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Properties

Label 1-837-837.173-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.201 + 0.979i$
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  + (−0.990 − 0.139i)2-s + (0.961 + 0.275i)4-s + (0.939 − 0.342i)5-s + (0.438 − 0.898i)7-s + (−0.913 − 0.406i)8-s + (−0.978 + 0.207i)10-s + (−0.438 + 0.898i)11-s + (−0.374 + 0.927i)13-s + (−0.559 + 0.829i)14-s + (0.848 + 0.529i)16-s + (0.809 − 0.587i)17-s + (0.669 + 0.743i)19-s + (0.997 − 0.0697i)20-s + (0.559 − 0.829i)22-s + (−0.559 + 0.829i)23-s + ⋯
L(s)  = 1  + (−0.990 − 0.139i)2-s + (0.961 + 0.275i)4-s + (0.939 − 0.342i)5-s + (0.438 − 0.898i)7-s + (−0.913 − 0.406i)8-s + (−0.978 + 0.207i)10-s + (−0.438 + 0.898i)11-s + (−0.374 + 0.927i)13-s + (−0.559 + 0.829i)14-s + (0.848 + 0.529i)16-s + (0.809 − 0.587i)17-s + (0.669 + 0.743i)19-s + (0.997 − 0.0697i)20-s + (0.559 − 0.829i)22-s + (−0.559 + 0.829i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.201 + 0.979i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.201 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5830322760 + 0.7150544477i\)
\(L(\frac12)\) \(\approx\) \(0.5830322760 + 0.7150544477i\)
\(L(1)\) \(\approx\) \(0.7813282852 + 0.01485200322i\)
\(L(1)\) \(\approx\) \(0.7813282852 + 0.01485200322i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.990 - 0.139i)T \)
5 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (0.438 - 0.898i)T \)
11 \( 1 + (-0.438 + 0.898i)T \)
13 \( 1 + (-0.374 + 0.927i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (-0.559 + 0.829i)T \)
29 \( 1 + (-0.990 - 0.139i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.848 + 0.529i)T \)
43 \( 1 + (-0.615 + 0.788i)T \)
47 \( 1 + (-0.0348 + 0.999i)T \)
53 \( 1 + (0.104 + 0.994i)T \)
59 \( 1 + (-0.990 + 0.139i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.913 - 0.406i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.241 + 0.970i)T \)
83 \( 1 + (0.615 - 0.788i)T \)
89 \( 1 + (0.104 - 0.994i)T \)
97 \( 1 + (-0.997 + 0.0697i)T \)
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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

−21.72868975463286762342623546364, −20.82356758542596479816493365507, −20.13150256476459352283611680231, −18.97410221783050060208560977637, −18.42546981511621406649736344882, −17.89341938547658854564459951894, −17.0299771474093814456382347783, −16.30045060843650052825794882929, −15.21911876691955056405510754704, −14.76113501251907283939879724928, −13.70589872637730681406582067935, −12.65093800668625754741938424790, −11.70981735891106338598480590235, −10.79973385263245300450716918915, −10.16216495253408535397332423243, −9.3057769216230973203661112326, −8.45748277255254388359655802128, −7.76986749858550979294504087711, −6.640630647962662678836330749719, −5.634154749097762751864713251515, −5.35736274422466507870097204806, −3.20982375068107521681752440709, −2.49955105764224676905406693912, −1.54073235981459041966022796117, −0.26764038085334254422750550781, 1.27316293629324090517264606315, 1.76771304511892402135047297737, 2.96589110066084790648246087494, 4.30403612830607921390158039153, 5.36760676575123564078490517725, 6.40443468669282159223259314179, 7.48386305344663060086702075415, 7.84509135277262248211247005159, 9.29456171135648525813146442027, 9.725818351692866627043201029156, 10.37440121616973173242945186905, 11.455801666289722367805529173439, 12.19507765244875538958343969949, 13.224163633965778783756275241896, 14.123295115820854507155065268177, 14.89569652519713465482947624536, 16.17533373597531622218928694372, 16.72794818158765216010182890699, 17.380984216700852391179919670546, 18.14781815071126196693915883685, 18.73516925115079060139472567412, 20.01083927701245774628207346067, 20.41923490917214401073752729613, 21.13181236016763911964615396493, 21.82106074516674756250207480341

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