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Properties

Label 1-147-147.44-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: $\chi_{147} (44, \cdot )$
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

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

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