Properties

Label 1-837-837.184-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: $\chi_{837} (184, \cdot )$
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

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

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