Properties

Label 1-837-837.661-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.0472 - 0.998i$
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  + (−0.615 − 0.788i)2-s + (−0.241 + 0.970i)4-s + (−0.939 − 0.342i)5-s + (0.438 + 0.898i)7-s + (0.913 − 0.406i)8-s + (0.309 + 0.951i)10-s + (−0.997 − 0.0697i)11-s + (−0.374 − 0.927i)13-s + (0.438 − 0.898i)14-s + (−0.882 − 0.469i)16-s + (0.913 − 0.406i)17-s + (0.309 + 0.951i)19-s + (0.559 − 0.829i)20-s + (0.559 + 0.829i)22-s + (−0.997 + 0.0697i)23-s + ⋯
L(s)  = 1  + (−0.615 − 0.788i)2-s + (−0.241 + 0.970i)4-s + (−0.939 − 0.342i)5-s + (0.438 + 0.898i)7-s + (0.913 − 0.406i)8-s + (0.309 + 0.951i)10-s + (−0.997 − 0.0697i)11-s + (−0.374 − 0.927i)13-s + (0.438 − 0.898i)14-s + (−0.882 − 0.469i)16-s + (0.913 − 0.406i)17-s + (0.309 + 0.951i)19-s + (0.559 − 0.829i)20-s + (0.559 + 0.829i)22-s + (−0.997 + 0.0697i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0472 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0472 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.0472 - 0.998i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (661, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.0472 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4458531083 - 0.4674363830i\)
\(L(\frac12)\) \(\approx\) \(0.4458531083 - 0.4674363830i\)
\(L(1)\) \(\approx\) \(0.5883972433 - 0.2225875620i\)
\(L(1)\) \(\approx\) \(0.5883972433 - 0.2225875620i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.615 - 0.788i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (0.438 + 0.898i)T \)
11 \( 1 + (-0.997 - 0.0697i)T \)
13 \( 1 + (-0.374 - 0.927i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.997 + 0.0697i)T \)
29 \( 1 + (-0.615 - 0.788i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.848 + 0.529i)T \)
43 \( 1 + (-0.615 - 0.788i)T \)
47 \( 1 + (0.848 - 0.529i)T \)
53 \( 1 + (0.913 - 0.406i)T \)
59 \( 1 + (-0.374 - 0.927i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.913 + 0.406i)T \)
79 \( 1 + (-0.241 - 0.970i)T \)
83 \( 1 + (0.990 - 0.139i)T \)
89 \( 1 + (-0.104 - 0.994i)T \)
97 \( 1 + (0.438 + 0.898i)T \)
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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

−22.684184449726194654965630857617, −21.58205650674785323395879291630, −20.47948439223609523765548308935, −19.76849447700656714832283520014, −19.07318562676379379880359275888, −18.28740576179933890537673739070, −17.57030218921890185707893498320, −16.56651545547221267109520253275, −16.08613300865900177064917297425, −15.19640812890009492758014264904, −14.40143332763540769528692156147, −13.82249646534501413017677174543, −12.61211787017034547077480826197, −11.44948277200518032598476082660, −10.74512749847357627174879743529, −10.04968710135270070356016125669, −8.959680570907097713242176330494, −7.9465407416034299976554612104, −7.45422276715188469113130960537, −6.81636440171946098017674009700, −5.5291161852058615367241755115, −4.60984625097073518201332460202, −3.77271382993527873013739882276, −2.26516496860841612694413371802, −0.88764194939455503370884780176, 0.48875147729762876864722143009, 1.877007156410623493214081082973, 2.90111560503881037241975006827, 3.716936236104065769367457862375, 4.91255724602041177123167012740, 5.6722244361417982057989332629, 7.470946002864766093877941094677, 7.97158012610190651167904509688, 8.52216983212218063168967683987, 9.708101053653969735336095598840, 10.38409124983324067383393953923, 11.432804656561709959426975833610, 12.0927837126581637789881369029, 12.56533304153172096899139218024, 13.594788944247050713510338633, 14.834161491120377404016100381587, 15.66801225352648586746567973594, 16.33149204530937793168163129930, 17.27555739351748970231012436755, 18.33099142859047620637469610849, 18.62887815068867509365295302266, 19.50469886675466742480175144046, 20.5272455964148421011477176492, 20.75731060831222767879252371643, 21.79416256100292419758620973309

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