Properties

Label 1-837-837.769-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: $\chi_{837} (769, \cdot )$
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

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

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