# Properties

 Label 2-1100-5.4-c3-0-40 Degree $2$ Conductor $1100$ Sign $-0.447 + 0.894i$ Analytic cond. $64.9021$ Root an. cond. $8.05618$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5i·3-s − 26i·7-s + 2·9-s − 11·11-s − 52i·13-s + 46i·17-s + 96·19-s + 130·21-s − 27i·23-s + 145i·27-s − 16·29-s − 293·31-s − 55i·33-s − 29i·37-s + 260·39-s + ⋯
 L(s)  = 1 + 0.962i·3-s − 1.40i·7-s + 0.0740·9-s − 0.301·11-s − 1.10i·13-s + 0.656i·17-s + 1.15·19-s + 1.35·21-s − 0.244i·23-s + 1.03i·27-s − 0.102·29-s − 1.69·31-s − 0.290i·33-s − 0.128i·37-s + 1.06·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1100$$    =    $$2^{2} \cdot 5^{2} \cdot 11$$ Sign: $-0.447 + 0.894i$ Analytic conductor: $$64.9021$$ Root analytic conductor: $$8.05618$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{1100} (749, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1100,\ (\ :3/2),\ -0.447 + 0.894i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.8895199104$$ $$L(\frac12)$$ $$\approx$$ $$0.8895199104$$ $$L(\frac{5}{2})$$ 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$$
11 $$1 + 11T$$
good3 $$1 - 5iT - 27T^{2}$$
7 $$1 + 26iT - 343T^{2}$$
13 $$1 + 52iT - 2.19e3T^{2}$$
17 $$1 - 46iT - 4.91e3T^{2}$$
19 $$1 - 96T + 6.85e3T^{2}$$
23 $$1 + 27iT - 1.21e4T^{2}$$
29 $$1 + 16T + 2.43e4T^{2}$$
31 $$1 + 293T + 2.97e4T^{2}$$
37 $$1 + 29iT - 5.06e4T^{2}$$
41 $$1 + 472T + 6.89e4T^{2}$$
43 $$1 - 110iT - 7.95e4T^{2}$$
47 $$1 + 224iT - 1.03e5T^{2}$$
53 $$1 + 754iT - 1.48e5T^{2}$$
59 $$1 + 825T + 2.05e5T^{2}$$
61 $$1 + 548T + 2.26e5T^{2}$$
67 $$1 + 123iT - 3.00e5T^{2}$$
71 $$1 - 1.00e3T + 3.57e5T^{2}$$
73 $$1 - 1.02e3iT - 3.89e5T^{2}$$
79 $$1 + 526T + 4.93e5T^{2}$$
83 $$1 - 158iT - 5.71e5T^{2}$$
89 $$1 - 1.21e3T + 7.04e5T^{2}$$
97 $$1 + 263iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$