# Properties

 Label 2-40e2-40.29-c1-0-26 Degree $2$ Conductor $1600$ Sign $-0.948 - 0.316i$ Analytic cond. $12.7760$ Root an. cond. $3.57436$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2·3-s + 9-s − 6i·11-s + 6i·17-s − 2i·19-s + 4·27-s + 12i·33-s − 6·41-s − 10·43-s + 7·49-s − 12i·51-s + 4i·57-s + 6i·59-s − 14·67-s + 2i·73-s + ⋯
 L(s)  = 1 − 1.15·3-s + 0.333·9-s − 1.80i·11-s + 1.45i·17-s − 0.458i·19-s + 0.769·27-s + 2.08i·33-s − 0.937·41-s − 1.52·43-s + 49-s − 1.68i·51-s + 0.529i·57-s + 0.781i·59-s − 1.71·67-s + 0.234i·73-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1600$$    =    $$2^{6} \cdot 5^{2}$$ Sign: $-0.948 - 0.316i$ Analytic conductor: $$12.7760$$ Root analytic conductor: $$3.57436$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1600} (1249, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$1$$ Selberg data: $$(2,\ 1600,\ (\ :1/2),\ -0.948 - 0.316i)$$

## 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 + 2T + 3T^{2}$$
7 $$1 - 7T^{2}$$
11 $$1 + 6iT - 11T^{2}$$
13 $$1 + 13T^{2}$$
17 $$1 - 6iT - 17T^{2}$$
19 $$1 + 2iT - 19T^{2}$$
23 $$1 - 23T^{2}$$
29 $$1 - 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 + 37T^{2}$$
41 $$1 + 6T + 41T^{2}$$
43 $$1 + 10T + 43T^{2}$$
47 $$1 - 47T^{2}$$
53 $$1 + 53T^{2}$$
59 $$1 - 6iT - 59T^{2}$$
61 $$1 - 61T^{2}$$
67 $$1 + 14T + 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 - 2iT - 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 + 18T + 83T^{2}$$
89 $$1 + 18T + 89T^{2}$$
97 $$1 + 10iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$