Properties

Label 1-837-837.493-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  + (−0.374 − 0.927i)2-s + (−0.719 + 0.694i)4-s + (0.766 + 0.642i)5-s + (0.559 + 0.829i)7-s + (0.913 + 0.406i)8-s + (0.309 − 0.951i)10-s + (0.438 − 0.898i)11-s + (0.990 − 0.139i)13-s + (0.559 − 0.829i)14-s + (0.0348 − 0.999i)16-s + (0.913 + 0.406i)17-s + (0.309 − 0.951i)19-s + (−0.997 + 0.0697i)20-s + (−0.997 − 0.0697i)22-s + (0.438 + 0.898i)23-s + ⋯
L(s)  = 1  + (−0.374 − 0.927i)2-s + (−0.719 + 0.694i)4-s + (0.766 + 0.642i)5-s + (0.559 + 0.829i)7-s + (0.913 + 0.406i)8-s + (0.309 − 0.951i)10-s + (0.438 − 0.898i)11-s + (0.990 − 0.139i)13-s + (0.559 − 0.829i)14-s + (0.0348 − 0.999i)16-s + (0.913 + 0.406i)17-s + (0.309 − 0.951i)19-s + (−0.997 + 0.0697i)20-s + (−0.997 − 0.0697i)22-s + (0.438 + 0.898i)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: $\chi_{837} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.888 - 0.458i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.536693549 - 0.3730775134i\)
\(L(\frac12)\) \(\approx\) \(1.536693549 - 0.3730775134i\)
\(L(1)\) \(\approx\) \(1.104540051 - 0.2615301570i\)
\(L(1)\) \(\approx\) \(1.104540051 - 0.2615301570i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.374 - 0.927i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (0.559 + 0.829i)T \)
11 \( 1 + (0.438 - 0.898i)T \)
13 \( 1 + (0.990 - 0.139i)T \)
17 \( 1 + (0.913 + 0.406i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.438 + 0.898i)T \)
29 \( 1 + (-0.374 - 0.927i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.882 - 0.469i)T \)
43 \( 1 + (-0.374 - 0.927i)T \)
47 \( 1 + (-0.882 + 0.469i)T \)
53 \( 1 + (0.913 + 0.406i)T \)
59 \( 1 + (0.990 - 0.139i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.913 - 0.406i)T \)
79 \( 1 + (-0.719 - 0.694i)T \)
83 \( 1 + (-0.615 + 0.788i)T \)
89 \( 1 + (-0.104 + 0.994i)T \)
97 \( 1 + (0.559 + 0.829i)T \)
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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

−22.53819525314591283983098509327, −21.256589536675275432929024875, −20.54244618236804116261291051099, −19.95590081206060185735651057767, −18.58177440591024832126145269875, −18.137353367552899410720353036545, −17.20751823641079380915728063590, −16.670338499493173055440698569877, −16.09707158258247334589026117749, −14.742555722522156697708468036379, −14.333326241788711793309108795997, −13.474732070467266686780905945753, −12.735110428687148833105095720061, −11.52343983156659294306331417103, −10.22314017463026341621653358846, −9.91120909774400732923628242696, −8.77778783834691518929527335379, −8.15561491478644108677558705901, −7.1184031714306034004079821158, −6.38468803021187917160949573253, −5.30714659381569583850686725363, −4.68722413169136477759646236153, −3.64477346414754884613727238771, −1.64965850899949048711273655650, −1.128074821253400632048852229017, 1.14547903353365968102012496322, 2.03391512775467615524514861369, 3.05354766556346407587145001408, 3.75913750681894280124439668779, 5.28133065394994790067809236164, 5.87271732961462340471054349364, 7.16202730465950946374635329208, 8.30392460845898261689267415373, 8.95831521140896523086172895536, 9.74249561167226092990178158374, 10.758862867642735317010539530939, 11.3238302341535475673426169799, 12.03444299690843426941465246122, 13.26446794379236490288814630988, 13.732875401429060463675206545347, 14.64040439221960262801046820426, 15.62132331123165050881742362254, 16.80213512834712790288883714052, 17.553094013097229351362693099678, 18.21117265968101477322680982074, 18.904703353361352434712544661183, 19.455307475420178938483135707599, 20.7794336687630273262848864275, 21.24284432613842577148701704010, 21.84220466695857598990506495151

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