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Properties

Label 2-147-7.4-c3-0-10
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: $\chi_{147} (67, \cdot )$
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

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

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