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Properties

Label 2-147-49.43-c1-0-5
Degree $2$
Conductor $147$
Sign $0.884 - 0.465i$
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  + (0.159 − 0.200i)2-s + (0.900 + 0.433i)3-s + (0.430 + 1.88i)4-s + (−1.50 − 0.726i)5-s + (0.231 − 0.111i)6-s + (2.63 − 0.216i)7-s + (0.909 + 0.437i)8-s + (0.623 + 0.781i)9-s + (−0.387 + 0.186i)10-s + (−2.04 + 2.56i)11-s + (−0.430 + 1.88i)12-s + (1.69 − 2.12i)13-s + (0.378 − 0.563i)14-s + (−1.04 − 1.30i)15-s + (−3.25 + 1.56i)16-s + (0.947 − 4.15i)17-s + ⋯
L(s)  = 1  + (0.113 − 0.141i)2-s + (0.520 + 0.250i)3-s + (0.215 + 0.942i)4-s + (−0.675 − 0.325i)5-s + (0.0943 − 0.0454i)6-s + (0.996 − 0.0819i)7-s + (0.321 + 0.154i)8-s + (0.207 + 0.260i)9-s + (−0.122 + 0.0589i)10-s + (−0.616 + 0.772i)11-s + (−0.124 + 0.544i)12-s + (0.469 − 0.589i)13-s + (0.101 − 0.150i)14-s + (−0.269 − 0.338i)15-s + (−0.813 + 0.391i)16-s + (0.229 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.884 - 0.465i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ 0.884 - 0.465i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32432 + 0.327399i\)
\(L(\frac12)\) \(\approx\) \(1.32432 + 0.327399i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.900 - 0.433i)T \)
7 \( 1 + (-2.63 + 0.216i)T \)
good2 \( 1 + (-0.159 + 0.200i)T + (-0.445 - 1.94i)T^{2} \)
5 \( 1 + (1.50 + 0.726i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (2.04 - 2.56i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-1.69 + 2.12i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (-0.947 + 4.15i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 2.59T + 19T^{2} \)
23 \( 1 + (1.32 + 5.82i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (0.403 - 1.76i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 - 1.19T + 31T^{2} \)
37 \( 1 + (-0.755 + 3.31i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (5.05 + 2.43i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (9.10 - 4.38i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-0.656 + 0.823i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-2.01 - 8.82i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (-10.7 + 5.18i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (1.71 - 7.49i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 - 11.8T + 67T^{2} \)
71 \( 1 + (-3.66 - 16.0i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (4.59 + 5.76i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + 0.219T + 79T^{2} \)
83 \( 1 + (-10.0 - 12.6i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (10.7 + 13.5i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 5.31T + 97T^{2} \)
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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

−12.99540699178575745222976928298, −12.17978564774699480089663549105, −11.26640908710514202254626352340, −10.20781639445867034369854468259, −8.568341435093029282367631878362, −8.078928089886540432040536380100, −7.12748656946103500238804633319, −4.93907150420521716980728585051, −4.00011870215413741671135654478, −2.47386238870178490583578126337, 1.78981853400052106966140308054, 3.76489152779146017281671364071, 5.28804768721496077421580361918, 6.52119409169435449650734435454, 7.79857066858903190169032493833, 8.581087273375012163461466973321, 10.03932682412493831557201062781, 11.09075930261666973087941979987, 11.69374273717737152876075552073, 13.32288497682525894711231072444

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