Properties

 Label 1-153-153.25-r0-0-0 Degree $1$ Conductor $153$ Sign $0.812 - 0.582i$ Analytic cond. $0.710529$ Root an. cond. $0.710529$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.965 + 0.258i)5-s + (−0.965 − 0.258i)7-s + i·8-s + (0.707 − 0.707i)10-s + (0.965 + 0.258i)11-s + (0.5 − 0.866i)13-s + (0.965 − 0.258i)14-s + (−0.5 − 0.866i)16-s − i·19-s + (−0.258 + 0.965i)20-s + (−0.965 + 0.258i)22-s + (−0.258 − 0.965i)23-s + (0.866 − 0.5i)25-s + i·26-s + ⋯
 L(s)  = 1 + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.965 + 0.258i)5-s + (−0.965 − 0.258i)7-s + i·8-s + (0.707 − 0.707i)10-s + (0.965 + 0.258i)11-s + (0.5 − 0.866i)13-s + (0.965 − 0.258i)14-s + (−0.5 − 0.866i)16-s − i·19-s + (−0.258 + 0.965i)20-s + (−0.965 + 0.258i)22-s + (−0.258 − 0.965i)23-s + (0.866 − 0.5i)25-s + i·26-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$1$$ Conductor: $$153$$    =    $$3^{2} \cdot 17$$ Sign: $0.812 - 0.582i$ Analytic conductor: $$0.710529$$ Root analytic conductor: $$0.710529$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{153} (25, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 153,\ (0:\ ),\ 0.812 - 0.582i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.4971262974 - 0.1598388905i$$ $$L(\frac12)$$ $$\approx$$ $$0.4971262974 - 0.1598388905i$$ $$L(1)$$ $$\approx$$ $$0.5857843683 + 0.007878766971i$$ $$L(1)$$ $$\approx$$ $$0.5857843683 + 0.007878766971i$$

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
17 $$1$$
good2 $$1 + (-0.866 + 0.5i)T$$
5 $$1 + (-0.965 + 0.258i)T$$
7 $$1 + (-0.965 - 0.258i)T$$
11 $$1 + (0.965 + 0.258i)T$$
13 $$1 + (0.5 - 0.866i)T$$
19 $$1 - iT$$
23 $$1 + (-0.258 - 0.965i)T$$
29 $$1 + (0.258 - 0.965i)T$$
31 $$1 + (0.965 - 0.258i)T$$
37 $$1 + (-0.707 - 0.707i)T$$
41 $$1 + (0.258 + 0.965i)T$$
43 $$1 + (0.866 - 0.5i)T$$
47 $$1 + (0.5 + 0.866i)T$$
53 $$1 - iT$$
59 $$1 + (-0.866 - 0.5i)T$$
61 $$1 + (-0.965 - 0.258i)T$$
67 $$1 + (-0.5 + 0.866i)T$$
71 $$1 + (-0.707 - 0.707i)T$$
73 $$1 + (0.707 + 0.707i)T$$
79 $$1 + (0.965 + 0.258i)T$$
83 $$1 + (-0.866 + 0.5i)T$$
89 $$1 - T$$
97 $$1 + (0.258 - 0.965i)T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$