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Properties

Label 1-147-147.125-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: $\chi_{147} (125, \cdot )$
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

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

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