# Properties

 Label 1-799-799.234-r1-0-0 Degree $1$ Conductor $799$ Sign $0.615 + 0.788i$ Analytic cond. $85.8644$ Root an. cond. $85.8644$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + i·3-s + 4-s − i·5-s − i·6-s − i·7-s − 8-s − 9-s + i·10-s + i·11-s + i·12-s − 13-s + i·14-s + 15-s + 16-s + ⋯
 L(s)  = 1 − 2-s + i·3-s + 4-s − i·5-s − i·6-s − i·7-s − 8-s − 9-s + i·10-s + i·11-s + i·12-s − 13-s + i·14-s + 15-s + 16-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$799$$    =    $$17 \cdot 47$$ Sign: $0.615 + 0.788i$ Analytic conductor: $$85.8644$$ Root analytic conductor: $$85.8644$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{799} (234, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 799,\ (1:\ ),\ 0.615 + 0.788i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.7602695902 + 0.3709570980i$$ $$L(\frac12)$$ $$\approx$$ $$0.7602695902 + 0.3709570980i$$ $$L(1)$$ $$\approx$$ $$0.6240010123 + 0.07681803500i$$ $$L(1)$$ $$\approx$$ $$0.6240010123 + 0.07681803500i$$

## Euler product

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

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

−22.00339765506706596404786957988, −21.10298480587214474381106666889, −19.87109870184798057207469665021, −19.310141765421263679729411581257, −18.642267850558834422067110211923, −18.155911641568984871126088635646, −17.481985337056238611801904256787, −16.42881652401664463281883078627, −15.61523050148431593221174819392, −14.52292425798318039655132514167, −14.12480323179222035378692169985, −12.59334957876014003395716083002, −12.03302646431070854732707051953, −11.19915048738758255276986792675, −10.49652550341685265139672643733, −9.27607427134125725482939571711, −8.60784424103795841997654398749, −7.630815409761533941025260146801, −7.01222019143744817335044125809, −6.12609662906039095894991177589, −5.43647384867946254780448521499, −3.14625693007173291720422989880, −2.69865072136989277334388109671, −1.73184478675425203729862992171, −0.40484774163314828889735177052, 0.647604883668800348056931468335, 1.828637567097981997201808224327, 3.12799634115316305660647088209, 4.299888459344135654930786858114, 5.013762130745605156622796374972, 6.1420282059495760731178391090, 7.54261449027531680866808523001, 7.88943142510251044459161100426, 9.28636520166707622959620825159, 9.65248873965510826468560663009, 10.238711484961016912345769453112, 11.41912870403548297338267806262, 11.98761750608379254423858774620, 13.14949964227127653558548149027, 14.270941067402638341711362555143, 15.26627641905883599883163991137, 15.85445855570517803380949177511, 16.78237415865645752709165350206, 17.19619670466762353542099693018, 17.7642641423595619490680103111, 19.25030463890714044047022357534, 20.02160141833073629659565126303, 20.389263987783494620286917611911, 21.010883409482448588178692858649, 22.00265371352066656322164666288