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Properties

Label 2-147-21.17-c1-0-8
Degree $2$
Conductor $147$
Sign $0.195 + 0.980i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.81 − 1.05i)2-s + (−0.562 − 1.63i)3-s + (1.20 − 2.09i)4-s + (0.804 + 1.39i)5-s + (−2.74 − 2.38i)6-s − 0.870i·8-s + (−2.36 + 1.84i)9-s + (2.92 + 1.68i)10-s + (−2.57 − 1.48i)11-s + (−4.10 − 0.800i)12-s + 0.317i·13-s + (1.82 − 2.10i)15-s + (1.49 + 2.59i)16-s + (−1.94 + 3.36i)17-s + (−2.36 + 5.84i)18-s + (4.52 − 2.61i)19-s + ⋯
L(s)  = 1  + (1.28 − 0.742i)2-s + (−0.325 − 0.945i)3-s + (0.603 − 1.04i)4-s + (0.359 + 0.622i)5-s + (−1.12 − 0.975i)6-s − 0.307i·8-s + (−0.788 + 0.614i)9-s + (0.925 + 0.534i)10-s + (−0.775 − 0.447i)11-s + (−1.18 − 0.230i)12-s + 0.0879i·13-s + (0.472 − 0.542i)15-s + (0.374 + 0.649i)16-s + (−0.470 + 0.815i)17-s + (−0.558 + 1.37i)18-s + (1.03 − 0.599i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.195 + 0.980i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ 0.195 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43955 - 1.18124i\)
\(L(\frac12)\) \(\approx\) \(1.43955 - 1.18124i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.562 + 1.63i)T \)
7 \( 1 \)
good2 \( 1 + (-1.81 + 1.05i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-0.804 - 1.39i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.57 + 1.48i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.317iT - 13T^{2} \)
17 \( 1 + (1.94 - 3.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.52 + 2.61i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.06 - 0.615i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.71iT - 29T^{2} \)
31 \( 1 + (2.26 + 1.30i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.53 + 7.85i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.15T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (6.62 + 11.4i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.50 - 0.870i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.21 - 5.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.67 - 3.85i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.24 - 2.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.71iT - 71T^{2} \)
73 \( 1 + (-8.93 - 5.16i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.414 - 0.717i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.60T + 83T^{2} \)
89 \( 1 + (-5.15 - 8.93i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.25iT - 97T^{2} \)
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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

−12.91784125389803445189586677355, −12.03293577808562667172834321092, −11.08941547390450580816297513062, −10.44343153089542612736871762583, −8.546813476573306052687208664603, −7.16490200294336951740502070044, −6.01121594941855782321708608768, −5.10694924666258488658727360831, −3.28843307348142896242758116767, −2.09452973834173561412889798452, 3.27230094310359044999942581245, 4.74865434616114661565966526607, 5.22524990982603196217873290869, 6.36472335894635270375944585320, 7.78102756230059545931386855213, 9.300072515707198963827707911816, 10.15735574775939167163571878000, 11.56810445938057072310077135283, 12.49509006308700286303144055191, 13.46830299915779995746476758392

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