Properties

Label 1-837-837.608-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: $\chi_{837} (608, \cdot )$
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

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

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