Properties

Label 1-147-147.137-r1-0-0
Degree $1$
Conductor $147$
Sign $0.243 - 0.969i$
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  + (−0.955 − 0.294i)2-s + (0.826 + 0.563i)4-s + (0.988 − 0.149i)5-s + (−0.623 − 0.781i)8-s + (−0.988 − 0.149i)10-s + (0.733 − 0.680i)11-s + (−0.222 − 0.974i)13-s + (0.365 + 0.930i)16-s + (−0.0747 − 0.997i)17-s + (−0.5 + 0.866i)19-s + (0.900 + 0.433i)20-s + (−0.900 + 0.433i)22-s + (−0.0747 + 0.997i)23-s + (0.955 − 0.294i)25-s + (−0.0747 + 0.997i)26-s + ⋯
L(s)  = 1  + (−0.955 − 0.294i)2-s + (0.826 + 0.563i)4-s + (0.988 − 0.149i)5-s + (−0.623 − 0.781i)8-s + (−0.988 − 0.149i)10-s + (0.733 − 0.680i)11-s + (−0.222 − 0.974i)13-s + (0.365 + 0.930i)16-s + (−0.0747 − 0.997i)17-s + (−0.5 + 0.866i)19-s + (0.900 + 0.433i)20-s + (−0.900 + 0.433i)22-s + (−0.0747 + 0.997i)23-s + (0.955 − 0.294i)25-s + (−0.0747 + 0.997i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.243 - 0.969i$
Analytic conductor: \(15.7973\)
Root analytic conductor: \(15.7973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 147,\ (1:\ ),\ 0.243 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.051062345 - 0.8199747729i\)
\(L(\frac12)\) \(\approx\) \(1.051062345 - 0.8199747729i\)
\(L(1)\) \(\approx\) \(0.8526311799 - 0.2639495770i\)
\(L(1)\) \(\approx\) \(0.8526311799 - 0.2639495770i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.955 - 0.294i)T \)
5 \( 1 + (0.988 - 0.149i)T \)
11 \( 1 + (0.733 - 0.680i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (-0.0747 - 0.997i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.0747 + 0.997i)T \)
29 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.826 - 0.563i)T \)
41 \( 1 + (-0.623 - 0.781i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (-0.955 - 0.294i)T \)
53 \( 1 + (-0.826 - 0.563i)T \)
59 \( 1 + (0.988 + 0.149i)T \)
61 \( 1 + (0.826 - 0.563i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (0.955 - 0.294i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (0.733 + 0.680i)T \)
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

−28.34118358858319968553453868772, −27.0252064024115415457859798101, −26.10800402544657070037656384977, −25.409547180830587068211475135082, −24.506682808692842736645666052973, −23.50429371993674295430226320625, −22.02346542782454471939491835718, −21.15348351424289233839559905814, −19.955839725557711867162799957951, −19.08439549848989455306546289991, −17.91802142963800510343625150447, −17.245445038713599739058216440429, −16.38295167918064952702201822390, −14.93812324071449385463767987913, −14.2552871847527918403192592664, −12.73365716017462529620358554194, −11.39674418871545732425992778447, −10.25529103202656493652372677840, −9.406707305394609470295259290265, −8.45581523974870154086764870567, −6.82593328574102083387538766575, −6.29441646099190003968884422359, −4.65690106539982931004680089352, −2.480544952957142468116320412760, −1.39976812497220240151777583506, 0.74054913408618869703803120131, 2.11014056436689785117807937563, 3.45336216316290617515505980408, 5.50233763333004120827075603076, 6.60955496504032508564285942150, 7.96275431850179195531294534803, 9.107891161470619655337779792635, 9.9276353620852368510972774981, 10.9827482953430284952416097073, 12.16674157200438583423129894828, 13.30021958849838380432594965839, 14.53473368681223609304555074922, 15.95190412823166289051095911988, 16.92232304963314557686537880008, 17.70304501247134924442704120534, 18.616478973150124421062058408378, 19.74368310409177107346526909438, 20.66769428780160403735694338174, 21.57932277851227184501214925051, 22.4944199671103818633347575972, 24.201963489547489485076075836404, 25.18340820725337702579069484315, 25.57841870505013379600724487962, 27.06568227934701003720520290906, 27.51583675388052480219786994519

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