Properties

Label 1-837-837.358-r1-0-0
Degree $1$
Conductor $837$
Sign $0.556 - 0.830i$
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  + (−0.997 + 0.0697i)2-s + (0.990 − 0.139i)4-s + (0.766 + 0.642i)5-s + (0.0348 − 0.999i)7-s + (−0.978 + 0.207i)8-s + (−0.809 − 0.587i)10-s + (0.882 − 0.469i)11-s + (−0.438 − 0.898i)13-s + (0.0348 + 0.999i)14-s + (0.961 − 0.275i)16-s + (0.978 − 0.207i)17-s + (−0.809 − 0.587i)19-s + (0.848 + 0.529i)20-s + (−0.848 + 0.529i)22-s + (0.882 + 0.469i)23-s + ⋯
L(s)  = 1  + (−0.997 + 0.0697i)2-s + (0.990 − 0.139i)4-s + (0.766 + 0.642i)5-s + (0.0348 − 0.999i)7-s + (−0.978 + 0.207i)8-s + (−0.809 − 0.587i)10-s + (0.882 − 0.469i)11-s + (−0.438 − 0.898i)13-s + (0.0348 + 0.999i)14-s + (0.961 − 0.275i)16-s + (0.978 − 0.207i)17-s + (−0.809 − 0.587i)19-s + (0.848 + 0.529i)20-s + (−0.848 + 0.529i)22-s + (0.882 + 0.469i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.556 - 0.830i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (358, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.556 - 0.830i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.509007201 - 0.8055401656i\)
\(L(\frac12)\) \(\approx\) \(1.509007201 - 0.8055401656i\)
\(L(1)\) \(\approx\) \(0.9097338858 - 0.1163222091i\)
\(L(1)\) \(\approx\) \(0.9097338858 - 0.1163222091i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.997 + 0.0697i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (0.0348 - 0.999i)T \)
11 \( 1 + (0.882 - 0.469i)T \)
13 \( 1 + (-0.438 - 0.898i)T \)
17 \( 1 + (0.978 - 0.207i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.882 + 0.469i)T \)
29 \( 1 + (0.997 - 0.0697i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.719 + 0.694i)T \)
43 \( 1 + (0.997 - 0.0697i)T \)
47 \( 1 + (-0.719 - 0.694i)T \)
53 \( 1 + (0.978 - 0.207i)T \)
59 \( 1 + (0.438 + 0.898i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (0.978 + 0.207i)T \)
79 \( 1 + (-0.990 - 0.139i)T \)
83 \( 1 + (-0.559 - 0.829i)T \)
89 \( 1 + (-0.669 + 0.743i)T \)
97 \( 1 + (0.0348 - 0.999i)T \)
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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

−21.55347057804929324246023098400, −21.33360869696242787114503873089, −20.478770029549093607744066694663, −19.4184272346053815869358175074, −18.9528295983539345424133657965, −18.02561989339751006323229409290, −17.21228136481729503472614342956, −16.72305760411241798807776297886, −15.92098832052391611030439955026, −14.77079968864856953314154144397, −14.29812412911534300325593728433, −12.66581999724526903031114773995, −12.31018863506178543167406198720, −11.46394236316870940957704239634, −10.27198034506991043638986238877, −9.55653376022332734490369903274, −8.91669063250200330225201998535, −8.29224014682846165092827141107, −7.00714362244373537892421432223, −6.232401765452310185510070290050, −5.39039976940874167370171267233, −4.164644214554986036634726607199, −2.6596193383944289210127744369, −1.8903682670174810943325336735, −1.0263037736857558896160005129, 0.61123650091439440824609253276, 1.42055941828317996470967499361, 2.72789949843018702397594126572, 3.46840699129418278826451776533, 5.04940327964169519943223660630, 6.177643789476485220525590887436, 6.831419775350887544892355826273, 7.61232213875918058886759239670, 8.582372587716251348224904297111, 9.62728953166530837632106421507, 10.1912891196373096586981407870, 10.901660806690903261782319794845, 11.69198149995202982021873553446, 12.89791698950419303672286442683, 13.86483713197451820775744700778, 14.66398573833093768738310766363, 15.3490586423928523713824476452, 16.68503587235522943507747737248, 17.03779670113043042105062052403, 17.7157854534134226347697036482, 18.5549529030368090903815697658, 19.47107079009118686491277679242, 19.9181352245405869496427170513, 21.025721591713427037240473554434, 21.554053962548415611863789697853

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