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Properties

Label 1-2793-2793.1058-r0-0-0
Degree $1$
Conductor $2793$
Sign $0.594 - 0.803i$
Analytic cond. $12.9706$
Root an. cond. $12.9706$
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  + (0.878 − 0.478i)2-s + (0.542 − 0.840i)4-s + (−0.698 + 0.715i)5-s + (0.0747 − 0.997i)8-s + (−0.270 + 0.962i)10-s + (0.988 − 0.149i)11-s + (−0.980 − 0.198i)13-s + (−0.411 − 0.911i)16-s + (0.998 − 0.0498i)17-s + (0.222 + 0.974i)20-s + (0.797 − 0.603i)22-s + (−0.542 + 0.840i)23-s + (−0.0249 − 0.999i)25-s + (−0.955 + 0.294i)26-s + (0.921 − 0.388i)29-s + ⋯
L(s)  = 1  + (0.878 − 0.478i)2-s + (0.542 − 0.840i)4-s + (−0.698 + 0.715i)5-s + (0.0747 − 0.997i)8-s + (−0.270 + 0.962i)10-s + (0.988 − 0.149i)11-s + (−0.980 − 0.198i)13-s + (−0.411 − 0.911i)16-s + (0.998 − 0.0498i)17-s + (0.222 + 0.974i)20-s + (0.797 − 0.603i)22-s + (−0.542 + 0.840i)23-s + (−0.0249 − 0.999i)25-s + (−0.955 + 0.294i)26-s + (0.921 − 0.388i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.594 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.594 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2793\)    =    \(3 \cdot 7^{2} \cdot 19\)
Sign: $0.594 - 0.803i$
Analytic conductor: \(12.9706\)
Root analytic conductor: \(12.9706\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2793} (1058, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2793,\ (0:\ ),\ 0.594 - 0.803i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.380634965 - 1.200115577i\)
\(L(\frac12)\) \(\approx\) \(2.380634965 - 1.200115577i\)
\(L(1)\) \(\approx\) \(1.586499069 - 0.4582778945i\)
\(L(1)\) \(\approx\) \(1.586499069 - 0.4582778945i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.878 - 0.478i)T \)
5 \( 1 + (-0.698 + 0.715i)T \)
11 \( 1 + (0.988 - 0.149i)T \)
13 \( 1 + (-0.980 - 0.198i)T \)
17 \( 1 + (0.998 - 0.0498i)T \)
23 \( 1 + (-0.542 + 0.840i)T \)
29 \( 1 + (0.921 - 0.388i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.222 + 0.974i)T \)
41 \( 1 + (-0.969 - 0.246i)T \)
43 \( 1 + (0.995 + 0.0995i)T \)
47 \( 1 + (0.661 - 0.749i)T \)
53 \( 1 + (0.542 - 0.840i)T \)
59 \( 1 + (-0.583 + 0.811i)T \)
61 \( 1 + (-0.124 + 0.992i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (-0.124 - 0.992i)T \)
73 \( 1 + (0.980 - 0.198i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.988 + 0.149i)T \)
89 \( 1 + (-0.0249 - 0.999i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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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

−19.50779206736638345243905042916, −18.76696556521024288097172090374, −17.53196799890631521404656670486, −16.98178457038233051623722103773, −16.47877257286400469943441881562, −15.7751828754940482932308159204, −15.03752461815652272186920048627, −14.34536015152314757779893898666, −13.89780063229773980436461086159, −12.681126419383237371758494634685, −12.34485307791361931958292551122, −11.843171868245055049182006064744, −11.03950110232543851905786732161, −9.92555510521221649302837339708, −9.072618152785942202264172184337, −8.263649563769141907829868346784, −7.62063783845682725319180541658, −6.92634748506094151151694741716, −6.07167969552930185130202241429, −5.25537507946385211580759584812, −4.42689005042775576524536898551, −4.034408990693372076476669099284, −3.05732620559307456020291491883, −2.10547125270774390515923830254, −0.918713377416702862152689129843, 0.76638243810237850907681296114, 1.82363098682453294201228582534, 2.862440175671168241064001992668, 3.39493708618557931552328494445, 4.17805235648529012057747971511, 4.94000188909105756018229172603, 5.87054529851345549279000345202, 6.631070444200718231339002097580, 7.28914660988382650569329764502, 8.05970744383401204464390463400, 9.2257976344441773850974760243, 10.13960190948090704729131484790, 10.47663057445555191605757885542, 11.67237184795693996562604951956, 11.88776859993273548576609641011, 12.43866994291221042899054028519, 13.700878778312438480789993954990, 14.05990979820351169748125124060, 14.881087321817091960423542372417, 15.262740732431517329257520020761, 16.125173917940334328785449989608, 16.88213215578109625196215248874, 17.82132089818692422508371230275, 18.68915840353913257025163570185, 19.50629548418223786590685208628

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