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Properties

Label 1-147-147.2-r1-0-0
Degree $1$
Conductor $147$
Sign $-0.462 - 0.886i$
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  + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (−0.955 + 0.294i)5-s + (0.222 − 0.974i)8-s + (0.955 + 0.294i)10-s + (−0.0747 + 0.997i)11-s + (−0.900 + 0.433i)13-s + (−0.733 + 0.680i)16-s + (0.988 − 0.149i)17-s + (−0.5 − 0.866i)19-s + (−0.623 − 0.781i)20-s + (0.623 − 0.781i)22-s + (0.988 + 0.149i)23-s + (0.826 − 0.563i)25-s + (0.988 + 0.149i)26-s + ⋯
L(s)  = 1  + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (−0.955 + 0.294i)5-s + (0.222 − 0.974i)8-s + (0.955 + 0.294i)10-s + (−0.0747 + 0.997i)11-s + (−0.900 + 0.433i)13-s + (−0.733 + 0.680i)16-s + (0.988 − 0.149i)17-s + (−0.5 − 0.866i)19-s + (−0.623 − 0.781i)20-s + (0.623 − 0.781i)22-s + (0.988 + 0.149i)23-s + (0.826 − 0.563i)25-s + (0.988 + 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.462 - 0.886i$
Analytic conductor: \(15.7973\)
Root analytic conductor: \(15.7973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 147,\ (1:\ ),\ -0.462 - 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2481336788 - 0.4093224569i\)
\(L(\frac12)\) \(\approx\) \(0.2481336788 - 0.4093224569i\)
\(L(1)\) \(\approx\) \(0.5394611045 - 0.1178613348i\)
\(L(1)\) \(\approx\) \(0.5394611045 - 0.1178613348i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.826 - 0.563i)T \)
5 \( 1 + (-0.955 + 0.294i)T \)
11 \( 1 + (-0.0747 + 0.997i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (0.988 - 0.149i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.988 + 0.149i)T \)
29 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.365 - 0.930i)T \)
41 \( 1 + (0.222 - 0.974i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.826 - 0.563i)T \)
53 \( 1 + (-0.365 - 0.930i)T \)
59 \( 1 + (-0.955 - 0.294i)T \)
61 \( 1 + (0.365 - 0.930i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (0.826 - 0.563i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (-0.0747 - 0.997i)T \)
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

−27.8197645931995964581054481241, −27.31102959009997661719104649220, −26.50356488592285611340358827954, −25.29978360677985346268086646482, −24.38768910446409961859024217887, −23.6256631904297382513635759338, −22.65769295645091699809055036513, −21.09163219005246743197619162736, −19.98428926904909242148473504409, −19.130993997060810590974587658401, −18.4329232877486765325554202542, −16.80268285582994538828282855422, −16.53347262275039417750411132278, −15.170528835165563347730780412926, −14.51301284626384754494443434475, −12.83968474226902092818227164867, −11.57963850014497212585292718773, −10.59889024929885193560723104889, −9.34134604988412491806681698521, −8.165061300861944691619977708259, −7.539173953837195377801486916701, −6.074408661427412870656635892703, −4.86527634003383853392882784246, −3.13617205244609264339110925342, −1.09912294441629375893471914599, 0.2841806127059842179658762373, 2.130730339066868395363212048364, 3.45060838188073524698006270883, 4.73411160397564946674927364185, 6.99978314765511872874276907735, 7.5482943335060399514332108891, 8.89553678450759384694624604736, 9.963707793914420720827553078746, 11.071786062766319898336718312403, 12.02223362136142369377532439968, 12.83047122214397416696388138180, 14.63017893284551845326127036481, 15.592703925690058499730431393529, 16.74220888201304400515329844268, 17.66246434893856976755552353183, 18.86492095614402345682528194977, 19.48854818028324261140410970441, 20.425017624270990336141071299978, 21.49289434157983922664856160029, 22.60813818871841123820553452557, 23.59803928795008613999913822275, 24.919051414072317488859602629595, 25.94053635533444434301680639576, 26.79862905522409890842383205608, 27.63950627545264303275368155449

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