# Properties

 Label 2-5e4-5.4-c1-0-16 Degree $2$ Conductor $625$ Sign $1$ Analytic cond. $4.99065$ Root an. cond. $2.23397$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.618i·2-s + i·3-s + 1.61·4-s + 0.618·6-s + 1.61i·7-s − 2.23i·8-s + 2·9-s − 0.763·11-s + 1.61i·12-s − 4.85i·13-s + 1.00·14-s + 1.85·16-s + 0.763i·17-s − 1.23i·18-s + 5.85·19-s + ⋯
 L(s)  = 1 − 0.437i·2-s + 0.577i·3-s + 0.809·4-s + 0.252·6-s + 0.611i·7-s − 0.790i·8-s + 0.666·9-s − 0.230·11-s + 0.467i·12-s − 1.34i·13-s + 0.267·14-s + 0.463·16-s + 0.185i·17-s − 0.291i·18-s + 1.34·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$625$$    =    $$5^{4}$$ Sign: $1$ Analytic conductor: $$4.99065$$ Root analytic conductor: $$2.23397$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{625} (624, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 625,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.94765$$ $$L(\frac12)$$ $$\approx$$ $$1.94765$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
good2 $$1 + 0.618iT - 2T^{2}$$
3 $$1 - iT - 3T^{2}$$
7 $$1 - 1.61iT - 7T^{2}$$
11 $$1 + 0.763T + 11T^{2}$$
13 $$1 + 4.85iT - 13T^{2}$$
17 $$1 - 0.763iT - 17T^{2}$$
19 $$1 - 5.85T + 19T^{2}$$
23 $$1 - 8.23iT - 23T^{2}$$
29 $$1 - 1.38T + 29T^{2}$$
31 $$1 + 3T + 31T^{2}$$
37 $$1 + 4.23iT - 37T^{2}$$
41 $$1 + 5.23T + 41T^{2}$$
43 $$1 - 1.85iT - 43T^{2}$$
47 $$1 - 1.61iT - 47T^{2}$$
53 $$1 - 5.47iT - 53T^{2}$$
59 $$1 - 4.14T + 59T^{2}$$
61 $$1 + 4.70T + 61T^{2}$$
67 $$1 + 9.23iT - 67T^{2}$$
71 $$1 + 4.38T + 71T^{2}$$
73 $$1 + 9iT - 73T^{2}$$
79 $$1 + 3.09T + 79T^{2}$$
83 $$1 + 1.76iT - 83T^{2}$$
89 $$1 + 8.94T + 89T^{2}$$
97 $$1 + 2.85iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$