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Properties

Label 2-147-21.20-c5-0-40
Degree $2$
Conductor $147$
Sign $0.788 + 0.614i$
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  + 1.50i·2-s + (−11.3 + 10.6i)3-s + 29.7·4-s + 9.66·5-s + (−15.9 − 17.1i)6-s + 92.7i·8-s + (16.4 − 242. i)9-s + 14.5i·10-s − 628. i·11-s + (−338. + 316. i)12-s − 282. i·13-s + (−110. + 102. i)15-s + 812.·16-s − 1.37e3·17-s + (364. + 24.6i)18-s + 158. i·19-s + ⋯
L(s)  = 1  + 0.265i·2-s + (−0.730 + 0.682i)3-s + 0.929·4-s + 0.172·5-s + (−0.181 − 0.193i)6-s + 0.512i·8-s + (0.0675 − 0.997i)9-s + 0.0459i·10-s − 1.56i·11-s + (−0.679 + 0.634i)12-s − 0.464i·13-s + (−0.126 + 0.118i)15-s + 0.793·16-s − 1.15·17-s + (0.264 + 0.0179i)18-s + 0.100i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.614i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.788 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.788 + 0.614i$
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: \((2,\ 147,\ (\ :5/2),\ 0.788 + 0.614i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.544560375\)
\(L(\frac12)\) \(\approx\) \(1.544560375\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (11.3 - 10.6i)T \)
7 \( 1 \)
good2 \( 1 - 1.50iT - 32T^{2} \)
5 \( 1 - 9.66T + 3.12e3T^{2} \)
11 \( 1 + 628. iT - 1.61e5T^{2} \)
13 \( 1 + 282. iT - 3.71e5T^{2} \)
17 \( 1 + 1.37e3T + 1.41e6T^{2} \)
19 \( 1 - 158. iT - 2.47e6T^{2} \)
23 \( 1 + 2.03e3iT - 6.43e6T^{2} \)
29 \( 1 + 4.90e3iT - 2.05e7T^{2} \)
31 \( 1 + 5.78e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.23e4T + 6.93e7T^{2} \)
41 \( 1 - 3.66e3T + 1.15e8T^{2} \)
43 \( 1 - 1.54e4T + 1.47e8T^{2} \)
47 \( 1 - 2.34e4T + 2.29e8T^{2} \)
53 \( 1 + 1.55e4iT - 4.18e8T^{2} \)
59 \( 1 + 4.72e4T + 7.14e8T^{2} \)
61 \( 1 - 2.23e4iT - 8.44e8T^{2} \)
67 \( 1 - 1.41e3T + 1.35e9T^{2} \)
71 \( 1 - 2.69e4iT - 1.80e9T^{2} \)
73 \( 1 + 4.50e4iT - 2.07e9T^{2} \)
79 \( 1 + 8.67e4T + 3.07e9T^{2} \)
83 \( 1 + 3.03e4T + 3.93e9T^{2} \)
89 \( 1 + 8.16e3T + 5.58e9T^{2} \)
97 \( 1 + 1.54e5iT - 8.58e9T^{2} \)
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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

−11.62773395492993833288558473633, −11.13131914693366010745285356996, −10.25968937957608543237554872247, −8.959352174235409748913079030914, −7.71766974871374802072984465369, −6.16381304290377395463041378680, −5.86387389099834378891634225825, −4.16489951468054597597158481586, −2.63896761205991186332774855424, −0.56451618502044662918034760690, 1.45440723635564222687910774386, 2.38905145130308401462772889393, 4.46578749046755659692340769280, 5.91184970366693343164590942628, 6.95147749164714010864053751802, 7.56412865664181737379297644379, 9.356273439821960094751425766342, 10.53051476075222098318640258891, 11.31706468497981867491546597688, 12.24085477081039599453731442904

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