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Properties

Label 1-147-147.26-r0-0-0
Degree $1$
Conductor $147$
Sign $0.725 + 0.687i$
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  + (0.988 + 0.149i)2-s + (0.955 + 0.294i)4-s + (0.0747 + 0.997i)5-s + (0.900 + 0.433i)8-s + (−0.0747 + 0.997i)10-s + (−0.365 − 0.930i)11-s + (−0.623 + 0.781i)13-s + (0.826 + 0.563i)16-s + (−0.733 − 0.680i)17-s + (0.5 + 0.866i)19-s + (−0.222 + 0.974i)20-s + (−0.222 − 0.974i)22-s + (0.733 − 0.680i)23-s + (−0.988 + 0.149i)25-s + (−0.733 + 0.680i)26-s + ⋯
L(s)  = 1  + (0.988 + 0.149i)2-s + (0.955 + 0.294i)4-s + (0.0747 + 0.997i)5-s + (0.900 + 0.433i)8-s + (−0.0747 + 0.997i)10-s + (−0.365 − 0.930i)11-s + (−0.623 + 0.781i)13-s + (0.826 + 0.563i)16-s + (−0.733 − 0.680i)17-s + (0.5 + 0.866i)19-s + (−0.222 + 0.974i)20-s + (−0.222 − 0.974i)22-s + (0.733 − 0.680i)23-s + (−0.988 + 0.149i)25-s + (−0.733 + 0.680i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.725 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.725 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.725 + 0.687i$
Analytic conductor: \(0.682665\)
Root analytic conductor: \(0.682665\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 147,\ (0:\ ),\ 0.725 + 0.687i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.801097697 + 0.7180074460i\)
\(L(\frac12)\) \(\approx\) \(1.801097697 + 0.7180074460i\)
\(L(1)\) \(\approx\) \(1.715471220 + 0.4294862353i\)
\(L(1)\) \(\approx\) \(1.715471220 + 0.4294862353i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.988 + 0.149i)T \)
5 \( 1 + (0.0747 + 0.997i)T \)
11 \( 1 + (-0.365 - 0.930i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
17 \( 1 + (-0.733 - 0.680i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.733 - 0.680i)T \)
29 \( 1 + (0.222 - 0.974i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.955 - 0.294i)T \)
41 \( 1 + (-0.900 - 0.433i)T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (-0.988 - 0.149i)T \)
53 \( 1 + (-0.955 - 0.294i)T \)
59 \( 1 + (0.0747 - 0.997i)T \)
61 \( 1 + (-0.955 + 0.294i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.988 - 0.149i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (0.365 - 0.930i)T \)
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

−28.48741276113409599352591507174, −27.26490290416708095847334001720, −25.70281727529050107132248292237, −24.92966597393937713944771581715, −24.01230343504858122517114521847, −23.22487926091133145440449589002, −22.073842136714816774889600716296, −21.23417834096443012432442735950, −20.059778885241521936766928753005, −19.77454153439614802686784258086, −17.86635421058662372366106345303, −16.87167570187167894818116048521, −15.632580061195583793341586224025, −14.995394526510944833175997204086, −13.49373979633569898942281310581, −12.81913793537627924578979285185, −11.959680204830051110962908493495, −10.64233834731833482280413691792, −9.46595933191351336874117956284, −7.94133328262238088923000807678, −6.72422103781628555324064256947, −5.19861226953295301087675163756, −4.65930697042267646077864465538, −3.03453215363169031232456568143, −1.568720752715189488389311068773, 2.276636033827608310415515436410, 3.285500888779439637478811709193, 4.64707652627465850219320008913, 6.01740180242262666601743757262, 6.89911273414998469841357770331, 8.04201676083750710400139444838, 9.85377942894357733599937941867, 11.1083348552836617622733587619, 11.77761597036597437805223653892, 13.266134902914200879047367441339, 14.07547624113400251931069980771, 14.92710137048431948692944334703, 15.999223561627860776606495085260, 16.9966549548045382324660220516, 18.46672661008066742545580814162, 19.347865802684286685416769893574, 20.67127001972250805492348691078, 21.609455937930736289814611546140, 22.3982874539916503271658191477, 23.21677218330295991319489579908, 24.3399157592110343591877359085, 25.09510151915797342800521049728, 26.45178485702061253615721429656, 26.842618385445702491062711051544, 28.8239064880093469805552885741

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