Properties

Label 1-837-837.169-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.999 + 0.0258i$
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  + (−0.997 + 0.0697i)2-s + (0.990 − 0.139i)4-s + (0.766 + 0.642i)5-s + (0.0348 − 0.999i)7-s + (−0.978 + 0.207i)8-s + (−0.809 − 0.587i)10-s + (−0.882 + 0.469i)11-s + (0.438 + 0.898i)13-s + (0.0348 + 0.999i)14-s + (0.961 − 0.275i)16-s + (−0.978 + 0.207i)17-s + (−0.809 − 0.587i)19-s + (0.848 + 0.529i)20-s + (0.848 − 0.529i)22-s + (−0.882 − 0.469i)23-s + ⋯
L(s)  = 1  + (−0.997 + 0.0697i)2-s + (0.990 − 0.139i)4-s + (0.766 + 0.642i)5-s + (0.0348 − 0.999i)7-s + (−0.978 + 0.207i)8-s + (−0.809 − 0.587i)10-s + (−0.882 + 0.469i)11-s + (0.438 + 0.898i)13-s + (0.0348 + 0.999i)14-s + (0.961 − 0.275i)16-s + (−0.978 + 0.207i)17-s + (−0.809 − 0.587i)19-s + (0.848 + 0.529i)20-s + (0.848 − 0.529i)22-s + (−0.882 − 0.469i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.999 + 0.0258i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.999 + 0.0258i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001264833757 + 0.09786554467i\)
\(L(\frac12)\) \(\approx\) \(0.001264833757 + 0.09786554467i\)
\(L(1)\) \(\approx\) \(0.5682925338 + 0.06285293521i\)
\(L(1)\) \(\approx\) \(0.5682925338 + 0.06285293521i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.997 + 0.0697i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (0.0348 - 0.999i)T \)
11 \( 1 + (-0.882 + 0.469i)T \)
13 \( 1 + (0.438 + 0.898i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.882 - 0.469i)T \)
29 \( 1 + (-0.997 + 0.0697i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.719 + 0.694i)T \)
43 \( 1 + (-0.997 + 0.0697i)T \)
47 \( 1 + (-0.719 - 0.694i)T \)
53 \( 1 + (-0.978 + 0.207i)T \)
59 \( 1 + (0.438 + 0.898i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.978 - 0.207i)T \)
79 \( 1 + (0.990 + 0.139i)T \)
83 \( 1 + (0.559 + 0.829i)T \)
89 \( 1 + (0.669 - 0.743i)T \)
97 \( 1 + (0.0348 - 0.999i)T \)
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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

−21.619268133682977736309384966143, −20.63869832996173892928166779081, −20.38334710006932832546534349038, −19.10223384587314706030209591091, −18.48629459312488357924411673435, −17.771806734399236311064189285475, −17.17222691427259472788305830398, −16.04451795466036358700728209559, −15.68371166340060984989504120550, −14.71073274351241972139976063249, −13.3490248935003908502982802556, −12.78287089311834639442094208413, −11.83332453587534953415963817853, −10.89952440075117170613300055934, −10.0884845897852252691092704752, −9.30962257725333288514027817902, −8.37329439737281405992100784329, −8.103556220825717730273682973887, −6.55146471888160286060440881036, −5.82187328499569515422867667798, −5.10968011739744193271556878421, −3.40751558445956028748673617507, −2.324761001897802935051372177144, −1.65660071759708209592638691618, −0.054165705437222504567303579804, 1.71959888345763969100512325636, 2.27232389324109603965089592694, 3.57174781700370137552312967118, 4.811660285342612498504434672603, 6.176186710197774134427931974336, 6.758229399153284305326166062665, 7.49670321951081805793736404124, 8.539846127540184266057169927564, 9.43893307772716877697176739487, 10.304422258741381054791138548909, 10.76840636926590586194077584050, 11.5407055235839546276239589974, 12.937339547252111476013189358161, 13.62687672242478863830429422958, 14.63300340103312890482884642911, 15.36940029831425337354038151256, 16.39983916943224217652714045924, 17.026815192828830387022533552121, 17.92795990138630551911221253196, 18.27498794702347659679838487131, 19.29194243840576871074961119453, 20.05277631932991813657800986313, 20.87531731614366101899603747380, 21.474075192942300389960126074709, 22.50877327162975853898657167506

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