# Properties

 Label 2-3332-3332.135-c0-0-2 Degree $2$ Conductor $3332$ Sign $-0.788 + 0.615i$ 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 + (0.826 − 0.563i)2-s + (−1.07 + 0.997i)3-s + (0.365 − 0.930i)4-s + (−0.326 + 1.42i)6-s + (0.222 + 0.974i)7-s + (−0.222 − 0.974i)8-s + (0.0858 − 1.14i)9-s + (−0.123 − 1.64i)11-s + (0.535 + 1.36i)12-s + (−1.48 − 0.716i)13-s + (0.733 + 0.680i)14-s + (−0.733 − 0.680i)16-s + (−0.988 − 0.149i)17-s + (−0.574 − 0.995i)18-s + (−1.21 − 0.825i)21-s + (−1.03 − 1.29i)22-s + ⋯
 L(s)  = 1 + (0.826 − 0.563i)2-s + (−1.07 + 0.997i)3-s + (0.365 − 0.930i)4-s + (−0.326 + 1.42i)6-s + (0.222 + 0.974i)7-s + (−0.222 − 0.974i)8-s + (0.0858 − 1.14i)9-s + (−0.123 − 1.64i)11-s + (0.535 + 1.36i)12-s + (−1.48 − 0.716i)13-s + (0.733 + 0.680i)14-s + (−0.733 − 0.680i)16-s + (−0.988 − 0.149i)17-s + (−0.574 − 0.995i)18-s + (−1.21 − 0.825i)21-s + (−1.03 − 1.29i)22-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3332$$    =    $$2^{2} \cdot 7^{2} \cdot 17$$ Sign: $-0.788 + 0.615i$ Analytic conductor: $$1.66288$$ Root analytic conductor: $$1.28952$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3332} (135, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3332,\ (\ :0),\ -0.788 + 0.615i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.5655712303$$ $$L(\frac12)$$ $$\approx$$ $$0.5655712303$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.826 + 0.563i)T$$
7 $$1 + (-0.222 - 0.974i)T$$
17 $$1 + (0.988 + 0.149i)T$$
good3 $$1 + (1.07 - 0.997i)T + (0.0747 - 0.997i)T^{2}$$
5 $$1 + (-0.826 - 0.563i)T^{2}$$
11 $$1 + (0.123 + 1.64i)T + (-0.988 + 0.149i)T^{2}$$
13 $$1 + (1.48 + 0.716i)T + (0.623 + 0.781i)T^{2}$$
19 $$1 + (0.5 + 0.866i)T^{2}$$
23 $$1 + (1.78 - 0.268i)T + (0.955 - 0.294i)T^{2}$$
29 $$1 + (0.222 + 0.974i)T^{2}$$
31 $$1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2}$$
37 $$1 + (0.733 - 0.680i)T^{2}$$
41 $$1 + (0.900 - 0.433i)T^{2}$$
43 $$1 + (0.900 + 0.433i)T^{2}$$
47 $$1 + (-0.365 + 0.930i)T^{2}$$
53 $$1 + (-0.698 + 1.77i)T + (-0.733 - 0.680i)T^{2}$$
59 $$1 + (-0.826 + 0.563i)T^{2}$$
61 $$1 + (0.733 - 0.680i)T^{2}$$
67 $$1 + (0.5 - 0.866i)T^{2}$$
71 $$1 + (1.19 + 1.49i)T + (-0.222 + 0.974i)T^{2}$$
73 $$1 + (-0.365 - 0.930i)T^{2}$$
79 $$1 + (-0.0747 + 0.129i)T + (-0.5 - 0.866i)T^{2}$$
83 $$1 + (-0.623 + 0.781i)T^{2}$$
89 $$1 + (0.109 - 1.46i)T + (-0.988 - 0.149i)T^{2}$$
97 $$1 - T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$