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Properties

Label 1-837-837.160-r0-0-0
Degree $1$
Conductor $837$
Sign $0.896 - 0.443i$
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  + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (0.766 + 0.642i)5-s + (−0.939 − 0.342i)7-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)10-s + (−0.939 − 0.342i)11-s + (0.173 − 0.984i)13-s + (0.766 + 0.642i)14-s + (0.173 + 0.984i)16-s + 17-s + (−0.5 + 0.866i)19-s + (0.173 + 0.984i)20-s + (0.766 + 0.642i)22-s + (0.766 + 0.642i)23-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (0.766 + 0.642i)5-s + (−0.939 − 0.342i)7-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)10-s + (−0.939 − 0.342i)11-s + (0.173 − 0.984i)13-s + (0.766 + 0.642i)14-s + (0.173 + 0.984i)16-s + 17-s + (−0.5 + 0.866i)19-s + (0.173 + 0.984i)20-s + (0.766 + 0.642i)22-s + (0.766 + 0.642i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.896 - 0.443i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.896 - 0.443i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8675434525 - 0.2028104009i\)
\(L(\frac12)\) \(\approx\) \(0.8675434525 - 0.2028104009i\)
\(L(1)\) \(\approx\) \(0.7315484566 - 0.08198731491i\)
\(L(1)\) \(\approx\) \(0.7315484566 - 0.08198731491i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 + (-0.939 - 0.342i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.766 - 0.642i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (-0.939 + 0.342i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (0.766 - 0.642i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
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

−22.06266802458014571464601966382, −21.13148664387172956149732110299, −20.6427300717271728835959272882, −19.62356697459887008569533692060, −18.85939346250970700446415764121, −18.270742931615842799174487535480, −17.33138545425852199681824245280, −16.52084832072088736728327725214, −16.15572500305507155433503331959, −15.13493220053351159889459235061, −14.27626582500249958652363075792, −13.10262766461184268306082422256, −12.586970089784598254667575144736, −11.38297760696064915010172805993, −10.44359608408376042684884074529, −9.557664739150084561965352815819, −9.18754727062488750651903639096, −8.236188379891783443797810604980, −7.17353538432938376495752597252, −6.32024898731673086507705974692, −5.58349128742684025757445868143, −4.61449466669156460031587423358, −2.88682654310237257972072659650, −2.11993745738328482556722684144, −0.88186999985175946826140427283, 0.7321786754074759796574231409, 2.09103119192730559633166492615, 3.04893317820333388404840270192, 3.61215546009945444963735976022, 5.6003831469648849027101812140, 6.124553266051859227656850509724, 7.325448648079944153743912883328, 7.83558341215842327451932619814, 9.086514554929881837005361250853, 9.8577862854023823075950700553, 10.4984661073114479586854661867, 10.98650031213678185868980585899, 12.39232212334508304911082999584, 12.99724350611176514988114608433, 13.82190979926564630781155523161, 15.0784985264283552475141842577, 15.75338093386969605905841838391, 16.80994362441732497258504398820, 17.2075804440721457804153885753, 18.38966319814660473273004777439, 18.696870465660502334225544124887, 19.49281371563627282053152180905, 20.57639023825584847215511455503, 21.05096019749993890985202495322, 21.93644855432106249051074030159

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