# Properties

 Label 1-5e2-25.17-r1-0-0 Degree $1$ Conductor $25$ Sign $0.248 + 0.968i$ Analytic cond. $2.68662$ Root an. cond. $2.68662$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.587 − 0.809i)2-s + (−0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (0.309 + 0.951i)6-s + i·7-s + (0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + (−0.809 + 0.587i)11-s + (0.587 − 0.809i)12-s + (−0.587 + 0.809i)13-s + (0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s + (−0.951 + 0.309i)17-s − i·18-s + (−0.309 − 0.951i)19-s + ⋯
 L(s)  = 1 + (−0.587 − 0.809i)2-s + (−0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (0.309 + 0.951i)6-s + i·7-s + (0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + (−0.809 + 0.587i)11-s + (0.587 − 0.809i)12-s + (−0.587 + 0.809i)13-s + (0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s + (−0.951 + 0.309i)17-s − i·18-s + (−0.309 − 0.951i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$25$$    =    $$5^{2}$$ Sign: $0.248 + 0.968i$ Analytic conductor: $$2.68662$$ Root analytic conductor: $$2.68662$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{25} (17, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 25,\ (1:\ ),\ 0.248 + 0.968i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.3058709774 + 0.2372578502i$$ $$L(\frac12)$$ $$\approx$$ $$0.3058709774 + 0.2372578502i$$ $$L(1)$$ $$\approx$$ $$0.4848512120 + 0.02760602250i$$ $$L(1)$$ $$\approx$$ $$0.4848512120 + 0.02760602250i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
good2 $$1 + (-0.587 - 0.809i)T$$
3 $$1 + (-0.951 - 0.309i)T$$
7 $$1 + iT$$
11 $$1 + (-0.809 + 0.587i)T$$
13 $$1 + (-0.587 + 0.809i)T$$
17 $$1 + (-0.951 + 0.309i)T$$
19 $$1 + (-0.309 - 0.951i)T$$
23 $$1 + (0.587 + 0.809i)T$$
29 $$1 + (-0.309 + 0.951i)T$$
31 $$1 + (0.309 + 0.951i)T$$
37 $$1 + (0.587 - 0.809i)T$$
41 $$1 + (-0.809 - 0.587i)T$$
43 $$1 - iT$$
47 $$1 + (0.951 + 0.309i)T$$
53 $$1 + (-0.951 - 0.309i)T$$
59 $$1 + (0.809 + 0.587i)T$$
61 $$1 + (-0.809 + 0.587i)T$$
67 $$1 + (-0.951 + 0.309i)T$$
71 $$1 + (0.309 - 0.951i)T$$
73 $$1 + (0.587 + 0.809i)T$$
79 $$1 + (-0.309 + 0.951i)T$$
83 $$1 + (0.951 - 0.309i)T$$
89 $$1 + (0.809 - 0.587i)T$$
97 $$1 + (0.951 + 0.309i)T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$