Properties

Label 1-837-837.176-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.894 + 0.446i$
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  + (0.615 + 0.788i)2-s + (−0.241 + 0.970i)4-s + (0.939 + 0.342i)5-s + (0.438 + 0.898i)7-s + (−0.913 + 0.406i)8-s + (0.309 + 0.951i)10-s + (−0.997 − 0.0697i)11-s + (0.374 + 0.927i)13-s + (−0.438 + 0.898i)14-s + (−0.882 − 0.469i)16-s + (0.913 − 0.406i)17-s + (0.309 + 0.951i)19-s + (−0.559 + 0.829i)20-s + (−0.559 − 0.829i)22-s + (−0.997 + 0.0697i)23-s + ⋯
L(s)  = 1  + (0.615 + 0.788i)2-s + (−0.241 + 0.970i)4-s + (0.939 + 0.342i)5-s + (0.438 + 0.898i)7-s + (−0.913 + 0.406i)8-s + (0.309 + 0.951i)10-s + (−0.997 − 0.0697i)11-s + (0.374 + 0.927i)13-s + (−0.438 + 0.898i)14-s + (−0.882 − 0.469i)16-s + (0.913 − 0.406i)17-s + (0.309 + 0.951i)19-s + (−0.559 + 0.829i)20-s + (−0.559 − 0.829i)22-s + (−0.997 + 0.0697i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.894 + 0.446i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.894 + 0.446i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5002185831 + 2.124478274i\)
\(L(\frac12)\) \(\approx\) \(0.5002185831 + 2.124478274i\)
\(L(1)\) \(\approx\) \(1.132684556 + 1.109157922i\)
\(L(1)\) \(\approx\) \(1.132684556 + 1.109157922i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.615 + 0.788i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (0.438 + 0.898i)T \)
11 \( 1 + (-0.997 - 0.0697i)T \)
13 \( 1 + (0.374 + 0.927i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.997 + 0.0697i)T \)
29 \( 1 + (-0.615 - 0.788i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.848 - 0.529i)T \)
43 \( 1 + (0.615 + 0.788i)T \)
47 \( 1 + (-0.848 + 0.529i)T \)
53 \( 1 + (0.913 - 0.406i)T \)
59 \( 1 + (0.374 + 0.927i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.913 - 0.406i)T \)
79 \( 1 + (0.241 + 0.970i)T \)
83 \( 1 + (0.990 - 0.139i)T \)
89 \( 1 + (-0.104 - 0.994i)T \)
97 \( 1 + (0.438 + 0.898i)T \)
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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

−21.7150409505480035897319910504, −20.90911302550675076851401232569, −20.39402728676183474443534381181, −19.82519268541058677966329478058, −18.46756622055572911573743069078, −18.03074841437315522550097891309, −17.14642304325810406007514593628, −16.13130465238162271240183815868, −15.10477665234857581048507738194, −14.28773716979040902665279059947, −13.394849706619536126913066300783, −13.14060214886219316132621023250, −12.13288994901879941027605407182, −11.02551341406169804190827307039, −10.30781589838132274561769731382, −9.88691022565666364899990462980, −8.65187787707517600632985001498, −7.66975991389752317942976961302, −6.414250128426784997398160304092, −5.40531622187296468420924684176, −4.96000547104014777529844472179, −3.752895068918726276304874488698, −2.79852438748121877192556982798, −1.73588006076829696065036519040, −0.79146318105923570184210352376, 1.86244749545137343235934919476, 2.68036408830904736853902872361, 3.78680265931278672873432909417, 5.02945874357672892662543844666, 5.71409944064509497022380277402, 6.2442613074721882067796847420, 7.48396868521167147789246840395, 8.17189030913028779517986408235, 9.20196150901530944583801968053, 9.96960684040536764814470670482, 11.243979612391546323264431160, 12.06392064909112620130668372201, 12.9019075876207529793140973758, 13.84683448196954217987920613475, 14.316658272052894300581235060713, 15.12772966915487216827509544481, 16.07912734640242215758871334270, 16.64635000417882668575554587574, 17.75109028703684643922180732822, 18.33661872331856610726375220911, 18.84622569468523983942025003145, 20.61366680133395421807273307323, 21.21368215444293570761811490059, 21.567463946002827011912248851813, 22.56896823516142036568774564381

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