Properties

 Label 1-145-145.17-r0-0-0 Degree $1$ Conductor $145$ Sign $0.934 + 0.355i$ Analytic cond. $0.673377$ Root an. cond. $0.673377$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + 2-s − 3-s + 4-s − 6-s + i·7-s + 8-s + 9-s − i·11-s − 12-s + i·13-s + i·14-s + 16-s + 17-s + 18-s + i·19-s + ⋯
 L(s)  = 1 + 2-s − 3-s + 4-s − 6-s + i·7-s + 8-s + 9-s − i·11-s − 12-s + i·13-s + i·14-s + 16-s + 17-s + 18-s + i·19-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$1$$ Conductor: $$145$$    =    $$5 \cdot 29$$ Sign: $0.934 + 0.355i$ Analytic conductor: $$0.673377$$ Root analytic conductor: $$0.673377$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{145} (17, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 145,\ (0:\ ),\ 0.934 + 0.355i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.526411005 + 0.2803797537i$$ $$L(\frac12)$$ $$\approx$$ $$1.526411005 + 0.2803797537i$$ $$L(1)$$ $$\approx$$ $$1.445877376 + 0.1421279466i$$ $$L(1)$$ $$\approx$$ $$1.445877376 + 0.1421279466i$$

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
29 $$1$$
good2 $$1$$
3 $$1 + T$$
7 $$1 - T$$
11 $$1 + T$$
13 $$1$$
17 $$1 - T$$
19 $$1 + iT$$
23 $$1 + T$$
31 $$1$$
37 $$1 - iT$$
41 $$1 - T$$
43 $$1 + iT$$
47 $$1 + iT$$
53 $$1$$
59 $$1 + T$$
61 $$1 + T$$
67 $$1 + T$$
71 $$1 + iT$$
73 $$1$$
79 $$1 - iT$$
83 $$1 - iT$$
89 $$1 - iT$$
97 $$1 - T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$