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Properties

Label 1-147-147.104-r0-0-0
Degree $1$
Conductor $147$
Sign $-0.695 + 0.718i$
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  + (0.222 + 0.974i)2-s + (−0.900 + 0.433i)4-s + (0.623 + 0.781i)5-s + (−0.623 − 0.781i)8-s + (−0.623 + 0.781i)10-s + (0.222 + 0.974i)11-s + (0.222 + 0.974i)13-s + (0.623 − 0.781i)16-s + (−0.900 − 0.433i)17-s − 19-s + (−0.900 − 0.433i)20-s + (−0.900 + 0.433i)22-s + (0.900 − 0.433i)23-s + (−0.222 + 0.974i)25-s + (−0.900 + 0.433i)26-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)2-s + (−0.900 + 0.433i)4-s + (0.623 + 0.781i)5-s + (−0.623 − 0.781i)8-s + (−0.623 + 0.781i)10-s + (0.222 + 0.974i)11-s + (0.222 + 0.974i)13-s + (0.623 − 0.781i)16-s + (−0.900 − 0.433i)17-s − 19-s + (−0.900 − 0.433i)20-s + (−0.900 + 0.433i)22-s + (0.900 − 0.433i)23-s + (−0.222 + 0.974i)25-s + (−0.900 + 0.433i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.695 + 0.718i$
Analytic conductor: \(0.682665\)
Root analytic conductor: \(0.682665\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 147,\ (0:\ ),\ -0.695 + 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4468563201 + 1.054816106i\)
\(L(\frac12)\) \(\approx\) \(0.4468563201 + 1.054816106i\)
\(L(1)\) \(\approx\) \(0.8191297092 + 0.7609787947i\)
\(L(1)\) \(\approx\) \(0.8191297092 + 0.7609787947i\)

Euler product

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

−27.87427878507422186510521676434, −27.26112927693634631080259530334, −25.91503401483571031741168717994, −24.69897119903311294095301829593, −23.82794366914505529358909520230, −22.66410107296341205571157640662, −21.61674344335203867541564976192, −20.99593018889495543993978643798, −19.93250903855735427043556496007, −19.10902450087057770318437170051, −17.7930014093210517314717311302, −17.08485757433238305596179603702, −15.60056716110673252440396221617, −14.25215162405412952698331235482, −13.21409167043958978599074858463, −12.65168061312307557794665234058, −11.235543601128927923267169628964, −10.368198488571250141512754542242, −9.06837101565436813793843680998, −8.38157532359489368812225382815, −6.16302549667013667328128689250, −5.175559623294655121720022148977, −3.89196334180534365518305598788, −2.45912640243477262677460225187, −1.003897211434890367776474994987, 2.23614327574732325482062679479, 3.94433569459842442160371935086, 5.13935576852966342939878668488, 6.650598028860241482513337133054, 6.97987046121215894033470294912, 8.69528306836988685425902272876, 9.6132630041594892888763650385, 10.932439116578897203180716390339, 12.466921896702879795294080494680, 13.55065483292354053286206008603, 14.489880646251667804726272314468, 15.22370009806365051076100144566, 16.48288800989539837871739319653, 17.50708138714554956133774253287, 18.22518928832645353577649826660, 19.31245079866558011757209304381, 20.96478873920803122740507060415, 21.87437777138214544011019469512, 22.758968592321840058904831540, 23.57291516432309258799282963315, 24.80951415815979017141322834764, 25.58309439674923743394170962137, 26.32747692494693088108693017570, 27.24830440963292649833477345575, 28.462938094034748280275077898040

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