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Properties

Label 1-147-147.20-r0-0-0
Degree $1$
Conductor $147$
Sign $-0.958 - 0.284i$
Analytic cond. $0.682665$
Root an. cond. $0.682665$
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

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

Invariants

Degree: \(1\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.958 - 0.284i$
Analytic conductor: \(0.682665\)
Root analytic conductor: \(0.682665\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 147,\ (0:\ ),\ -0.958 - 0.284i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04367432126 - 0.3006509494i\)
\(L(\frac12)\) \(\approx\) \(0.04367432126 - 0.3006509494i\)
\(L(1)\) \(\approx\) \(0.4518709343 - 0.2276369302i\)
\(L(1)\) \(\approx\) \(0.4518709343 - 0.2276369302i\)

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

−28.30034755949668545075933110451, −27.61238700830908553967494513359, −26.61025042818590006734021995209, −25.80319772279451418483414479711, −24.71628592089883194647284814787, −23.600072410765475470740292312846, −23.3662708482054456388800014178, −21.79185212501363788408196527045, −20.36334945163325749462002183905, −19.45938216620515306713034988812, −18.705415863978141810350843571914, −17.34304116575243292283231450181, −16.684486017041961838047134347, −15.38695162780882806497453307051, −15.00275960707140349169322092158, −13.449294158830631994686818443514, −12.22148954358175703387030654467, −10.93211969368261334992803372947, −9.75437120586664071161521667837, −8.60389625595776787819112574334, −7.68485310812550392075965514797, −6.69237561656621575423014264568, −5.156129907878887175020463981838, −4.138004498721211724893578697027, −1.85416283910367866587079193432, 0.31271717640168216305218463642, 2.51309923861670835530920210887, 3.48466231181894282003140903831, 4.90795887131029998460614797537, 6.92371786302477146710377100897, 7.9513839643350391456143529131, 8.875396206469597001382991578227, 10.40690571558011701619397430136, 11.01593609244555545755517832917, 12.16710227962782653105432618984, 13.08510080219041067732660716169, 14.53620625181892740980906245601, 15.8236513824291308229768048197, 16.720887947074024950165132018348, 18.07828117032802151489201102811, 18.76992616768089123760971688557, 19.734843914268334249058490631816, 20.536801723613770202138249994539, 21.75353503798954183455279107985, 22.595781703710254861339143592093, 23.65720468616919294267631435368, 24.98109227578820736399946753437, 26.131626825985418945935981389789, 27.06804513773965348232842879923, 27.4789916676263385685566714909

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