Properties

 Label 2-3332-476.167-c0-0-2 Degree $2$ Conductor $3332$ Sign $0.974 - 0.222i$ Analytic cond. $1.66288$ Root an. cond. $1.28952$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (0.923 − 0.382i)2-s + (0.707 − 0.707i)4-s + (1.08 + 1.63i)5-s + (0.382 − 0.923i)8-s + (0.923 + 0.382i)9-s + (1.63 + 1.08i)10-s + (−1 + i)13-s − i·16-s + (−0.382 − 0.923i)17-s + 18-s + (1.92 + 0.382i)20-s + (−1.08 + 2.63i)25-s + (−0.541 + 1.30i)26-s + (−1.08 − 1.63i)29-s + (−0.382 − 0.923i)32-s + ⋯
 L(s)  = 1 + (0.923 − 0.382i)2-s + (0.707 − 0.707i)4-s + (1.08 + 1.63i)5-s + (0.382 − 0.923i)8-s + (0.923 + 0.382i)9-s + (1.63 + 1.08i)10-s + (−1 + i)13-s − i·16-s + (−0.382 − 0.923i)17-s + 18-s + (1.92 + 0.382i)20-s + (−1.08 + 2.63i)25-s + (−0.541 + 1.30i)26-s + (−1.08 − 1.63i)29-s + (−0.382 − 0.923i)32-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$3332$$    =    $$2^{2} \cdot 7^{2} \cdot 17$$ Sign: $0.974 - 0.222i$ Analytic conductor: $$1.66288$$ Root analytic conductor: $$1.28952$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3332} (2547, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3332,\ (\ :0),\ 0.974 - 0.222i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$2.732340466$$ $$L(\frac12)$$ $$\approx$$ $$2.732340466$$ $$L(1)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.923 + 0.382i)T$$
7 $$1$$
17 $$1 + (0.382 + 0.923i)T$$
good3 $$1 + (-0.923 - 0.382i)T^{2}$$
5 $$1 + (-1.08 - 1.63i)T + (-0.382 + 0.923i)T^{2}$$
11 $$1 + (-0.923 + 0.382i)T^{2}$$
13 $$1 + (1 - i)T - iT^{2}$$
19 $$1 + (0.707 - 0.707i)T^{2}$$
23 $$1 + (0.923 - 0.382i)T^{2}$$
29 $$1 + (1.08 + 1.63i)T + (-0.382 + 0.923i)T^{2}$$
31 $$1 + (0.923 + 0.382i)T^{2}$$
37 $$1 + (0.0761 - 0.382i)T + (-0.923 - 0.382i)T^{2}$$
41 $$1 + (-0.617 + 0.923i)T + (-0.382 - 0.923i)T^{2}$$
43 $$1 + (-0.707 - 0.707i)T^{2}$$
47 $$1 + iT^{2}$$
53 $$1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2}$$
59 $$1 + (-0.707 - 0.707i)T^{2}$$
61 $$1 + (1.38 + 0.923i)T + (0.382 + 0.923i)T^{2}$$
67 $$1 + T^{2}$$
71 $$1 + (0.923 + 0.382i)T^{2}$$
73 $$1 + (-0.216 - 0.324i)T + (-0.382 + 0.923i)T^{2}$$
79 $$1 + (-0.923 + 0.382i)T^{2}$$
83 $$1 + (-0.707 + 0.707i)T^{2}$$
89 $$1 + iT^{2}$$
97 $$1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$