# Properties

 Label 2-3200-1.1-c1-0-58 Degree $2$ Conductor $3200$ Sign $-1$ Analytic cond. $25.5521$ Root an. cond. $5.05491$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.806·3-s + 2.15·7-s − 2.35·9-s + 0.387·11-s + 0.962·13-s + 1.61·17-s − 6.31·19-s − 1.73·21-s − 6.15·23-s + 4.31·27-s + 2·29-s + 9.92·31-s − 0.312·33-s − 6.57·37-s − 0.775·39-s + 4.57·41-s − 11.5·43-s − 4.54·47-s − 2.35·49-s − 1.29·51-s − 8.96·53-s + 5.08·57-s + 6.31·59-s + 0.261·61-s − 5.06·63-s − 9.89·67-s + 4.96·69-s + ⋯
 L(s)  = 1 − 0.465·3-s + 0.815·7-s − 0.783·9-s + 0.116·11-s + 0.266·13-s + 0.390·17-s − 1.44·19-s − 0.379·21-s − 1.28·23-s + 0.829·27-s + 0.371·29-s + 1.78·31-s − 0.0544·33-s − 1.08·37-s − 0.124·39-s + 0.714·41-s − 1.75·43-s − 0.662·47-s − 0.335·49-s − 0.181·51-s − 1.23·53-s + 0.673·57-s + 0.821·59-s + 0.0335·61-s − 0.638·63-s − 1.20·67-s + 0.597·69-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3200$$    =    $$2^{7} \cdot 5^{2}$$ Sign: $-1$ Analytic conductor: $$25.5521$$ Root analytic conductor: $$5.05491$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{3200} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 3200,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
good3 $$1 + 0.806T + 3T^{2}$$
7 $$1 - 2.15T + 7T^{2}$$
11 $$1 - 0.387T + 11T^{2}$$
13 $$1 - 0.962T + 13T^{2}$$
17 $$1 - 1.61T + 17T^{2}$$
19 $$1 + 6.31T + 19T^{2}$$
23 $$1 + 6.15T + 23T^{2}$$
29 $$1 - 2T + 29T^{2}$$
31 $$1 - 9.92T + 31T^{2}$$
37 $$1 + 6.57T + 37T^{2}$$
41 $$1 - 4.57T + 41T^{2}$$
43 $$1 + 11.5T + 43T^{2}$$
47 $$1 + 4.54T + 47T^{2}$$
53 $$1 + 8.96T + 53T^{2}$$
59 $$1 - 6.31T + 59T^{2}$$
61 $$1 - 0.261T + 61T^{2}$$
67 $$1 + 9.89T + 67T^{2}$$
71 $$1 + 10.7T + 71T^{2}$$
73 $$1 - 13.0T + 73T^{2}$$
79 $$1 + 1.92T + 79T^{2}$$
83 $$1 + 2.88T + 83T^{2}$$
89 $$1 - 10.6T + 89T^{2}$$
97 $$1 - 9.61T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$