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Properties

Label 2-147-21.20-c5-0-32
Degree $2$
Conductor $147$
Sign $0.791 - 0.610i$
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  + 3.65i·2-s + (−15.5 − 0.882i)3-s + 18.6·4-s + 74.3·5-s + (3.22 − 56.8i)6-s + 185. i·8-s + (241. + 27.4i)9-s + 271. i·10-s − 39.3i·11-s + (−290. − 16.4i)12-s − 589. i·13-s + (−1.15e3 − 65.5i)15-s − 80.0·16-s + 1.23e3·17-s + (−100. + 882. i)18-s − 2.76e3i·19-s + ⋯
L(s)  = 1  + 0.646i·2-s + (−0.998 − 0.0566i)3-s + 0.582·4-s + 1.32·5-s + (0.0365 − 0.645i)6-s + 1.02i·8-s + (0.993 + 0.113i)9-s + 0.859i·10-s − 0.0981i·11-s + (−0.581 − 0.0329i)12-s − 0.967i·13-s + (−1.32 − 0.0752i)15-s − 0.0781·16-s + 1.03·17-s + (−0.0730 + 0.641i)18-s − 1.75i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.610i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.791 - 0.610i$
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: \((2,\ 147,\ (\ :5/2),\ 0.791 - 0.610i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.278875503\)
\(L(\frac12)\) \(\approx\) \(2.278875503\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (15.5 + 0.882i)T \)
7 \( 1 \)
good2 \( 1 - 3.65iT - 32T^{2} \)
5 \( 1 - 74.3T + 3.12e3T^{2} \)
11 \( 1 + 39.3iT - 1.61e5T^{2} \)
13 \( 1 + 589. iT - 3.71e5T^{2} \)
17 \( 1 - 1.23e3T + 1.41e6T^{2} \)
19 \( 1 + 2.76e3iT - 2.47e6T^{2} \)
23 \( 1 + 463. iT - 6.43e6T^{2} \)
29 \( 1 - 5.29e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.84e3iT - 2.86e7T^{2} \)
37 \( 1 - 7.84e3T + 6.93e7T^{2} \)
41 \( 1 - 1.21e4T + 1.15e8T^{2} \)
43 \( 1 - 5.35e3T + 1.47e8T^{2} \)
47 \( 1 - 6.75e3T + 2.29e8T^{2} \)
53 \( 1 - 1.60e4iT - 4.18e8T^{2} \)
59 \( 1 - 7.74e3T + 7.14e8T^{2} \)
61 \( 1 + 5.61e3iT - 8.44e8T^{2} \)
67 \( 1 + 1.28e4T + 1.35e9T^{2} \)
71 \( 1 + 6.06e4iT - 1.80e9T^{2} \)
73 \( 1 - 5.16e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.73e4T + 3.07e9T^{2} \)
83 \( 1 + 8.71e3T + 3.93e9T^{2} \)
89 \( 1 + 1.81e4T + 5.58e9T^{2} \)
97 \( 1 - 1.86e4iT - 8.58e9T^{2} \)
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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

−12.30445895295014706096854522385, −11.04973152693513383150982774417, −10.42037848502579719911401659570, −9.245431609465227630964859622883, −7.65557132266682524858943195806, −6.64722611333583331223942500576, −5.76596741463625709248591818297, −5.08965930467320408387427134155, −2.64495143247003017007130337632, −1.09493869236322679610720811882, 1.18124574455535341243572340895, 2.16117087481376137877233415593, 4.00992348684378122452635006838, 5.73193108965855283236047892931, 6.25929242146237116735292661325, 7.56310190345241475989017526819, 9.695328304707950804565339439422, 9.960219152450794461927677051527, 11.06867632953839198497499053929, 11.97107596825555740924815548369

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