Properties

Label 1-837-837.97-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.814 + 0.580i$
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  + (−0.719 − 0.694i)2-s + (0.0348 + 0.999i)4-s + (0.766 + 0.642i)5-s + (−0.615 + 0.788i)7-s + (0.669 − 0.743i)8-s + (−0.104 − 0.994i)10-s + (−0.374 − 0.927i)11-s + (−0.719 + 0.694i)13-s + (0.990 − 0.139i)14-s + (−0.997 + 0.0697i)16-s + (−0.978 − 0.207i)17-s + (0.913 + 0.406i)19-s + (−0.615 + 0.788i)20-s + (−0.374 + 0.927i)22-s + (−0.615 − 0.788i)23-s + ⋯
L(s)  = 1  + (−0.719 − 0.694i)2-s + (0.0348 + 0.999i)4-s + (0.766 + 0.642i)5-s + (−0.615 + 0.788i)7-s + (0.669 − 0.743i)8-s + (−0.104 − 0.994i)10-s + (−0.374 − 0.927i)11-s + (−0.719 + 0.694i)13-s + (0.990 − 0.139i)14-s + (−0.997 + 0.0697i)16-s + (−0.978 − 0.207i)17-s + (0.913 + 0.406i)19-s + (−0.615 + 0.788i)20-s + (−0.374 + 0.927i)22-s + (−0.615 − 0.788i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.814 + 0.580i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.814 + 0.580i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08468297636 + 0.2645860402i\)
\(L(\frac12)\) \(\approx\) \(0.08468297636 + 0.2645860402i\)
\(L(1)\) \(\approx\) \(0.6064247115 + 0.005581243438i\)
\(L(1)\) \(\approx\) \(0.6064247115 + 0.005581243438i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.719 - 0.694i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (-0.615 + 0.788i)T \)
11 \( 1 + (-0.374 - 0.927i)T \)
13 \( 1 + (-0.719 + 0.694i)T \)
17 \( 1 + (-0.978 - 0.207i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.615 - 0.788i)T \)
29 \( 1 + (-0.719 - 0.694i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.438 + 0.898i)T \)
43 \( 1 + (-0.241 + 0.970i)T \)
47 \( 1 + (0.559 + 0.829i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.241 - 0.970i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (-0.978 + 0.207i)T \)
79 \( 1 + (-0.882 + 0.469i)T \)
83 \( 1 + (-0.719 - 0.694i)T \)
89 \( 1 + (-0.978 + 0.207i)T \)
97 \( 1 + (-0.374 - 0.927i)T \)
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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

−22.04806952876220641159912340394, −20.6602413702111802775977820527, −20.04198960617036281330648863838, −19.63707931941944184443730858974, −18.30429868956988350460885112409, −17.5556454998250862662841786403, −17.26598126590886659803537500165, −16.220585367713322185544449634807, −15.640938713154273353636948579834, −14.69256967432230805065990184037, −13.669176216619520036867948627336, −13.15460350707810695480583845076, −12.12589464373083771811143096407, −10.70764802643149661385867172123, −10.08309889215877574539599531237, −9.45010041291863547797498391591, −8.66947705201126521072415615975, −7.36280234769790014670285859930, −7.10241009671005071557541780667, −5.77387920703361493351362697095, −5.18043807905235546738662542339, −4.12324894992422371434761458179, −2.47967484414022710323510579060, −1.48566154383468889033380408724, −0.15139972789979515952966750491, 1.6520882458500337128219971593, 2.62714839260327865930930388301, 3.10629150483684413144403689494, 4.48560067470653327617437252250, 5.84281877992938144018908142210, 6.573240890349548174218073068169, 7.59408800964439896553639333018, 8.65309652536055767685260941241, 9.46856899283261748198052887445, 9.96148863728943019766695745408, 11.00045261825189213920090753642, 11.66241987490309712350422301727, 12.597725806726839267657486659860, 13.46061510263542663536766179831, 14.16463258957428913590368074023, 15.3924700017747450571254440998, 16.25172765819438168020904444125, 16.96112581836003196495425265086, 17.95823243767177786963435848366, 18.584357673080360831057166099690, 19.029009031936007770029590658877, 19.9787152504714996345346735734, 20.90601379390386903709503349907, 21.76079315526871384227290103595, 22.09128368802605022099570169591

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