Properties

Label 1-837-837.731-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.747 - 0.664i$
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  + (0.615 − 0.788i)2-s + (−0.241 − 0.970i)4-s + (−0.766 − 0.642i)5-s + (0.559 + 0.829i)7-s + (−0.913 − 0.406i)8-s + (−0.978 + 0.207i)10-s + (−0.559 − 0.829i)11-s + (0.990 − 0.139i)13-s + (0.997 + 0.0697i)14-s + (−0.882 + 0.469i)16-s + (0.809 − 0.587i)17-s + (0.669 + 0.743i)19-s + (−0.438 + 0.898i)20-s + (−0.997 − 0.0697i)22-s + (0.997 + 0.0697i)23-s + ⋯
L(s)  = 1  + (0.615 − 0.788i)2-s + (−0.241 − 0.970i)4-s + (−0.766 − 0.642i)5-s + (0.559 + 0.829i)7-s + (−0.913 − 0.406i)8-s + (−0.978 + 0.207i)10-s + (−0.559 − 0.829i)11-s + (0.990 − 0.139i)13-s + (0.997 + 0.0697i)14-s + (−0.882 + 0.469i)16-s + (0.809 − 0.587i)17-s + (0.669 + 0.743i)19-s + (−0.438 + 0.898i)20-s + (−0.997 − 0.0697i)22-s + (0.997 + 0.0697i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.747 - 0.664i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (731, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.747 - 0.664i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9631272924 - 2.534476337i\)
\(L(\frac12)\) \(\approx\) \(0.9631272924 - 2.534476337i\)
\(L(1)\) \(\approx\) \(1.129784289 - 0.8692574735i\)
\(L(1)\) \(\approx\) \(1.129784289 - 0.8692574735i\)

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 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (0.559 + 0.829i)T \)
11 \( 1 + (-0.559 - 0.829i)T \)
13 \( 1 + (0.990 - 0.139i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (0.997 + 0.0697i)T \)
29 \( 1 + (0.615 - 0.788i)T \)
37 \( 1 + T \)
41 \( 1 + (0.882 + 0.469i)T \)
43 \( 1 + (-0.374 - 0.927i)T \)
47 \( 1 + (-0.848 - 0.529i)T \)
53 \( 1 + (0.104 + 0.994i)T \)
59 \( 1 + (0.615 + 0.788i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.913 - 0.406i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.719 - 0.694i)T \)
83 \( 1 + (0.374 + 0.927i)T \)
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

−22.56456292777737579134921218599, −21.51355715395841900191771428288, −20.79256174149080754402428055569, −20.04683969553192683275371051364, −18.942709033442941311431288769875, −17.96327077084105229489379569679, −17.530737770835103138853212870219, −16.287612126611527922639877191684, −15.88657801822616334155765594105, −14.69084265784087697513952587139, −14.58432775730622944088877093948, −13.37096122974939495670815502399, −12.75524567952615894090341458082, −11.59369953392303399718478881620, −11.023876888536919724743408045255, −9.98122270385950865595678243531, −8.62090163392311654852909257213, −7.83343658453543951469581089468, −7.219667219981487140851441528567, −6.499898613063940617401326660003, −5.24419801399718140691025220335, −4.43150927144456417501219737039, −3.6221392472044064617596143468, −2.735671161517961529049105364402, −1.02144481623276286205580955665, 0.60782693285924741326157037100, 1.39714654468693155135139520956, 2.80542148928953802831803323723, 3.50717085962982581519458731513, 4.60948114545850157775067438826, 5.41211707033022879908188267250, 6.03298771205694494242560896683, 7.65728995258805032261872097833, 8.47752744061110322718202966192, 9.203740726552533341124194295803, 10.33030823332349958301504073136, 11.407771910833403651256694556461, 11.66224914562679056537440708271, 12.61975669370395853186705831527, 13.36778492716137303575773401498, 14.22898613083859895635064770461, 15.170638455388462128021153943668, 15.8364182288000763304061779202, 16.57741369922423894798180913814, 18.050795126238101444788926145054, 18.653645670182805947610333318982, 19.23205535340437078452607154638, 20.276809219515013636579603119222, 21.03247836412272983214469684066, 21.25930936255379972604894039090

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