Properties

Label 2-147-21.20-c3-0-12
Degree $2$
Conductor $147$
Sign $0.969 - 0.243i$
Analytic cond. $8.67328$
Root an. cond. $2.94504$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.84i·2-s + (−5.11 − 0.905i)3-s − 15.4·4-s − 17.3·5-s + (4.38 − 24.7i)6-s − 36.2i·8-s + (25.3 + 9.26i)9-s − 83.8i·10-s + 27.8i·11-s + (79.1 + 14.0i)12-s + 16.3i·13-s + (88.5 + 15.6i)15-s + 51.6·16-s + 40.8·17-s + (−44.8 + 122. i)18-s − 68.1i·19-s + ⋯
L(s)  = 1  + 1.71i·2-s + (−0.984 − 0.174i)3-s − 1.93·4-s − 1.54·5-s + (0.298 − 1.68i)6-s − 1.59i·8-s + (0.939 + 0.343i)9-s − 2.65i·10-s + 0.763i·11-s + (1.90 + 0.336i)12-s + 0.347i·13-s + (1.52 + 0.269i)15-s + 0.806·16-s + 0.582·17-s + (−0.587 + 1.60i)18-s − 0.823i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.969 - 0.243i$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ 0.969 - 0.243i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.324731 + 0.0402115i\)
\(L(\frac12)\) \(\approx\) \(0.324731 + 0.0402115i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.11 + 0.905i)T \)
7 \( 1 \)
good2 \( 1 - 4.84iT - 8T^{2} \)
5 \( 1 + 17.3T + 125T^{2} \)
11 \( 1 - 27.8iT - 1.33e3T^{2} \)
13 \( 1 - 16.3iT - 2.19e3T^{2} \)
17 \( 1 - 40.8T + 4.91e3T^{2} \)
19 \( 1 + 68.1iT - 6.85e3T^{2} \)
23 \( 1 + 71.7iT - 1.21e4T^{2} \)
29 \( 1 + 216. iT - 2.43e4T^{2} \)
31 \( 1 - 157. iT - 2.97e4T^{2} \)
37 \( 1 + 348.T + 5.06e4T^{2} \)
41 \( 1 - 153.T + 6.89e4T^{2} \)
43 \( 1 - 427.T + 7.95e4T^{2} \)
47 \( 1 - 16.8T + 1.03e5T^{2} \)
53 \( 1 - 192. iT - 1.48e5T^{2} \)
59 \( 1 + 287.T + 2.05e5T^{2} \)
61 \( 1 + 224. iT - 2.26e5T^{2} \)
67 \( 1 + 172.T + 3.00e5T^{2} \)
71 \( 1 + 1.06e3iT - 3.57e5T^{2} \)
73 \( 1 + 1.07e3iT - 3.89e5T^{2} \)
79 \( 1 - 52.0T + 4.93e5T^{2} \)
83 \( 1 + 1.02e3T + 5.71e5T^{2} \)
89 \( 1 - 668.T + 7.04e5T^{2} \)
97 \( 1 - 1.35e3iT - 9.12e5T^{2} \)
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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

−12.56530378857834371436215615295, −11.85498959728862114970117649286, −10.65618371568083240907330568903, −9.138431566634838925836679798838, −7.84898909367097784565890026540, −7.27702261084749254869988941341, −6.35308698895640075460115715502, −4.96970903023232150018878087500, −4.19283044530656424357051345113, −0.24097789429681944898969593324, 1.02844468943354894560117112620, 3.36638121940613575019725956316, 4.13887354589075250358655474821, 5.49928359482668561238692782801, 7.39391616381435982007047530623, 8.663575247275815886069074487950, 9.999567963628435548175623215561, 10.91375148067236241810799734708, 11.46075334870433909011814604687, 12.24516258852110534126807867504

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