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Properties

Label 1-837-837.772-r0-0-0
Degree $1$
Conductor $837$
Sign $0.888 - 0.458i$
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  + (0.990 + 0.139i)2-s + (0.961 + 0.275i)4-s + (0.173 − 0.984i)5-s + (−0.997 + 0.0697i)7-s + (0.913 + 0.406i)8-s + (0.309 − 0.951i)10-s + (0.559 + 0.829i)11-s + (−0.615 − 0.788i)13-s + (−0.997 − 0.0697i)14-s + (0.848 + 0.529i)16-s + (0.913 + 0.406i)17-s + (0.309 − 0.951i)19-s + (0.438 − 0.898i)20-s + (0.438 + 0.898i)22-s + (0.559 − 0.829i)23-s + ⋯
L(s)  = 1  + (0.990 + 0.139i)2-s + (0.961 + 0.275i)4-s + (0.173 − 0.984i)5-s + (−0.997 + 0.0697i)7-s + (0.913 + 0.406i)8-s + (0.309 − 0.951i)10-s + (0.559 + 0.829i)11-s + (−0.615 − 0.788i)13-s + (−0.997 − 0.0697i)14-s + (0.848 + 0.529i)16-s + (0.913 + 0.406i)17-s + (0.309 − 0.951i)19-s + (0.438 − 0.898i)20-s + (0.438 + 0.898i)22-s + (0.559 − 0.829i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.888 - 0.458i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (772, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.888 - 0.458i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.698585765 - 0.6551609901i\)
\(L(\frac12)\) \(\approx\) \(2.698585765 - 0.6551609901i\)
\(L(1)\) \(\approx\) \(1.914322568 - 0.1741854478i\)
\(L(1)\) \(\approx\) \(1.914322568 - 0.1741854478i\)

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 \( 1 + (0.173 - 0.984i)T \)
7 \( 1 + (-0.997 + 0.0697i)T \)
11 \( 1 + (0.559 + 0.829i)T \)
13 \( 1 + (-0.615 - 0.788i)T \)
17 \( 1 + (0.913 + 0.406i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.559 - 0.829i)T \)
29 \( 1 + (0.990 + 0.139i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.0348 + 0.999i)T \)
43 \( 1 + (0.990 + 0.139i)T \)
47 \( 1 + (0.0348 - 0.999i)T \)
53 \( 1 + (0.913 + 0.406i)T \)
59 \( 1 + (-0.615 - 0.788i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.913 - 0.406i)T \)
79 \( 1 + (0.961 - 0.275i)T \)
83 \( 1 + (-0.374 - 0.927i)T \)
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

−22.447854009905286507897411075840, −21.470472585492528640727892409890, −20.98419739281704561346291037299, −19.599236092026564359287382143205, −19.24824798241694352909322328552, −18.572418014507813896339377276294, −17.1204238830415017702792880855, −16.439297178495696433015256276493, −15.658071117206751184481663527013, −14.73510955831779735366098284408, −13.92937882348742918853452937319, −13.65324838627737665178877291204, −12.26269139407828399801778412210, −11.8583824208507956167511243298, −10.8062947469809909857044222973, −10.05284880402993184593703386726, −9.27048084860902782154100512194, −7.648216419562293175316602674078, −6.91850375930483928401259414050, −6.17730750627428650504221471535, −5.47338159289022530174012587945, −4.08675568589042283378807944048, −3.292822348623220503979697639177, −2.7020661981657610808045745066, −1.35277070713183402865582749389, 1.00671241221110206129647011982, 2.356539755125990929366208899319, 3.26522918748139169347235538792, 4.35625106177477105809906822647, 5.08278788537042193751750827419, 5.95260657263198919172468701693, 6.86058278429742958023996911567, 7.70915708978151839813048587300, 8.86183556479427441317102086252, 9.78825746258091174419098560621, 10.58861266568802794269821456954, 11.99747221363254262603690808925, 12.44920959690306041361700819527, 12.98947057131014875663531343652, 13.88156127020579237474043305596, 14.86335246461708167974627952638, 15.55433236514416312981013647771, 16.40912921076318724838895473761, 16.9964140876551697499731332461, 17.81451107140446995113188789570, 19.37529864502782877851218310792, 19.84159426822669152124172990805, 20.50900218985823673466284368617, 21.42464852689256285643797314942, 22.11872972722190042124470487066

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