Properties

 Label 2-3332-68.67-c0-0-17 Degree $2$ Conductor $3332$ Sign $i$ 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 + 2-s + 4-s − 2i·5-s + 8-s − 9-s − 2i·10-s + 16-s − i·17-s − 18-s − 2i·20-s − 3·25-s + 32-s − i·34-s − 36-s − 2i·40-s + 2i·41-s + ⋯
 L(s)  = 1 + 2-s + 4-s − 2i·5-s + 8-s − 9-s − 2i·10-s + 16-s − i·17-s − 18-s − 2i·20-s − 3·25-s + 32-s − i·34-s − 36-s − 2i·40-s + 2i·41-s + ⋯

Functional equation

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

Invariants

 Degree: $$2$$ Conductor: $$3332$$    =    $$2^{2} \cdot 7^{2} \cdot 17$$ Sign: $i$ Analytic conductor: $$1.66288$$ Root analytic conductor: $$1.28952$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3332} (883, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3332,\ (\ :0),\ i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$2.214623940$$ $$L(\frac12)$$ $$\approx$$ $$2.214623940$$ $$L(1)$$ not available $$L(1)$$ not available

Euler product

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