Properties

 Label 2-799-799.234-c0-0-2 Degree $2$ Conductor $799$ Sign $0.559 + 0.828i$ Analytic cond. $0.398752$ Root an. cond. $0.631468$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 1.61i·2-s + (1.26 + 1.26i)3-s − 1.61·4-s + (2.03 − 2.03i)6-s + (0.221 − 0.221i)7-s + i·8-s + 2.17i·9-s + (−2.03 − 2.03i)12-s + (−0.357 − 0.357i)14-s + (0.309 − 0.951i)17-s + 3.52·18-s + 0.557·21-s + (−1.26 + 1.26i)24-s + i·25-s + (−1.48 + 1.48i)27-s + (−0.357 + 0.357i)28-s + ⋯
 L(s)  = 1 − 1.61i·2-s + (1.26 + 1.26i)3-s − 1.61·4-s + (2.03 − 2.03i)6-s + (0.221 − 0.221i)7-s + i·8-s + 2.17i·9-s + (−2.03 − 2.03i)12-s + (−0.357 − 0.357i)14-s + (0.309 − 0.951i)17-s + 3.52·18-s + 0.557·21-s + (−1.26 + 1.26i)24-s + i·25-s + (−1.48 + 1.48i)27-s + (−0.357 + 0.357i)28-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$799$$    =    $$17 \cdot 47$$ Sign: $0.559 + 0.828i$ Analytic conductor: $$0.398752$$ Root analytic conductor: $$0.631468$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{799} (234, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 799,\ (\ :0),\ 0.559 + 0.828i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.346132447$$ $$L(\frac12)$$ $$\approx$$ $$1.346132447$$ $$L(1)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad17 $$1 + (-0.309 + 0.951i)T$$
47 $$1 + T$$
good2 $$1 + 1.61iT - T^{2}$$
3 $$1 + (-1.26 - 1.26i)T + iT^{2}$$
5 $$1 - iT^{2}$$
7 $$1 + (-0.221 + 0.221i)T - iT^{2}$$
11 $$1 + iT^{2}$$
13 $$1 - T^{2}$$
19 $$1 + T^{2}$$
23 $$1 + iT^{2}$$
29 $$1 - iT^{2}$$
31 $$1 - iT^{2}$$
37 $$1 + (1.26 + 1.26i)T + iT^{2}$$
41 $$1 + iT^{2}$$
43 $$1 + T^{2}$$
53 $$1 + 1.17iT - T^{2}$$
59 $$1 + 1.61iT - T^{2}$$
61 $$1 + (1.39 - 1.39i)T - iT^{2}$$
67 $$1 - T^{2}$$
71 $$1 + (0.642 + 0.642i)T + iT^{2}$$
73 $$1 - iT^{2}$$
79 $$1 + (0.642 - 0.642i)T - iT^{2}$$
83 $$1 - T^{2}$$
89 $$1 + 1.61T + T^{2}$$
97 $$1 + (-1.39 - 1.39i)T + iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$