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Properties

Label 1-837-837.202-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.997 - 0.0729i$
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  + (0.0348 + 0.999i)2-s + (−0.997 + 0.0697i)4-s + (0.173 + 0.984i)5-s + (−0.241 − 0.970i)7-s + (−0.104 − 0.994i)8-s + (−0.978 + 0.207i)10-s + (−0.719 + 0.694i)11-s + (0.0348 − 0.999i)13-s + (0.961 − 0.275i)14-s + (0.990 − 0.139i)16-s + (0.913 + 0.406i)17-s + (0.669 + 0.743i)19-s + (−0.241 − 0.970i)20-s + (−0.719 − 0.694i)22-s + (−0.241 + 0.970i)23-s + ⋯
L(s)  = 1  + (0.0348 + 0.999i)2-s + (−0.997 + 0.0697i)4-s + (0.173 + 0.984i)5-s + (−0.241 − 0.970i)7-s + (−0.104 − 0.994i)8-s + (−0.978 + 0.207i)10-s + (−0.719 + 0.694i)11-s + (0.0348 − 0.999i)13-s + (0.961 − 0.275i)14-s + (0.990 − 0.139i)16-s + (0.913 + 0.406i)17-s + (0.669 + 0.743i)19-s + (−0.241 − 0.970i)20-s + (−0.719 − 0.694i)22-s + (−0.241 + 0.970i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.997 - 0.0729i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.997 - 0.0729i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03060874611 + 0.8382632356i\)
\(L(\frac12)\) \(\approx\) \(0.03060874611 + 0.8382632356i\)
\(L(1)\) \(\approx\) \(0.6469499734 + 0.5609857775i\)
\(L(1)\) \(\approx\) \(0.6469499734 + 0.5609857775i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.0348 + 0.999i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (-0.241 - 0.970i)T \)
11 \( 1 + (-0.719 + 0.694i)T \)
13 \( 1 + (0.0348 - 0.999i)T \)
17 \( 1 + (0.913 + 0.406i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (-0.241 + 0.970i)T \)
29 \( 1 + (0.0348 + 0.999i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.615 + 0.788i)T \)
43 \( 1 + (-0.882 - 0.469i)T \)
47 \( 1 + (-0.374 + 0.927i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.882 + 0.469i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (0.913 + 0.406i)T \)
73 \( 1 + (0.913 - 0.406i)T \)
79 \( 1 + (0.559 - 0.829i)T \)
83 \( 1 + (0.0348 + 0.999i)T \)
89 \( 1 + (0.913 - 0.406i)T \)
97 \( 1 + (-0.719 + 0.694i)T \)
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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

−21.55783616257595887337166646075, −20.96031479868397004887982014586, −20.331183709360097015606726499384, −19.26205481290084791642847963828, −18.73077408427376058644725185213, −18.033208328777520455972229167443, −16.924259058862237915227689494381, −16.2434913527228497694447998670, −15.322978093128690612472039692107, −14.063016390357933516699047183779, −13.50498562524925482872685458710, −12.630572411097966964939980477166, −11.91955284409077420173594153902, −11.3683869529363450082458166347, −10.06154291438332852288001736336, −9.44867915943553894841262789471, −8.63677264086037825078184627356, −8.04232884199929843707986706446, −6.36653689533792722628878328502, −5.28922360063698486914790213635, −4.85775020566908698615855493362, −3.56820588382406788530613690192, −2.61167059321094864750368688661, −1.69210593019302842937402004682, −0.39301528783638608472084732887, 1.387480781474691500285972521258, 3.16144827100090873267133565507, 3.69037717499466137364225867514, 5.04181992492273611740855075007, 5.801045638823193292958166942904, 6.76636344476571820525107425078, 7.65612200820116023843242745155, 7.87673401181492793353248842729, 9.50701471107614310178552138954, 10.17301048264798078612627676397, 10.69830525886112170227523618958, 12.21394943769423345214271980799, 13.08147824708497142556646931391, 13.845911756840577999377849971871, 14.54960166366954734462485013973, 15.28072666415712203650409641734, 16.05869865125233301451231251794, 16.94150096651232887199863963777, 17.79811397464698469115358834395, 18.22235110510404804786120786343, 19.17413198896052330964340577087, 20.09600779802444511735020811967, 21.09371145492418558187396401900, 22.04724245346730311675697153257, 22.85139570329786568381517386978

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