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Properties

Label 2-147-7.2-c5-0-7
Degree $2$
Conductor $147$
Sign $0.0725 - 0.997i$
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  + (1.54 − 2.67i)2-s + (−4.5 − 7.79i)3-s + (11.2 + 19.4i)4-s + (−6.89 + 11.9i)5-s − 27.8·6-s + 168.·8-s + (−40.5 + 70.1i)9-s + (21.3 + 36.9i)10-s + (1.83 + 3.17i)11-s + (100. − 174. i)12-s − 780.·13-s + 124.·15-s + (−98.8 + 171. i)16-s + (−25.0 − 43.3i)17-s + (125. + 216. i)18-s + (−531. + 920. i)19-s + ⋯
L(s)  = 1  + (0.273 − 0.473i)2-s + (−0.288 − 0.499i)3-s + (0.350 + 0.607i)4-s + (−0.123 + 0.213i)5-s − 0.315·6-s + 0.929·8-s + (−0.166 + 0.288i)9-s + (0.0674 + 0.116i)10-s + (0.00456 + 0.00791i)11-s + (0.202 − 0.350i)12-s − 1.28·13-s + 0.142·15-s + (−0.0965 + 0.167i)16-s + (−0.0210 − 0.0363i)17-s + (0.0910 + 0.157i)18-s + (−0.337 + 0.585i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.0725 - 0.997i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ 0.0725 - 0.997i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.269773703\)
\(L(\frac12)\) \(\approx\) \(1.269773703\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.5 + 7.79i)T \)
7 \( 1 \)
good2 \( 1 + (-1.54 + 2.67i)T + (-16 - 27.7i)T^{2} \)
5 \( 1 + (6.89 - 11.9i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-1.83 - 3.17i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 780.T + 3.71e5T^{2} \)
17 \( 1 + (25.0 + 43.3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (531. - 920. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (2.05e3 - 3.55e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 1.48e3T + 2.05e7T^{2} \)
31 \( 1 + (2.75e3 + 4.78e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (3.07e3 - 5.32e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 1.07e4T + 1.15e8T^{2} \)
43 \( 1 - 1.76e4T + 1.47e8T^{2} \)
47 \( 1 + (1.47e4 - 2.55e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-9.62e3 - 1.66e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (3.30e3 + 5.73e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.83e4 - 3.18e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (2.34e4 + 4.06e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 4.16e4T + 1.80e9T^{2} \)
73 \( 1 + (1.47e4 + 2.55e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (1.10e4 - 1.91e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 3.89e3T + 3.93e9T^{2} \)
89 \( 1 + (1.02e4 - 1.77e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 1.77e4T + 8.58e9T^{2} \)
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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

−12.32573954201946914776331483726, −11.57007486370243428607757930070, −10.68735350549158935835144808650, −9.424190380004572070381880940177, −7.77167114068840606292140538132, −7.31476707754188228629466641364, −5.85486199494106596798094652277, −4.34820668800424078627033154184, −2.95892122442717097740966620336, −1.70875169586454314612705475455, 0.38197842950764457984267666476, 2.32268495874871532115886534374, 4.34442500230938741191871968561, 5.20066754143186533046198425737, 6.37592711351547245764752747362, 7.39120932916551362209199690972, 8.823380543755808115024585209302, 10.06164554920929649365110187462, 10.71748805495290615541570289604, 11.89837666039436610753212056180

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