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Properties

Label 1-147-147.29-r1-0-0
Degree $1$
Conductor $147$
Sign $0.284 - 0.958i$
Analytic cond. $15.7973$
Root an. cond. $15.7973$
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  + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.900 − 0.433i)5-s + (0.900 − 0.433i)8-s + (−0.900 − 0.433i)10-s + (−0.623 − 0.781i)11-s + (0.623 + 0.781i)13-s + (−0.900 − 0.433i)16-s + (0.222 + 0.974i)17-s + 19-s + (0.222 + 0.974i)20-s + (−0.222 + 0.974i)22-s + (0.222 − 0.974i)23-s + (0.623 − 0.781i)25-s + (0.222 − 0.974i)26-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.900 − 0.433i)5-s + (0.900 − 0.433i)8-s + (−0.900 − 0.433i)10-s + (−0.623 − 0.781i)11-s + (0.623 + 0.781i)13-s + (−0.900 − 0.433i)16-s + (0.222 + 0.974i)17-s + 19-s + (0.222 + 0.974i)20-s + (−0.222 + 0.974i)22-s + (0.222 − 0.974i)23-s + (0.623 − 0.781i)25-s + (0.222 − 0.974i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.284 - 0.958i$
Analytic conductor: \(15.7973\)
Root analytic conductor: \(15.7973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 147,\ (1:\ ),\ 0.284 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.219652185 - 0.9102500826i\)
\(L(\frac12)\) \(\approx\) \(1.219652185 - 0.9102500826i\)
\(L(1)\) \(\approx\) \(0.9038644626 - 0.3925975818i\)
\(L(1)\) \(\approx\) \(0.9038644626 - 0.3925975818i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.623 - 0.781i)T \)
5 \( 1 + (0.900 - 0.433i)T \)
11 \( 1 + (-0.623 - 0.781i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + (0.222 + 0.974i)T \)
19 \( 1 + T \)
23 \( 1 + (0.222 - 0.974i)T \)
29 \( 1 + (0.222 + 0.974i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.222 - 0.974i)T \)
41 \( 1 + (0.900 - 0.433i)T \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (-0.623 - 0.781i)T \)
53 \( 1 + (0.222 - 0.974i)T \)
59 \( 1 + (0.900 + 0.433i)T \)
61 \( 1 + (-0.222 - 0.974i)T \)
67 \( 1 + T \)
71 \( 1 + (0.222 - 0.974i)T \)
73 \( 1 + (0.623 - 0.781i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.623 + 0.781i)T \)
89 \( 1 + (-0.623 + 0.781i)T \)
97 \( 1 + T \)
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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

−28.08262830171997910932535166892, −26.965381859992417394834327100828, −26.04098101334388010046298317094, −25.30015974884969979644418750326, −24.58624342121550399539338890581, −23.1341974087899829646055250545, −22.61652859475812803142486496603, −21.08222135475075872526476027363, −20.13348558674777273346941609063, −18.73397948902628187128308900482, −17.96904298287511344102490385920, −17.34434919820037332918018487846, −15.96428179469367293256382865720, −15.19366740791898415261468496937, −13.97346434640634940593673981266, −13.224341513874787975185253803482, −11.37948008693342891080458696476, −10.07839500990124649819281957184, −9.58112279470382385285403497269, −8.06800216372280144212055681177, −7.07987991371198341739698932099, −5.886419805729266949931442783881, −4.96465728301693316344549511373, −2.7708118428619932208098835354, −1.16479338495475741322263601014, 0.89643219470382860681351894080, 2.16489111688356202898283693798, 3.56638157548100251959127937470, 5.12201320137962859839804210789, 6.55768573587974566985587625923, 8.20938628091208579115107348071, 8.9900828533654905696972070552, 10.132837149668176424281975970, 11.01292655593038245107100427161, 12.30428074490307810147836921375, 13.28084542624787559603351975471, 14.13378347233710220854755583403, 16.088876275253987799647210404670, 16.77190343722071638499765031666, 17.92327486365335700641082375988, 18.661285705480357297826646897160, 19.77299611160369968380293745012, 20.999377481780721684840440926665, 21.3436567586817284799520328087, 22.47000292279971868012153309259, 23.9157604537244679613663848173, 24.97902124534904725050931857811, 26.10657631168013350997116677049, 26.62649502499334326934005948571, 28.08506031896688452015546448819

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