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Properties

Label 2-147-21.20-c5-0-11
Degree $2$
Conductor $147$
Sign $-0.606 + 0.795i$
Analytic cond. $23.5764$
Root an. cond. $4.85555$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.23i·2-s + (−13.6 + 7.44i)3-s − 35.8·4-s + 95.6·5-s + (−61.3 − 112. i)6-s − 31.5i·8-s + (132. − 203. i)9-s + 788. i·10-s + 61.6i·11-s + (490. − 266. i)12-s + 701. i·13-s + (−1.31e3 + 712. i)15-s − 886.·16-s − 2.09e3·17-s + (1.67e3 + 1.08e3i)18-s + 766. i·19-s + ⋯
L(s)  = 1  + 1.45i·2-s + (−0.878 + 0.477i)3-s − 1.11·4-s + 1.71·5-s + (−0.695 − 1.27i)6-s − 0.174i·8-s + (0.543 − 0.839i)9-s + 2.49i·10-s + 0.153i·11-s + (0.983 − 0.534i)12-s + 1.15i·13-s + (−1.50 + 0.817i)15-s − 0.865·16-s − 1.76·17-s + (1.22 + 0.791i)18-s + 0.487i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.606 + 0.795i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ -0.606 + 0.795i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.307534831\)
\(L(\frac12)\) \(\approx\) \(1.307534831\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (13.6 - 7.44i)T \)
7 \( 1 \)
good2 \( 1 - 8.23iT - 32T^{2} \)
5 \( 1 - 95.6T + 3.12e3T^{2} \)
11 \( 1 - 61.6iT - 1.61e5T^{2} \)
13 \( 1 - 701. iT - 3.71e5T^{2} \)
17 \( 1 + 2.09e3T + 1.41e6T^{2} \)
19 \( 1 - 766. iT - 2.47e6T^{2} \)
23 \( 1 - 1.16e3iT - 6.43e6T^{2} \)
29 \( 1 + 3.79e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.89e3iT - 2.86e7T^{2} \)
37 \( 1 - 4.07e3T + 6.93e7T^{2} \)
41 \( 1 + 1.78e4T + 1.15e8T^{2} \)
43 \( 1 + 4.91e3T + 1.47e8T^{2} \)
47 \( 1 - 1.36e4T + 2.29e8T^{2} \)
53 \( 1 + 7.16e3iT - 4.18e8T^{2} \)
59 \( 1 - 7.81e3T + 7.14e8T^{2} \)
61 \( 1 + 4.03e3iT - 8.44e8T^{2} \)
67 \( 1 + 3.57e4T + 1.35e9T^{2} \)
71 \( 1 + 6.74e3iT - 1.80e9T^{2} \)
73 \( 1 - 1.69e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.74e4T + 3.07e9T^{2} \)
83 \( 1 - 4.05e4T + 3.93e9T^{2} \)
89 \( 1 + 7.48e3T + 5.58e9T^{2} \)
97 \( 1 - 3.50e4iT - 8.58e9T^{2} \)
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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

−13.23011096167374554781379009508, −11.73002687075144707296031727354, −10.55470030733810146077544285529, −9.497883811990144270726362778385, −8.784486237610092739682400946557, −6.86085409021983988678622660386, −6.40338169525904363783615319006, −5.43844604047364818090959968132, −4.51856003824736790721437455748, −1.89123775691043203846403637851, 0.46997541488758973267669141706, 1.75381228134521014871905337419, 2.64979081338984729923770262516, 4.69080881020977081621641161153, 5.85914730783978257567550162196, 6.85580807504737844167222377529, 8.813373417055501737622319201323, 9.922230168556191708253961663567, 10.61157428589420917198320993072, 11.31784411007455083701329303905

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