Properties

 Label 1-4235-4235.1319-r0-0-0 Degree $1$ Conductor $4235$ Sign $-0.959 - 0.282i$ Analytic cond. $19.6672$ Root an. cond. $19.6672$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (0.580 + 0.814i)2-s + (−0.5 − 0.866i)3-s + (−0.327 + 0.945i)4-s + (0.415 − 0.909i)6-s + (−0.959 + 0.281i)8-s + (−0.5 + 0.866i)9-s + (0.981 − 0.189i)12-s + (0.654 + 0.755i)13-s + (−0.786 − 0.618i)16-s + (−0.0475 + 0.998i)17-s + (−0.995 + 0.0950i)18-s + (0.0475 + 0.998i)19-s + (0.786 + 0.618i)23-s + (0.723 + 0.690i)24-s + (−0.235 + 0.971i)26-s + 27-s + ⋯
 L(s)  = 1 + (0.580 + 0.814i)2-s + (−0.5 − 0.866i)3-s + (−0.327 + 0.945i)4-s + (0.415 − 0.909i)6-s + (−0.959 + 0.281i)8-s + (−0.5 + 0.866i)9-s + (0.981 − 0.189i)12-s + (0.654 + 0.755i)13-s + (−0.786 − 0.618i)16-s + (−0.0475 + 0.998i)17-s + (−0.995 + 0.0950i)18-s + (0.0475 + 0.998i)19-s + (0.786 + 0.618i)23-s + (0.723 + 0.690i)24-s + (−0.235 + 0.971i)26-s + 27-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$1$$ Conductor: $$4235$$    =    $$5 \cdot 7 \cdot 11^{2}$$ Sign: $-0.959 - 0.282i$ Analytic conductor: $$19.6672$$ Root analytic conductor: $$19.6672$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{4235} (1319, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 4235,\ (0:\ ),\ -0.959 - 0.282i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$-0.1329075383 + 0.9202767794i$$ $$L(\frac12)$$ $$\approx$$ $$-0.1329075383 + 0.9202767794i$$ $$L(1)$$ $$\approx$$ $$0.8757620576 + 0.4875731243i$$ $$L(1)$$ $$\approx$$ $$0.8757620576 + 0.4875731243i$$

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
7 $$1$$
11 $$1$$
good2 $$1 + (0.580 + 0.814i)T$$
3 $$1 + (-0.5 - 0.866i)T$$
13 $$1 + (0.654 + 0.755i)T$$
17 $$1 + (-0.0475 + 0.998i)T$$
19 $$1 + (0.0475 + 0.998i)T$$
23 $$1 + (0.786 + 0.618i)T$$
29 $$1 + (-0.841 + 0.540i)T$$
31 $$1 + (-0.981 - 0.189i)T$$
37 $$1 + (0.327 + 0.945i)T$$
41 $$1 + (0.415 - 0.909i)T$$
43 $$1 + (-0.959 + 0.281i)T$$
47 $$1 + (-0.995 - 0.0950i)T$$
53 $$1 + (0.786 - 0.618i)T$$
59 $$1 + (-0.580 + 0.814i)T$$
61 $$1 + (-0.995 - 0.0950i)T$$
67 $$1 + (0.995 - 0.0950i)T$$
71 $$1 + (0.841 - 0.540i)T$$
73 $$1 + (-0.928 + 0.371i)T$$
79 $$1 + (-0.723 + 0.690i)T$$
83 $$1 + (0.142 + 0.989i)T$$
89 $$1 + (0.888 - 0.458i)T$$
97 $$1 + (-0.959 + 0.281i)T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$