# Properties

 Label 2-5e4-5.4-c1-0-28 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 − 1.61i·2-s − i·3-s − 0.618·4-s − 1.61·6-s + 0.618i·7-s − 2.23i·8-s + 2·9-s − 5.23·11-s + 0.618i·12-s − 1.85i·13-s + 1.00·14-s − 4.85·16-s − 5.23i·17-s − 3.23i·18-s − 0.854·19-s + ⋯
 L(s)  = 1 − 1.14i·2-s − 0.577i·3-s − 0.309·4-s − 0.660·6-s + 0.233i·7-s − 0.790i·8-s + 0.666·9-s − 1.57·11-s + 0.178i·12-s − 0.514i·13-s + 0.267·14-s − 1.21·16-s − 1.26i·17-s − 0.762i·18-s − 0.195·19-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

 $$L(1)$$ $$\approx$$ $$1.35345i$$ $$L(\frac12)$$ $$\approx$$ $$1.35345i$$ $$L(\frac{3}{2})$$ 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 + 1.61iT - 2T^{2}$$
3 $$1 + iT - 3T^{2}$$
7 $$1 - 0.618iT - 7T^{2}$$
11 $$1 + 5.23T + 11T^{2}$$
13 $$1 + 1.85iT - 13T^{2}$$
17 $$1 + 5.23iT - 17T^{2}$$
19 $$1 + 0.854T + 19T^{2}$$
23 $$1 + 3.76iT - 23T^{2}$$
29 $$1 - 3.61T + 29T^{2}$$
31 $$1 + 3T + 31T^{2}$$
37 $$1 + 0.236iT - 37T^{2}$$
41 $$1 + 0.763T + 41T^{2}$$
43 $$1 - 4.85iT - 43T^{2}$$
47 $$1 - 0.618iT - 47T^{2}$$
53 $$1 - 3.47iT - 53T^{2}$$
59 $$1 - 10.8T + 59T^{2}$$
61 $$1 - 8.70T + 61T^{2}$$
67 $$1 - 4.76iT - 67T^{2}$$
71 $$1 + 6.61T + 71T^{2}$$
73 $$1 - 9iT - 73T^{2}$$
79 $$1 - 8.09T + 79T^{2}$$
83 $$1 - 6.23iT - 83T^{2}$$
89 $$1 - 8.94T + 89T^{2}$$
97 $$1 + 3.85iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$