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Properties

Label 1-147-147.89-r0-0-0
Degree $1$
Conductor $147$
Sign $0.886 + 0.462i$
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  + (−0.0747 − 0.997i)2-s + (−0.988 + 0.149i)4-s + (−0.733 + 0.680i)5-s + (0.222 + 0.974i)8-s + (0.733 + 0.680i)10-s + (−0.826 + 0.563i)11-s + (0.900 + 0.433i)13-s + (0.955 − 0.294i)16-s + (0.365 + 0.930i)17-s + (0.5 + 0.866i)19-s + (0.623 − 0.781i)20-s + (0.623 + 0.781i)22-s + (−0.365 + 0.930i)23-s + (0.0747 − 0.997i)25-s + (0.365 − 0.930i)26-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)2-s + (−0.988 + 0.149i)4-s + (−0.733 + 0.680i)5-s + (0.222 + 0.974i)8-s + (0.733 + 0.680i)10-s + (−0.826 + 0.563i)11-s + (0.900 + 0.433i)13-s + (0.955 − 0.294i)16-s + (0.365 + 0.930i)17-s + (0.5 + 0.866i)19-s + (0.623 − 0.781i)20-s + (0.623 + 0.781i)22-s + (−0.365 + 0.930i)23-s + (0.0747 − 0.997i)25-s + (0.365 − 0.930i)26-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.886 + 0.462i$
Analytic conductor: \(0.682665\)
Root analytic conductor: \(0.682665\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 147,\ (0:\ ),\ 0.886 + 0.462i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6797984017 + 0.1666664401i\)
\(L(\frac12)\) \(\approx\) \(0.6797984017 + 0.1666664401i\)
\(L(1)\) \(\approx\) \(0.7662262566 - 0.09318091594i\)
\(L(1)\) \(\approx\) \(0.7662262566 - 0.09318091594i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.0747 - 0.997i)T \)
5 \( 1 + (-0.733 + 0.680i)T \)
11 \( 1 + (-0.826 + 0.563i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
17 \( 1 + (0.365 + 0.930i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.365 + 0.930i)T \)
29 \( 1 + (-0.623 + 0.781i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.988 - 0.149i)T \)
41 \( 1 + (-0.222 - 0.974i)T \)
43 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (0.0747 + 0.997i)T \)
53 \( 1 + (0.988 - 0.149i)T \)
59 \( 1 + (-0.733 - 0.680i)T \)
61 \( 1 + (0.988 + 0.149i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (-0.0747 + 0.997i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (0.826 + 0.563i)T \)
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

−27.993941046547068758048275613113, −26.9415672369031038954448493588, −26.213326425088130242339937289255, −24.99389470102230099509607351225, −24.23245582567364113266218155969, −23.365254540881718957389420415471, −22.60215325138239205153984186963, −21.18425117619446323669543646720, −20.12666243806704452469726491409, −18.83760527272949524907775866320, −18.075145584901558146375735863620, −16.75474882819394154737516260134, −15.93857797639988439671098012694, −15.34347434936073348672593017167, −13.83173320621462955281157649153, −13.08021605393524091049001878338, −11.7724204316750442055761807758, −10.34060512690083272508136468175, −8.89691766793706601073954548716, −8.175462161975900332519186130919, −7.118748269290827372821724897025, −5.65256196278466901888499633225, −4.72444805883112983639280121647, −3.34418531938874769572821939137, −0.66591864309968318007130576661, 1.73333817853876823967958962538, 3.24965799763036231860742633815, 4.12995008656560834267192244585, 5.70575775039067920116019334975, 7.46728342766447183345027514950, 8.41826389634721695698576066791, 9.88490533502741143585310790706, 10.7636367162218035628243575653, 11.704409805331431061779953501113, 12.70385624304852973340673298458, 13.86959701328812141135821184927, 14.95172376736383865316003324646, 16.127439087023744305309424676233, 17.60513134107403884754706922606, 18.56642818925936660346399571331, 19.19423873419514783988155854641, 20.358876607610490128863583830689, 21.17136257568070748428939304712, 22.31926566918966981172966056678, 23.17928113556375610121819130439, 23.86999885421660996594386601760, 25.86132547129084411537328360969, 26.29953648234468826534558790823, 27.56955700206783922789633878478, 28.15240619546929079231085338424

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