Properties

Label 1-837-837.232-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.374 + 0.927i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.374 + 0.927i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (232, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.374 + 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.925126874 + 4.335020902i\)
\(L(\frac12)\) \(\approx\) \(2.925126874 + 4.335020902i\)
\(L(1)\) \(\approx\) \(2.158843762 + 1.055450832i\)
\(L(1)\) \(\approx\) \(2.158843762 + 1.055450832i\)

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

−21.92698144707355604670590987058, −20.77919837104529523470145329574, −20.3727756026087444168021141863, −19.729229566769762560238077959889, −18.5909912464759309604277421348, −17.21955101673092463449337706594, −17.01419208830476757281726803956, −16.11648845922746071455266150547, −14.849897821237676898599862034139, −14.34203156845253117522740818649, −13.6408003602061672081614059814, −12.82132230281869034213893961045, −12.05259370022228372994838504010, −11.22933891135329643574348450129, −10.23053199525175077995013539270, −9.542834542942789933714794397006, −8.32340562737019519480719999138, −7.196002204082412981038290388805, −6.504974188990274059236470842749, −5.419419787894955117673228902531, −4.63470684233640733289470683021, −4.02132776372869246144232760164, −2.60233691740393027373151518019, −1.73080045440343861343081650814, −0.70749129725814981767054540436, 1.803929675000508075280664969682, 2.13782969704523850007946691513, 3.40729078969367686530595574159, 4.32585119879562376559024721686, 5.484596159212913841169632909998, 6.02116871829599270037192724129, 6.883007104700562690922096071237, 7.84059978195464899111717119950, 8.96435260718954362733318266015, 9.98282006978731562813869246320, 10.93630127912580031206858176230, 11.72773594216056474643667748461, 12.45169892167008741013362209948, 13.3471518300322150942735037042, 14.32314766325423374859822462095, 14.80453311488766319184010511978, 15.27251227847531823391137404866, 16.70182766236308454463195123227, 17.19502105511326425194232333336, 18.08602681830482924239221815292, 19.22558713624780923529544116628, 19.769426933229909551245988552157, 21.099826710870723948248527925553, 21.5191209105180413913709077559, 22.10446832135658857042799735293

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