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Properties

Label 1-837-837.491-r0-0-0
Degree $1$
Conductor $837$
Sign $0.185 + 0.982i$
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  + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.173 − 0.984i)5-s + (0.766 + 0.642i)7-s + (0.5 − 0.866i)8-s + 10-s + (−0.939 + 0.342i)11-s + (0.939 + 0.342i)13-s + (−0.766 + 0.642i)14-s + (0.766 + 0.642i)16-s + (−0.5 + 0.866i)17-s + 19-s + (−0.173 + 0.984i)20-s + (−0.173 − 0.984i)22-s + (−0.939 − 0.342i)23-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.173 − 0.984i)5-s + (0.766 + 0.642i)7-s + (0.5 − 0.866i)8-s + 10-s + (−0.939 + 0.342i)11-s + (0.939 + 0.342i)13-s + (−0.766 + 0.642i)14-s + (0.766 + 0.642i)16-s + (−0.5 + 0.866i)17-s + 19-s + (−0.173 + 0.984i)20-s + (−0.173 − 0.984i)22-s + (−0.939 − 0.342i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.185 + 0.982i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.185 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9104992175 + 0.7545543595i\)
\(L(\frac12)\) \(\approx\) \(0.9104992175 + 0.7545543595i\)
\(L(1)\) \(\approx\) \(0.8562881773 + 0.3891145832i\)
\(L(1)\) \(\approx\) \(0.8562881773 + 0.3891145832i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + (-0.939 + 0.342i)T \)
13 \( 1 + (0.939 + 0.342i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 - T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (0.766 + 0.642i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−22.06459475531178737796297351985, −20.88106851165494004753918213912, −20.53536681731470707504639174952, −19.65713803225697018458687661724, −18.66513624676348767812488995484, −18.01228266121899763060098313335, −17.77326029534246308154313891663, −16.34099328002070843475193121932, −15.55964480456902431036454556385, −14.32739997138956832041091283395, −13.80762056944934400058904972699, −13.14474903013043073878898005407, −11.81385345757445408332433256239, −11.25390993954636670192794158892, −10.586204928068482894217873454347, −9.97086299834241255229467848890, −8.74135578358592753553770686820, −7.85915561963582469102628566420, −7.23644405407090254920960434527, −5.78019376159461257741226595506, −4.81325691879157445970291154166, −3.73074217736484347886379409136, −3.01985065148802649222286736295, −2.00950126047809333178839564108, −0.7397822999675356322017001605, 1.0137155248552639463582468162, 2.163404003759168126509141362336, 3.97113125040953178670922919955, 4.64938353233242654125500399848, 5.57660233301491149489251401221, 6.155758749761489961989620601165, 7.645342681972622471757164525069, 8.11025963134790684402614880286, 8.85636075206532555479203029527, 9.64166541366002593216572950319, 10.76504299275201654106033432537, 11.832607830196985108324628171829, 12.774753082469693790258152551168, 13.46423325661718915845747526855, 14.35148658058227363177202925599, 15.35621554688977914100889742000, 15.86320126782027519391760836394, 16.517617030087567508514502832793, 17.66347160345641361041013735499, 17.99968902059418980207588560311, 18.895129404247692994207610932473, 19.90005675828374779401658659701, 20.83983119235135300117816393636, 21.474055244731166109960839513856, 22.46527371311268551424942443930

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