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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 $0$

Origins

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Normalization:  

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 \( 1 - iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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

Graph of the $Z$-function along the critical line