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Properties

Label 2-147-21.5-c3-0-31
Degree $2$
Conductor $147$
Sign $-0.444 + 0.895i$
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  + (4.5 − 2.59i)3-s + (−4 − 6.92i)4-s + (13.5 − 23.3i)9-s + (−36 − 20.7i)12-s − 62.3i·13-s + (−31.9 + 55.4i)16-s + (−135 − 77.9i)19-s + (62.5 + 108. i)25-s − 140. i·27-s + (135 − 77.9i)31-s − 216·36-s + (55 − 95.2i)37-s + (−162 − 280. i)39-s + 520·43-s + 332. i·48-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)9-s + (−0.866 − 0.499i)12-s − 1.33i·13-s + (−0.499 + 0.866i)16-s + (−1.63 − 0.941i)19-s + (0.5 + 0.866i)25-s − 1.00i·27-s + (0.782 − 0.451i)31-s − 36-s + (0.244 − 0.423i)37-s + (−0.665 − 1.15i)39-s + 1.84·43-s + 0.999i·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.444 + 0.895i$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ -0.444 + 0.895i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.909492 - 1.46597i\)
\(L(\frac12)\) \(\approx\) \(0.909492 - 1.46597i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.5 + 2.59i)T \)
7 \( 1 \)
good2 \( 1 + (4 + 6.92i)T^{2} \)
5 \( 1 + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 62.3iT - 2.19e3T^{2} \)
17 \( 1 + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (135 + 77.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 + (-135 + 77.9i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-55 + 95.2i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 - 520T + 7.95e4T^{2} \)
47 \( 1 + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-810 - 467. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-440 - 762. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 3.57e5T^{2} \)
73 \( 1 + (-324 + 187. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (442 - 765. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.37e3iT - 9.12e5T^{2} \)
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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

−12.76429235265236511575299091310, −11.07084031480458158138030545482, −10.09833285523190914002744566774, −9.055513163319753054120133996004, −8.231482053800568090433631156479, −6.90874503228987059552471789283, −5.68430571937809836570203800509, −4.21038479497882429479439891498, −2.52046545616122236116475578433, −0.78743987367623522268325718696, 2.34254653060397117794089619841, 3.85000308351963722934647797393, 4.61938350253875917193651749385, 6.65184892002337987773017839813, 7.998988012514976192620873524052, 8.700510170234253245939204825906, 9.605011104500332028826719638495, 10.76276685109471222525457516020, 12.11467778887022233795233403886, 12.97237788051604285921999944468

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