Properties

Label 2-147-147.101-c1-0-2
Degree $2$
Conductor $147$
Sign $0.354 - 0.934i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.236 − 0.0928i)2-s + (−0.145 + 1.72i)3-s + (−1.41 − 1.31i)4-s + (1.66 + 1.13i)5-s + (0.194 − 0.394i)6-s + (1.28 + 2.31i)7-s + (0.434 + 0.901i)8-s + (−2.95 − 0.502i)9-s + (−0.288 − 0.422i)10-s + (0.730 + 4.84i)11-s + (2.47 − 2.25i)12-s + (1.15 − 0.920i)13-s + (−0.0892 − 0.666i)14-s + (−2.19 + 2.70i)15-s + (0.270 + 3.60i)16-s + (−4.96 − 1.53i)17-s + ⋯
L(s)  = 1  + (−0.167 − 0.0656i)2-s + (−0.0839 + 0.996i)3-s + (−0.709 − 0.658i)4-s + (0.743 + 0.506i)5-s + (0.0794 − 0.161i)6-s + (0.485 + 0.874i)7-s + (0.153 + 0.318i)8-s + (−0.985 − 0.167i)9-s + (−0.0911 − 0.133i)10-s + (0.220 + 1.46i)11-s + (0.715 − 0.651i)12-s + (0.319 − 0.255i)13-s + (−0.0238 − 0.178i)14-s + (−0.567 + 0.698i)15-s + (0.0675 + 0.901i)16-s + (−1.20 − 0.371i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.354 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.354 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.354 - 0.934i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ 0.354 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.798554 + 0.550997i\)
\(L(\frac12)\) \(\approx\) \(0.798554 + 0.550997i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.145 - 1.72i)T \)
7 \( 1 + (-1.28 - 2.31i)T \)
good2 \( 1 + (0.236 + 0.0928i)T + (1.46 + 1.36i)T^{2} \)
5 \( 1 + (-1.66 - 1.13i)T + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (-0.730 - 4.84i)T + (-10.5 + 3.24i)T^{2} \)
13 \( 1 + (-1.15 + 0.920i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (4.96 + 1.53i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (-6.46 + 3.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.23 + 4.00i)T + (-19.0 + 12.9i)T^{2} \)
29 \( 1 + (-5.37 + 1.22i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-0.635 - 0.366i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.23 + 2.07i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (5.15 - 2.48i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (3.97 + 1.91i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-3.45 + 8.79i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (5.92 - 6.38i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (-2.11 + 1.43i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-0.698 - 0.753i)T + (-4.55 + 60.8i)T^{2} \)
67 \( 1 + (3.22 - 5.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.69 - 1.98i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-10.3 + 4.04i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 + (-4.64 - 8.04i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.39 + 1.74i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (4.40 + 0.664i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 + 13.7iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.57608997319123023992283536453, −12.04583205782888775670451496297, −10.97081554858622394397440793324, −9.991463972838788413011786139313, −9.418112189514009970080223678999, −8.490137977045255509354299112567, −6.54197175927520293579658524715, −5.29643127354009883754778788118, −4.53540916228949727714876677907, −2.40452134283793143622041026030, 1.22800261533115508991678259873, 3.51992873327767013480952392652, 5.18917684495127809566844288093, 6.45103383009636888451441839241, 7.79213785274317813851601064408, 8.501304805813118452903473687451, 9.519820151062278701832785157380, 11.04645356265735441484857888192, 11.96212363426694821667031116086, 13.19676020770376168386757117912

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