Properties

 Label 2-145-145.57-c0-0-0 Degree $2$ Conductor $145$ Sign $0.850 - 0.525i$ Analytic cond. $0.0723644$ Root an. cond. $0.269006$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + i·4-s + i·5-s + (−1 − i)7-s − i·9-s + (1 − i)13-s − 16-s − 20-s + (−1 + i)23-s − 25-s + (1 − i)28-s + i·29-s + (1 − i)35-s + 36-s + 45-s + i·49-s + ⋯
 L(s)  = 1 + i·4-s + i·5-s + (−1 − i)7-s − i·9-s + (1 − i)13-s − 16-s − 20-s + (−1 + i)23-s − 25-s + (1 − i)28-s + i·29-s + (1 − i)35-s + 36-s + 45-s + i·49-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$145$$    =    $$5 \cdot 29$$ Sign: $0.850 - 0.525i$ Analytic conductor: $$0.0723644$$ Root analytic conductor: $$0.269006$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{145} (57, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 145,\ (\ :0),\ 0.850 - 0.525i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.6005376324$$ $$L(\frac12)$$ $$\approx$$ $$0.6005376324$$ $$L(1)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 - iT$$
29 $$1 - iT$$
good2 $$1 - iT^{2}$$
3 $$1 + iT^{2}$$
7 $$1 + (1 + i)T + iT^{2}$$
11 $$1 - T^{2}$$
13 $$1 + (-1 + i)T - iT^{2}$$
17 $$1 - iT^{2}$$
19 $$1 + T^{2}$$
23 $$1 + (1 - i)T - iT^{2}$$
31 $$1 - T^{2}$$
37 $$1 - iT^{2}$$
41 $$1 - T^{2}$$
43 $$1 + iT^{2}$$
47 $$1 - iT^{2}$$
53 $$1 + (1 - i)T - iT^{2}$$
59 $$1 + 2iT - T^{2}$$
61 $$1 - T^{2}$$
67 $$1 + (-1 - i)T + iT^{2}$$
71 $$1 + T^{2}$$
73 $$1 + iT^{2}$$
79 $$1 + T^{2}$$
83 $$1 + (-1 + i)T - iT^{2}$$
89 $$1 + T^{2}$$
97 $$1 - iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$