25e18T^{2}$$
73 $$1 - 2.70e9iT - 4.29e18T^{2}$$
79 $$1 - 9.46e18T^{2}$$
83 $$1 - 7.57e9iT - 1.55e19T^{2}$$
89 $$1 - 3.11e19T^{2}$$
97 $$1 + 7.37e19T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$