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Properties

Label 1-828-828.571-r0-0-0
Degree $1$
Conductor $828$
Sign $0.887 - 0.460i$
Analytic cond. $3.84521$
Root an. cond. $3.84521$
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  + (−0.580 + 0.814i)5-s + (0.235 − 0.971i)7-s + (0.981 − 0.189i)11-s + (0.235 + 0.971i)13-s + (0.142 − 0.989i)17-s + (−0.142 − 0.989i)19-s + (−0.327 − 0.945i)25-s + (−0.786 − 0.618i)29-s + (−0.0475 + 0.998i)31-s + (0.654 + 0.755i)35-s + (−0.415 + 0.909i)37-s + (0.580 − 0.814i)41-s + (0.0475 + 0.998i)43-s + (0.5 − 0.866i)47-s + (−0.888 − 0.458i)49-s + ⋯
L(s)  = 1  + (−0.580 + 0.814i)5-s + (0.235 − 0.971i)7-s + (0.981 − 0.189i)11-s + (0.235 + 0.971i)13-s + (0.142 − 0.989i)17-s + (−0.142 − 0.989i)19-s + (−0.327 − 0.945i)25-s + (−0.786 − 0.618i)29-s + (−0.0475 + 0.998i)31-s + (0.654 + 0.755i)35-s + (−0.415 + 0.909i)37-s + (0.580 − 0.814i)41-s + (0.0475 + 0.998i)43-s + (0.5 − 0.866i)47-s + (−0.888 − 0.458i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(828\)    =    \(2^{2} \cdot 3^{2} \cdot 23\)
Sign: $0.887 - 0.460i$
Analytic conductor: \(3.84521\)
Root analytic conductor: \(3.84521\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{828} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 828,\ (0:\ ),\ 0.887 - 0.460i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.317871147 - 0.3215153280i\)
\(L(\frac12)\) \(\approx\) \(1.317871147 - 0.3215153280i\)
\(L(1)\) \(\approx\) \(1.054195123 - 0.05388215781i\)
\(L(1)\) \(\approx\) \(1.054195123 - 0.05388215781i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 \)
good5 \( 1 + (-0.580 + 0.814i)T \)
7 \( 1 + (0.235 - 0.971i)T \)
11 \( 1 + (0.981 - 0.189i)T \)
13 \( 1 + (0.235 + 0.971i)T \)
17 \( 1 + (0.142 - 0.989i)T \)
19 \( 1 + (-0.142 - 0.989i)T \)
29 \( 1 + (-0.786 - 0.618i)T \)
31 \( 1 + (-0.0475 + 0.998i)T \)
37 \( 1 + (-0.415 + 0.909i)T \)
41 \( 1 + (0.580 - 0.814i)T \)
43 \( 1 + (0.0475 + 0.998i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.959 - 0.281i)T \)
59 \( 1 + (-0.235 - 0.971i)T \)
61 \( 1 + (0.888 - 0.458i)T \)
67 \( 1 + (0.981 + 0.189i)T \)
71 \( 1 + (0.654 - 0.755i)T \)
73 \( 1 + (-0.142 - 0.989i)T \)
79 \( 1 + (0.723 - 0.690i)T \)
83 \( 1 + (0.580 + 0.814i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (0.995 + 0.0950i)T \)
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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

−22.303563097305558408729117761403, −21.36151587756312960045125099283, −20.609603235197314431830412664505, −19.86170889829496869933495555708, −19.11065192853500812843459677956, −18.30125612552201524377819589390, −17.29564559014246933501156514937, −16.681415668617356346308905282946, −15.70476847179312521887060816022, −15.03963413603300571716876900522, −14.35772060807866528022229847920, −12.94989897774359538743553356848, −12.50101300494122804156936441633, −11.75401913545488620729051038612, −10.876546920541815570994285444644, −9.71951550619898248071968155489, −8.82802018714853284655692351232, −8.2575704700785844618602575321, −7.37209824647510598129508783489, −5.94929985879541867942549135343, −5.49320156662691365494713500540, −4.22072227375706957557084928284, −3.55218404987765482857149886670, −2.11095631740091067469580116099, −1.101762487992240061808381414752, 0.755597692034933096658431110920, 2.10631840482603921183278599734, 3.38339406410224285290755309764, 4.04694187276862627253411875514, 4.97178581612394924574641736073, 6.550430074165501501884319906161, 6.90580166894590085690795022118, 7.78299640371505223266993526680, 8.89410263450342265333493096807, 9.750781236019502629121287158236, 10.82915546471146165129424327708, 11.395216117728954218163927705760, 12.04500733394033842463158748655, 13.46120731469889900205433198259, 14.05032474756028738320106788729, 14.69446376738455811061725271032, 15.69863644833649521320079969235, 16.50235232230936171838455027405, 17.27808983516532589718890284813, 18.14776390053336800447822194300, 19.05051087340552434411095620493, 19.63163597343050138474046965655, 20.4125061630307063964517103133, 21.359295852947255214025779513522, 22.23937047359447685518032858171

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