Properties

Label 1-837-837.392-r1-0-0
Degree $1$
Conductor $837$
Sign $0.852 - 0.522i$
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  + (−0.559 + 0.829i)2-s + (−0.374 − 0.927i)4-s + (0.939 + 0.342i)5-s + (−0.882 − 0.469i)7-s + (0.978 + 0.207i)8-s + (−0.809 + 0.587i)10-s + (−0.848 + 0.529i)11-s + (−0.997 + 0.0697i)13-s + (0.882 − 0.469i)14-s + (−0.719 + 0.694i)16-s + (0.978 + 0.207i)17-s + (−0.809 + 0.587i)19-s + (−0.0348 − 0.999i)20-s + (0.0348 − 0.999i)22-s + (−0.848 − 0.529i)23-s + ⋯
L(s)  = 1  + (−0.559 + 0.829i)2-s + (−0.374 − 0.927i)4-s + (0.939 + 0.342i)5-s + (−0.882 − 0.469i)7-s + (0.978 + 0.207i)8-s + (−0.809 + 0.587i)10-s + (−0.848 + 0.529i)11-s + (−0.997 + 0.0697i)13-s + (0.882 − 0.469i)14-s + (−0.719 + 0.694i)16-s + (0.978 + 0.207i)17-s + (−0.809 + 0.587i)19-s + (−0.0348 − 0.999i)20-s + (0.0348 − 0.999i)22-s + (−0.848 − 0.529i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.852 - 0.522i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (392, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.852 - 0.522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6389635131 - 0.1800915108i\)
\(L(\frac12)\) \(\approx\) \(0.6389635131 - 0.1800915108i\)
\(L(1)\) \(\approx\) \(0.6462398644 + 0.2355681585i\)
\(L(1)\) \(\approx\) \(0.6462398644 + 0.2355681585i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.559 + 0.829i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (-0.882 - 0.469i)T \)
11 \( 1 + (-0.848 + 0.529i)T \)
13 \( 1 + (-0.997 + 0.0697i)T \)
17 \( 1 + (0.978 + 0.207i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.848 - 0.529i)T \)
29 \( 1 + (-0.559 + 0.829i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.241 - 0.970i)T \)
43 \( 1 + (0.559 - 0.829i)T \)
47 \( 1 + (0.241 + 0.970i)T \)
53 \( 1 + (0.978 + 0.207i)T \)
59 \( 1 + (0.997 - 0.0697i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (-0.978 + 0.207i)T \)
79 \( 1 + (-0.374 + 0.927i)T \)
83 \( 1 + (-0.438 - 0.898i)T \)
89 \( 1 + (-0.669 - 0.743i)T \)
97 \( 1 + (-0.882 - 0.469i)T \)
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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

−21.63944427209485785630929940099, −21.37446124966571221770947331563, −20.42630082951681598362479617219, −19.49826459303135702522671072445, −18.943788736418152506502229462157, −18.08500528153098556650203040822, −17.36141938019034559463490215181, −16.555980251428920699649856315, −15.91630346089752460629877699829, −14.6125008921026938428536971370, −13.52946436151330888761194234532, −12.96402361833471116365815671051, −12.31029034688194032055984108326, −11.355792753105183429488808952726, −10.13589690683768049296149621173, −9.878268337343412746588574679991, −9.00277371216340483072996242954, −8.13422980126446889747023070773, −7.11558862110271338040575832234, −5.87763284597959190514212747960, −5.13073855717589213617181857815, −3.82889853926029792525744089842, −2.65522794306982598883246274420, −2.226798320844214913334232723052, −0.75541919181080836982526984336, 0.23193375232157142186158845011, 1.68602871157952148072624334364, 2.67278285670485850999437115731, 4.10546072967438535552461848735, 5.30835922514566891470607238735, 5.94091010876102706186797822078, 6.93303073118852696347489729043, 7.474430331132637234197396092676, 8.5999250781574306708114756978, 9.657170979585905170828790446772, 10.18812754643450919291699962021, 10.599081386119200935084070923437, 12.36575158019328541483407655484, 13.048322752067625710652921240906, 14.05872183853270245590687389794, 14.582725550487839256467077505123, 15.49965490102197857516085456678, 16.475387586181026507261749417305, 17.01196278435166207572397805252, 17.73219176174497641351567525278, 18.66521894226344136818983353632, 19.136128484756845051771416720852, 20.19659469690620151215159314626, 20.99757003753570296851003742313, 22.20152972507065065967630304168

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