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Properties

Label 1-837-837.601-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.221 + 0.975i$
Analytic cond. $89.9481$
Root an. cond. $89.9481$
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  + (0.848 + 0.529i)2-s + (0.438 + 0.898i)4-s + (0.766 + 0.642i)5-s + (0.961 − 0.275i)7-s + (−0.104 + 0.994i)8-s + (0.309 + 0.951i)10-s + (0.719 + 0.694i)11-s + (0.882 + 0.469i)13-s + (0.961 + 0.275i)14-s + (−0.615 + 0.788i)16-s + (0.104 − 0.994i)17-s + (0.309 + 0.951i)19-s + (−0.241 + 0.970i)20-s + (0.241 + 0.970i)22-s + (0.719 − 0.694i)23-s + ⋯
L(s)  = 1  + (0.848 + 0.529i)2-s + (0.438 + 0.898i)4-s + (0.766 + 0.642i)5-s + (0.961 − 0.275i)7-s + (−0.104 + 0.994i)8-s + (0.309 + 0.951i)10-s + (0.719 + 0.694i)11-s + (0.882 + 0.469i)13-s + (0.961 + 0.275i)14-s + (−0.615 + 0.788i)16-s + (0.104 − 0.994i)17-s + (0.309 + 0.951i)19-s + (−0.241 + 0.970i)20-s + (0.241 + 0.970i)22-s + (0.719 − 0.694i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.221 + 0.975i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.221 + 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.582428970 + 4.487660616i\)
\(L(\frac12)\) \(\approx\) \(3.582428970 + 4.487660616i\)
\(L(1)\) \(\approx\) \(2.107088077 + 1.290125669i\)
\(L(1)\) \(\approx\) \(2.107088077 + 1.290125669i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.848 + 0.529i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (0.961 - 0.275i)T \)
11 \( 1 + (0.719 + 0.694i)T \)
13 \( 1 + (0.882 + 0.469i)T \)
17 \( 1 + (0.104 - 0.994i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.719 - 0.694i)T \)
29 \( 1 + (-0.848 - 0.529i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.990 - 0.139i)T \)
43 \( 1 + (-0.848 - 0.529i)T \)
47 \( 1 + (0.990 + 0.139i)T \)
53 \( 1 + (0.104 - 0.994i)T \)
59 \( 1 + (-0.882 - 0.469i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.104 + 0.994i)T \)
79 \( 1 + (-0.438 + 0.898i)T \)
83 \( 1 + (-0.0348 + 0.999i)T \)
89 \( 1 + (-0.913 - 0.406i)T \)
97 \( 1 + (0.961 - 0.275i)T \)
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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

−21.57132949951117514429491398607, −21.15714795357957935768354250461, −20.2851889191781526625336052956, −19.62354717837963720952935781256, −18.578310463307175937047165963619, −17.75936111668548861359805320104, −16.92417919818371638409459755630, −15.947408772650587509026372873284, −15.01096339213707738510681309058, −14.326424418558028418342319035255, −13.43964609466756876174049825809, −12.9732723675557124315272689899, −11.922466771822140908457137241303, −11.13554374003728955201456055361, −10.54174578402395305090148739162, −9.21265217721694540910859015740, −8.76362455431535044418290592187, −7.43028493755937960865099452055, −6.041482822670778708665420639044, −5.688929480477164210487842385263, −4.71516918199460357825412341017, −3.80358929218635092823748654350, −2.68978054428660367821556997805, −1.5014576079320968875214180099, −1.02091111321202800919710442856, 1.41988431585571486504685048061, 2.30034670174345678064814417684, 3.481750923343155733015982643930, 4.39538915385338255154065455463, 5.285750754100404677564311554900, 6.21603094181547256268937145870, 6.97024980487483309648346125475, 7.7341890333353349831689689135, 8.822337454661407781725968828924, 9.81488082498553492657092308649, 11.01613920585285664071856103955, 11.52953081908173458276741697074, 12.527757659446403895688106163105, 13.60116961628613571244508470582, 14.12581039147222640645172410633, 14.70050204361182813188999669568, 15.489608491972661559731303478403, 16.70510037398095540334627297083, 17.12479889877451349778491372617, 18.12793477285410159734085644418, 18.65204760304266540227848058316, 20.25572777215058492052532613430, 20.78603388640582876572352190479, 21.36293735826341226804718980456, 22.41447047370007348006360678721

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