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Properties

Label 1-837-837.752-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.580 + 0.814i$
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  + (0.241 + 0.970i)2-s + (−0.882 + 0.469i)4-s + (0.939 + 0.342i)5-s + (0.990 − 0.139i)7-s + (−0.669 − 0.743i)8-s + (−0.104 + 0.994i)10-s + (0.615 + 0.788i)11-s + (−0.241 + 0.970i)13-s + (0.374 + 0.927i)14-s + (0.559 − 0.829i)16-s + (0.978 − 0.207i)17-s + (0.913 − 0.406i)19-s + (−0.990 + 0.139i)20-s + (−0.615 + 0.788i)22-s + (−0.990 − 0.139i)23-s + ⋯
L(s)  = 1  + (0.241 + 0.970i)2-s + (−0.882 + 0.469i)4-s + (0.939 + 0.342i)5-s + (0.990 − 0.139i)7-s + (−0.669 − 0.743i)8-s + (−0.104 + 0.994i)10-s + (0.615 + 0.788i)11-s + (−0.241 + 0.970i)13-s + (0.374 + 0.927i)14-s + (0.559 − 0.829i)16-s + (0.978 − 0.207i)17-s + (0.913 − 0.406i)19-s + (−0.990 + 0.139i)20-s + (−0.615 + 0.788i)22-s + (−0.990 − 0.139i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.580 + 0.814i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (752, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.580 + 0.814i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.561578873 + 3.031694434i\)
\(L(\frac12)\) \(\approx\) \(1.561578873 + 3.031694434i\)
\(L(1)\) \(\approx\) \(1.253473942 + 1.001535870i\)
\(L(1)\) \(\approx\) \(1.253473942 + 1.001535870i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.241 + 0.970i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (0.990 - 0.139i)T \)
11 \( 1 + (0.615 + 0.788i)T \)
13 \( 1 + (-0.241 + 0.970i)T \)
17 \( 1 + (0.978 - 0.207i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (-0.990 - 0.139i)T \)
29 \( 1 + (0.241 + 0.970i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.997 - 0.0697i)T \)
43 \( 1 + (0.961 + 0.275i)T \)
47 \( 1 + (-0.438 - 0.898i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.961 + 0.275i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.978 - 0.207i)T \)
73 \( 1 + (-0.978 - 0.207i)T \)
79 \( 1 + (0.848 - 0.529i)T \)
83 \( 1 + (0.241 + 0.970i)T \)
89 \( 1 + (0.978 + 0.207i)T \)
97 \( 1 + (-0.615 - 0.788i)T \)
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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

−21.57515797671220251425934624537, −20.83374898953464827088658786542, −20.37593015376676738215100743302, −19.346258951010577955271833874343, −18.51994514846931307554306770001, −17.67033581511499178057137680381, −17.28242341107339975806593707756, −16.09735451625195439501745408393, −14.80030792175958491502490465488, −14.13353392494704022281070759121, −13.6528630630643828903542758772, −12.55442148969524348208297521448, −11.92112243570389758637781611197, −11.05909301289951972825296487920, −10.12537662021393913244333844963, −9.55572409614997318398746760140, −8.48668093293751614665110179121, −7.83928423394441085830734957720, −5.97817913472002552484311349960, −5.58350197331631890562658161658, −4.64049945396839255318750022481, −3.52916589923465100451000156700, −2.53049054003423442177143225802, −1.4725389404962202374705632270, −0.80506343865253297836407951282, 1.150893554188424800516872160253, 2.21585808228442149811088831942, 3.6165242655234879464659941399, 4.665582438146029714954045464378, 5.33573902837382049052123802428, 6.31433374058657068409392890908, 7.16821338154333699260808274140, 7.80052813960539308089056831371, 9.107977739102544867181318108528, 9.52965696523763855447514925924, 10.61361682264542509491995097520, 11.84253233444335152575773652658, 12.51287034851030089992948044147, 13.80244873294242156804959400950, 14.26225326779539877085180086138, 14.632023724460165312608353587564, 15.808627315072904293858390108202, 16.70436937406539587055132195433, 17.38578147760470816812072868539, 18.02037399548997570359816643518, 18.594485401448126044720204263502, 19.88600759387005651297215551958, 20.95282874953348430668730963771, 21.55830995039542142438701815564, 22.25651689654576525495416392373

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