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Properties

Label 1-837-837.605-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.843 + 0.537i$
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  + (−0.990 − 0.139i)2-s + (0.961 + 0.275i)4-s + (−0.766 − 0.642i)5-s + (0.559 + 0.829i)7-s + (−0.913 − 0.406i)8-s + (0.669 + 0.743i)10-s + (0.997 − 0.0697i)11-s + (0.990 − 0.139i)13-s + (−0.438 − 0.898i)14-s + (0.848 + 0.529i)16-s + (0.104 + 0.994i)17-s + (−0.978 + 0.207i)19-s + (−0.559 − 0.829i)20-s + (−0.997 − 0.0697i)22-s + (−0.559 + 0.829i)23-s + ⋯
L(s)  = 1  + (−0.990 − 0.139i)2-s + (0.961 + 0.275i)4-s + (−0.766 − 0.642i)5-s + (0.559 + 0.829i)7-s + (−0.913 − 0.406i)8-s + (0.669 + 0.743i)10-s + (0.997 − 0.0697i)11-s + (0.990 − 0.139i)13-s + (−0.438 − 0.898i)14-s + (0.848 + 0.529i)16-s + (0.104 + 0.994i)17-s + (−0.978 + 0.207i)19-s + (−0.559 − 0.829i)20-s + (−0.997 − 0.0697i)22-s + (−0.559 + 0.829i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.843 + 0.537i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (605, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.843 + 0.537i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1287206151 + 0.4418461298i\)
\(L(\frac12)\) \(\approx\) \(0.1287206151 + 0.4418461298i\)
\(L(1)\) \(\approx\) \(0.6282048163 + 0.04552059626i\)
\(L(1)\) \(\approx\) \(0.6282048163 + 0.04552059626i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.990 - 0.139i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (0.559 + 0.829i)T \)
11 \( 1 + (0.997 - 0.0697i)T \)
13 \( 1 + (0.990 - 0.139i)T \)
17 \( 1 + (0.104 + 0.994i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.559 + 0.829i)T \)
29 \( 1 + (-0.990 - 0.139i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.882 + 0.469i)T \)
43 \( 1 + (-0.374 - 0.927i)T \)
47 \( 1 + (-0.0348 + 0.999i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.374 - 0.927i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.104 + 0.994i)T \)
73 \( 1 + (-0.104 + 0.994i)T \)
79 \( 1 + (-0.719 - 0.694i)T \)
83 \( 1 + (-0.990 - 0.139i)T \)
89 \( 1 + (0.104 - 0.994i)T \)
97 \( 1 + (-0.997 + 0.0697i)T \)
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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

−21.36993454921016042596127190004, −20.54589023837449328245391301958, −19.88593257387643481739538034530, −19.21164413959055723505167406296, −18.32381558466206360011596962327, −17.808000704425467005559018737855, −16.691936968784718679765652946621, −16.29090481456067620582423716084, −15.19136267572303702664415953402, −14.56365462955318447347985831369, −13.77605640662075883355972802528, −12.31571549586169164542435715505, −11.42859781302964500557149269994, −10.95880433754477136613596438991, −10.20389975381807120894585880514, −9.04062168822479664556362368464, −8.34798049212680085712248986916, −7.37300082560473879946983537017, −6.866450416397108282638438301825, −5.93911678013837397234413361354, −4.36058201790115912429745892771, −3.6027561235307980225903362318, −2.31954672831573041247229851879, −1.16501615328123447339799649377, −0.1578041716346261294669194789, 1.26379959402478523769640862371, 1.88142177308705633630870753684, 3.42534972393149429089540553045, 4.16645762440459564303823015071, 5.64137698314422199389532635440, 6.377753655988881922215209969194, 7.611899443478113866118851798929, 8.451584163282285151742788472167, 8.7522203866370302949359193672, 9.74881631992158836731600331737, 10.98528159558132076885671932455, 11.494330783014167555467397167681, 12.262949263671114580859684337572, 13.00632139037879461867259141532, 14.50709281267227624766275327629, 15.31205558363164474474150719240, 15.84938636438372918789299850589, 16.89670943139511641926016482330, 17.34216501655501808426014541127, 18.4219683292888766556323933702, 19.08213670451213326377256907830, 19.72053608817187082309537708791, 20.565600423807752081865184772363, 21.242679296166934018682631929244, 22.00997287239189880591834297150

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