Properties

Label 2-147-21.20-c3-0-6
Degree $2$
Conductor $147$
Sign $0.967 - 0.252i$
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  − 5.37i·2-s + (2.08 + 4.75i)3-s − 20.8·4-s − 2.78·5-s + (25.5 − 11.2i)6-s + 69.3i·8-s + (−18.3 + 19.8i)9-s + 14.9i·10-s + 17.5i·11-s + (−43.5 − 99.4i)12-s + 47.8i·13-s + (−5.81 − 13.2i)15-s + 205.·16-s + 89.4·17-s + (106. + 98.3i)18-s + 42.2i·19-s + ⋯
L(s)  = 1  − 1.90i·2-s + (0.401 + 0.915i)3-s − 2.61·4-s − 0.249·5-s + (1.74 − 0.762i)6-s + 3.06i·8-s + (−0.677 + 0.735i)9-s + 0.473i·10-s + 0.481i·11-s + (−1.04 − 2.39i)12-s + 1.02i·13-s + (−0.100 − 0.228i)15-s + 3.21·16-s + 1.27·17-s + (1.39 + 1.28i)18-s + 0.510i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.967 - 0.252i$
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: \((2,\ 147,\ (\ :3/2),\ 0.967 - 0.252i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.03978 + 0.133682i\)
\(L(\frac12)\) \(\approx\) \(1.03978 + 0.133682i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.08 - 4.75i)T \)
7 \( 1 \)
good2 \( 1 + 5.37iT - 8T^{2} \)
5 \( 1 + 2.78T + 125T^{2} \)
11 \( 1 - 17.5iT - 1.33e3T^{2} \)
13 \( 1 - 47.8iT - 2.19e3T^{2} \)
17 \( 1 - 89.4T + 4.91e3T^{2} \)
19 \( 1 - 42.2iT - 6.85e3T^{2} \)
23 \( 1 - 87.6iT - 1.21e4T^{2} \)
29 \( 1 + 40.8iT - 2.43e4T^{2} \)
31 \( 1 - 95.6iT - 2.97e4T^{2} \)
37 \( 1 + 64.5T + 5.06e4T^{2} \)
41 \( 1 + 403.T + 6.89e4T^{2} \)
43 \( 1 + 230.T + 7.95e4T^{2} \)
47 \( 1 - 365.T + 1.03e5T^{2} \)
53 \( 1 - 598. iT - 1.48e5T^{2} \)
59 \( 1 - 236.T + 2.05e5T^{2} \)
61 \( 1 + 430. iT - 2.26e5T^{2} \)
67 \( 1 + 428.T + 3.00e5T^{2} \)
71 \( 1 + 519. iT - 3.57e5T^{2} \)
73 \( 1 - 764. iT - 3.89e5T^{2} \)
79 \( 1 + 227.T + 4.93e5T^{2} \)
83 \( 1 - 1.13e3T + 5.71e5T^{2} \)
89 \( 1 - 1.13e3T + 7.04e5T^{2} \)
97 \( 1 - 1.48e3iT - 9.12e5T^{2} \)
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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

−12.16621836377149729964770203118, −11.66422664088282890150364561675, −10.51680865112125115269833896201, −9.824061453954500794392963410852, −9.062190066390007072825445317404, −7.915166822429085608785468512319, −5.30996463701369865931849391784, −4.15645250223464672764762844951, −3.30480069504554163239866323165, −1.80203082625476453169755656396, 0.50801823185913200291146157977, 3.49705529385178316064860365928, 5.28625758151875534105241918035, 6.21535672789914949443553057112, 7.29877203803085347484023456951, 8.053516886844936886509387719589, 8.757634121764528227096289585684, 10.05866162467706864916422518152, 11.98401394215032710227926327326, 13.06875312600494565605137501430

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