# Properties

 Label 2-799-799.798-c0-0-3 Degree $2$ Conductor $799$ Sign $1$ Analytic cond. $0.398752$ Root an. cond. $0.631468$ Motivic weight $0$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 2·2-s + 3·4-s − 4·8-s + 9-s + 5·16-s + 17-s − 2·18-s − 25-s − 6·32-s − 2·34-s + 3·36-s + 47-s + 49-s + 2·50-s + 2·53-s − 2·59-s + 7·64-s + 3·68-s − 4·72-s + 81-s + 2·83-s + 2·89-s − 2·94-s − 2·98-s − 3·100-s − 2·101-s − 2·103-s + ⋯
 L(s)  = 1 − 2·2-s + 3·4-s − 4·8-s + 9-s + 5·16-s + 17-s − 2·18-s − 25-s − 6·32-s − 2·34-s + 3·36-s + 47-s + 49-s + 2·50-s + 2·53-s − 2·59-s + 7·64-s + 3·68-s − 4·72-s + 81-s + 2·83-s + 2·89-s − 2·94-s − 2·98-s − 3·100-s − 2·101-s − 2·103-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$799$$    =    $$17 \cdot 47$$ Sign: $1$ Analytic conductor: $$0.398752$$ Root analytic conductor: $$0.631468$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: $\chi_{799} (798, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 799,\ (\ :0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.4586100528$$ $$L(\frac12)$$ $$\approx$$ $$0.4586100528$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

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

## Imaginary part of the first few zeros on the critical line

−10.29820576207840850537714049679, −9.595906964192881264702390295351, −8.936588683823187090954747564502, −7.86870164786907834379830867003, −7.44783713661188941128173603628, −6.53198685684828429728004907793, −5.58468925574343620531674961962, −3.70806646406166267541434071666, −2.35894187413246931141314718096, −1.19290530175140966528755188267, 1.19290530175140966528755188267, 2.35894187413246931141314718096, 3.70806646406166267541434071666, 5.58468925574343620531674961962, 6.53198685684828429728004907793, 7.44783713661188941128173603628, 7.86870164786907834379830867003, 8.936588683823187090954747564502, 9.595906964192881264702390295351, 10.29820576207840850537714049679