Properties

Label 1-837-837.401-r0-0-0
Degree $1$
Conductor $837$
Sign $0.927 - 0.374i$
Analytic cond. $3.88701$
Root an. cond. $3.88701$
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  + (0.374 − 0.927i)2-s + (−0.719 − 0.694i)4-s + (0.939 + 0.342i)5-s + (0.438 + 0.898i)7-s + (−0.913 + 0.406i)8-s + (0.669 − 0.743i)10-s + (0.559 − 0.829i)11-s + (0.374 + 0.927i)13-s + (0.997 − 0.0697i)14-s + (0.0348 + 0.999i)16-s + (−0.104 + 0.994i)17-s + (−0.978 − 0.207i)19-s + (−0.438 − 0.898i)20-s + (−0.559 − 0.829i)22-s + (0.438 − 0.898i)23-s + ⋯
L(s)  = 1  + (0.374 − 0.927i)2-s + (−0.719 − 0.694i)4-s + (0.939 + 0.342i)5-s + (0.438 + 0.898i)7-s + (−0.913 + 0.406i)8-s + (0.669 − 0.743i)10-s + (0.559 − 0.829i)11-s + (0.374 + 0.927i)13-s + (0.997 − 0.0697i)14-s + (0.0348 + 0.999i)16-s + (−0.104 + 0.994i)17-s + (−0.978 − 0.207i)19-s + (−0.438 − 0.898i)20-s + (−0.559 − 0.829i)22-s + (0.438 − 0.898i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.927 - 0.374i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.927 - 0.374i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.994975875 - 0.3874169863i\)
\(L(\frac12)\) \(\approx\) \(1.994975875 - 0.3874169863i\)
\(L(1)\) \(\approx\) \(1.400409056 - 0.4045867732i\)
\(L(1)\) \(\approx\) \(1.400409056 - 0.4045867732i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.374 - 0.927i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (0.438 + 0.898i)T \)
11 \( 1 + (0.559 - 0.829i)T \)
13 \( 1 + (0.374 + 0.927i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (0.438 - 0.898i)T \)
29 \( 1 + (-0.374 + 0.927i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.848 - 0.529i)T \)
43 \( 1 + (0.615 + 0.788i)T \)
47 \( 1 + (0.882 + 0.469i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.615 - 0.788i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.104 - 0.994i)T \)
73 \( 1 + (0.104 + 0.994i)T \)
79 \( 1 + (0.241 + 0.970i)T \)
83 \( 1 + (-0.374 + 0.927i)T \)
89 \( 1 + (-0.104 - 0.994i)T \)
97 \( 1 + (0.559 - 0.829i)T \)
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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

−22.375845252749193971886596524407, −21.45962226371337871563448816155, −20.68419206073718842083094138210, −20.12689544522747667490656312127, −18.71773227549435378320413949884, −17.714870239046023466362690755744, −17.363820236539487944307632592385, −16.75130983209998363641905928299, −15.70651251263763745610961663227, −14.868875347906585064435731780372, −14.11684092615529683713390523453, −13.38175834448744245434233365780, −12.83759886878742468336448462094, −11.76235261264494419780271520255, −10.54630552730242121158705854424, −9.65278946057125740847699042037, −8.91048053914451752414478940853, −7.84440686918673317699303053117, −7.12436295198516546190263139567, −6.211657899678079561267806566799, −5.32290129921921852490156581177, −4.54034901240423487746243170211, −3.65594432839318349255007443258, −2.25346973237795526636812061326, −0.89667281437107686701710377575, 1.37330002636659681677001268355, 2.090044752136615511941437173104, 3.00551013935498629441004606971, 4.10854183336406815595222838877, 5.10097452684054008931695289447, 6.08169428229223068232323840656, 6.535739133269831819471490550493, 8.557353961194850553362928087228, 8.85558650653820298468200176121, 9.8294334162862360307281707186, 10.92943213083131791190657344839, 11.230709098224999496511280339770, 12.40892134295214408691575683256, 13.03506402466617517434425032715, 14.05842859032147270591075834701, 14.5178081700327315832929797440, 15.3039731067070515292274882555, 16.716135490177529393629151374407, 17.42655207270297010137353200395, 18.491940258468494354863713545612, 18.80204484360713991676660090023, 19.637923569278460249241607566610, 20.90042351608306202634352995447, 21.27022990744690080147453526156, 22.01037379831530812685262847778

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