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Properties

Label 2-147-7.2-c3-0-4
Degree $2$
Conductor $147$
Sign $-0.386 - 0.922i$
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  + (−1.13 + 1.97i)2-s + (−1.5 − 2.59i)3-s + (1.41 + 2.44i)4-s + (2.27 − 3.94i)5-s + 6.82·6-s − 24.6·8-s + (−4.5 + 7.79i)9-s + (5.17 + 8.96i)10-s + (20.3 + 35.2i)11-s + (4.23 − 7.33i)12-s + 53.2·13-s − 13.6·15-s + (16.7 − 28.9i)16-s + (−2.27 − 3.94i)17-s + (−10.2 − 17.7i)18-s + (−61.2 + 106. i)19-s + ⋯
L(s)  = 1  + (−0.402 + 0.696i)2-s + (−0.288 − 0.499i)3-s + (0.176 + 0.305i)4-s + (0.203 − 0.352i)5-s + 0.464·6-s − 1.08·8-s + (−0.166 + 0.288i)9-s + (0.163 + 0.283i)10-s + (0.558 + 0.967i)11-s + (0.101 − 0.176i)12-s + 1.13·13-s − 0.234·15-s + (0.261 − 0.452i)16-s + (−0.0324 − 0.0562i)17-s + (−0.134 − 0.232i)18-s + (−0.740 + 1.28i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.386 - 0.922i$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.627763 + 0.943746i\)
\(L(\frac12)\) \(\approx\) \(0.627763 + 0.943746i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 2.59i)T \)
7 \( 1 \)
good2 \( 1 + (1.13 - 1.97i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (-2.27 + 3.94i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-20.3 - 35.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 53.2T + 2.19e3T^{2} \)
17 \( 1 + (2.27 + 3.94i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (61.2 - 106. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (65.6 - 113. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 216.T + 2.43e4T^{2} \)
31 \( 1 + (-125. - 218. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (5.94 - 10.3i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 111.T + 6.89e4T^{2} \)
43 \( 1 - 369.T + 7.95e4T^{2} \)
47 \( 1 + (-131. + 227. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-283. - 491. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (419. + 727. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-242. + 420. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-166. - 288. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 590.T + 3.57e5T^{2} \)
73 \( 1 + (245. + 424. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (60.8 - 105. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 609.T + 5.71e5T^{2} \)
89 \( 1 + (359. - 622. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 637.T + 9.12e5T^{2} \)
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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

−12.71081044043296503003670707448, −12.11026315579936299733671212830, −10.97893431328025090804607300329, −9.532187768809379123092058292825, −8.546333973051242560177809555081, −7.56656589518672363562460124831, −6.57400547805011098065439205085, −5.61399278249297958231216988588, −3.75157181735387410894784269478, −1.66913228257120325637199557086, 0.66922079192933098214161963031, 2.54313998316529335130109602507, 4.02414526520975542504704162540, 5.85245536681459585870757937992, 6.52530525755590830632894217856, 8.541444901186451992417462657827, 9.316304869679924384025661580411, 10.53151240959775896078371737765, 11.02317145889284886776606453718, 11.82088710663114433605106961171

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