Properties

Label 1-837-837.329-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: $\chi_{837} (329, \cdot )$
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

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

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