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Properties

Label 2-147-7.4-c3-0-0
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  + (−2 − 3.46i)2-s + (1.5 − 2.59i)3-s + (−3.99 + 6.92i)4-s + (2 + 3.46i)5-s − 12·6-s + (−4.5 − 7.79i)9-s + (7.99 − 13.8i)10-s + (−31 + 53.6i)11-s + (12.0 + 20.7i)12-s − 62·13-s + 12·15-s + (31.9 + 55.4i)16-s + (−42 + 72.7i)17-s + (−18 + 31.1i)18-s + (−50 − 86.6i)19-s − 31.9·20-s + ⋯
L(s)  = 1  + (−0.707 − 1.22i)2-s + (0.288 − 0.499i)3-s + (−0.499 + 0.866i)4-s + (0.178 + 0.309i)5-s − 0.816·6-s + (−0.166 − 0.288i)9-s + (0.252 − 0.438i)10-s + (−0.849 + 1.47i)11-s + (0.288 + 0.499i)12-s − 1.32·13-s + 0.206·15-s + (0.499 + 0.866i)16-s + (−0.599 + 1.03i)17-s + (−0.235 + 0.408i)18-s + (−0.603 − 1.04i)19-s − 0.357·20-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

\(L(2)\) \(\approx\) \(0.148591 + 0.0988404i\)
\(L(\frac12)\) \(\approx\) \(0.148591 + 0.0988404i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2 + 3.46i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-2 - 3.46i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (31 - 53.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 62T + 2.19e3T^{2} \)
17 \( 1 + (42 - 72.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (50 + 86.6i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-21 - 36.3i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 10T + 2.43e4T^{2} \)
31 \( 1 + (-24 + 41.5i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-123 - 213. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 248T + 6.89e4T^{2} \)
43 \( 1 - 68T + 7.95e4T^{2} \)
47 \( 1 + (162 + 280. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (129 - 223. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (60 - 103. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (311 + 538. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (452 - 782. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 678T + 3.57e5T^{2} \)
73 \( 1 + (-321 + 555. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (370 + 640. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 468T + 5.71e5T^{2} \)
89 \( 1 + (100 + 173. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.26e3T + 9.12e5T^{2} \)
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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

−12.61470786232972562683692514314, −11.76835401640349597338102020651, −10.54362934522020778409019562264, −9.948175012576847352764150136184, −8.891285712655997752088833343703, −7.71074881953860271323024444356, −6.57095636602869056213432905851, −4.65404875301625184896231634293, −2.73763070421543801096851456966, −1.94777566006664212690097161507, 0.097314680124257407902454939799, 2.91958294407979352431406125571, 4.92638736247532255902526139887, 5.90090738042782909077235837136, 7.26607796081666511336836652554, 8.232014661377769703748803048471, 9.033886359141865791267846441741, 9.952220939393282829205379476760, 11.12118772807426015873389571861, 12.52511080465949130820948904160

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