# Properties

 Label 2-40e2-20.7-c1-0-26 Degree $2$ Conductor $1600$ Sign $-0.525 + 0.850i$ Analytic cond. $12.7760$ Root an. cond. $3.57436$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3i·9-s + (−1 + i)13-s + (−3 − 3i)17-s − 4i·29-s + (−7 − 7i)37-s − 8·41-s + 7i·49-s + (9 − 9i)53-s − 12·61-s + (11 − 11i)73-s − 9·81-s − 16i·89-s + (−13 − 13i)97-s − 2·101-s + 6i·109-s + ⋯
 L(s)  = 1 − i·9-s + (−0.277 + 0.277i)13-s + (−0.727 − 0.727i)17-s − 0.742i·29-s + (−1.15 − 1.15i)37-s − 1.24·41-s + i·49-s + (1.23 − 1.23i)53-s − 1.53·61-s + (1.28 − 1.28i)73-s − 81-s − 1.69i·89-s + (−1.31 − 1.31i)97-s − 0.199·101-s + 0.574i·109-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1600$$    =    $$2^{6} \cdot 5^{2}$$ Sign: $-0.525 + 0.850i$ Analytic conductor: $$12.7760$$ Root analytic conductor: $$3.57436$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1600} (1407, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1600,\ (\ :1/2),\ -0.525 + 0.850i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.9589279285$$ $$L(\frac12)$$ $$\approx$$ $$0.9589279285$$ $$L(\frac{3}{2})$$ 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 + 3iT^{2}$$
7 $$1 - 7iT^{2}$$
11 $$1 - 11T^{2}$$
13 $$1 + (1 - i)T - 13iT^{2}$$
17 $$1 + (3 + 3i)T + 17iT^{2}$$
19 $$1 + 19T^{2}$$
23 $$1 + 23iT^{2}$$
29 $$1 + 4iT - 29T^{2}$$
31 $$1 - 31T^{2}$$
37 $$1 + (7 + 7i)T + 37iT^{2}$$
41 $$1 + 8T + 41T^{2}$$
43 $$1 + 43iT^{2}$$
47 $$1 - 47iT^{2}$$
53 $$1 + (-9 + 9i)T - 53iT^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 + 12T + 61T^{2}$$
67 $$1 - 67iT^{2}$$
71 $$1 - 71T^{2}$$
73 $$1 + (-11 + 11i)T - 73iT^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 + 83iT^{2}$$
89 $$1 + 16iT - 89T^{2}$$
97 $$1 + (13 + 13i)T + 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$