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Properties

Label 1-837-837.484-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.746 - 0.665i$
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 + (−0.939 + 0.342i)5-s + (−0.719 − 0.694i)7-s + (−0.104 + 0.994i)8-s + (0.309 + 0.951i)10-s + (−0.241 + 0.970i)11-s + (0.848 − 0.529i)13-s + (−0.719 + 0.694i)14-s + (0.990 + 0.139i)16-s + (−0.104 + 0.994i)17-s + (0.309 + 0.951i)19-s + (0.961 − 0.275i)20-s + (0.961 + 0.275i)22-s + (−0.241 − 0.970i)23-s + ⋯
L(s)  = 1  + (0.0348 − 0.999i)2-s + (−0.997 − 0.0697i)4-s + (−0.939 + 0.342i)5-s + (−0.719 − 0.694i)7-s + (−0.104 + 0.994i)8-s + (0.309 + 0.951i)10-s + (−0.241 + 0.970i)11-s + (0.848 − 0.529i)13-s + (−0.719 + 0.694i)14-s + (0.990 + 0.139i)16-s + (−0.104 + 0.994i)17-s + (0.309 + 0.951i)19-s + (0.961 − 0.275i)20-s + (0.961 + 0.275i)22-s + (−0.241 − 0.970i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.746 - 0.665i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (484, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.746 - 0.665i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2421464816 - 0.6354712002i\)
\(L(\frac12)\) \(\approx\) \(0.2421464816 - 0.6354712002i\)
\(L(1)\) \(\approx\) \(0.6205484910 - 0.3651016659i\)
\(L(1)\) \(\approx\) \(0.6205484910 - 0.3651016659i\)

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 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (-0.719 - 0.694i)T \)
11 \( 1 + (-0.241 + 0.970i)T \)
13 \( 1 + (0.848 - 0.529i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.241 - 0.970i)T \)
29 \( 1 + (0.0348 - 0.999i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.374 + 0.927i)T \)
43 \( 1 + (0.0348 - 0.999i)T \)
47 \( 1 + (-0.374 - 0.927i)T \)
53 \( 1 + (-0.104 + 0.994i)T \)
59 \( 1 + (0.848 - 0.529i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.104 - 0.994i)T \)
79 \( 1 + (-0.997 + 0.0697i)T \)
83 \( 1 + (-0.882 + 0.469i)T \)
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

−22.57510704942456927282371210157, −21.91170511808005618962913451432, −21.014937480072355155659492405342, −19.8302705829830453874605452741, −19.02237758194377038486943360976, −18.53756269530497470318206519719, −17.596736337763516622836482239253, −16.39572529026215329370225220234, −15.947528705679731030732811696, −15.63104410712211703872362365557, −14.49310769444496975077852693562, −13.52011869770915899414798640684, −12.988229098593929279168298714255, −11.85827678733567691828396974213, −11.20869507842774582144835509401, −9.76157005295833415474307926149, −8.86188274106509757705608342798, −8.495085007324121454291184207671, −7.342928248361212793983845177352, −6.630826372827835452080877085710, −5.61177873500504019648573163248, −4.87674077861423758031397051359, −3.6941627959776166899414759402, −3.02702219670584099433162004115, −0.96510048244790882481172706556, 0.402554737487827795615578135545, 1.76680773831816946819633066123, 2.997498957819756233001917837625, 3.85126143400103014579728270064, 4.314684541296810407429912271643, 5.70978144794749881190126694450, 6.81642959647845664773307379827, 7.89715566657202964942707669329, 8.52309040651647611838821917803, 9.87368702628863455475505759464, 10.37286416795554490574394427226, 11.07869189585560229205700591614, 12.16154151243082198392234849615, 12.67933681781051377569419356021, 13.49901273243222809007518071471, 14.49615975611034685237863835842, 15.28512374424345708764641116502, 16.21415503175567746760539036364, 17.18423157521837816158431859222, 18.10226217637881588477570345819, 18.85405105720290196736582154680, 19.50196776145353305089090312523, 20.377915768633607036915392282834, 20.62362792614997634999660769232, 21.92834372942449371592420071565

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