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Properties

Label 2-147-21.11-c2-0-12
Degree $2$
Conductor $147$
Sign $0.787 - 0.616i$
Analytic cond. $4.00545$
Root an. cond. $2.00136$
Motivic weight $2$
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 − 0.653i)2-s + (1.15 + 2.76i)3-s + (−1.14 + 1.98i)4-s + (6.39 − 3.68i)5-s + (3.11 + 2.38i)6-s + 8.22i·8-s + (−6.34 + 6.38i)9-s + (4.82 − 8.35i)10-s + (−2.26 − 1.30i)11-s + (−6.81 − 0.887i)12-s + 6.35·13-s + (17.5 + 13.4i)15-s + (0.791 + 1.37i)16-s + (10.5 + 6.07i)17-s + (−3.01 + 11.3i)18-s + (−5.11 − 8.85i)19-s + ⋯
L(s)  = 1  + (0.565 − 0.326i)2-s + (0.384 + 0.923i)3-s + (−0.286 + 0.496i)4-s + (1.27 − 0.737i)5-s + (0.519 + 0.397i)6-s + 1.02i·8-s + (−0.705 + 0.709i)9-s + (0.482 − 0.835i)10-s + (−0.205 − 0.118i)11-s + (−0.568 − 0.0739i)12-s + 0.488·13-s + (1.17 + 0.896i)15-s + (0.0494 + 0.0856i)16-s + (0.618 + 0.357i)17-s + (−0.167 + 0.631i)18-s + (−0.269 − 0.466i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.787 - 0.616i$
Analytic conductor: \(4.00545\)
Root analytic conductor: \(2.00136\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1),\ 0.787 - 0.616i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.16358 + 0.746195i\)
\(L(\frac12)\) \(\approx\) \(2.16358 + 0.746195i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.15 - 2.76i)T \)
7 \( 1 \)
good2 \( 1 + (-1.13 + 0.653i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (-6.39 + 3.68i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (2.26 + 1.30i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 6.35T + 169T^{2} \)
17 \( 1 + (-10.5 - 6.07i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (5.11 + 8.85i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (3.72 - 2.15i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 17.3iT - 841T^{2} \)
31 \( 1 + (-19.6 + 34.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (20.5 + 35.5i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 30.2iT - 1.68e3T^{2} \)
43 \( 1 + 55.8T + 1.84e3T^{2} \)
47 \( 1 + (34.6 - 19.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-90.9 - 52.5i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (35.8 + 20.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (10.2 + 17.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-13.5 + 23.5i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 67.8iT - 5.04e3T^{2} \)
73 \( 1 + (-30.3 + 52.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-31.6 - 54.7i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 89.9iT - 6.88e3T^{2} \)
89 \( 1 + (-54.7 + 31.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 19.1T + 9.40e3T^{2} \)
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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

−13.26293282422868338420310131934, −12.07955344524723345314194923835, −10.85331983068384072791490845067, −9.775681348485612039729869120715, −8.927205299264501145870885051031, −8.062385377578621289052053767730, −5.85380947803644860815876399048, −4.98045683662758695616866662554, −3.79957732149822334843306515255, −2.33094204063900646725351377420, 1.57264823040542560027782327911, 3.20603678398107839940461246781, 5.26326057723053721059561991606, 6.26579383218814827157976875805, 6.92382383477969234843579486835, 8.486661699910404299555495897436, 9.733110830782893296120893400164, 10.47318395471583631551092107196, 12.03093145119991648448123944050, 13.21553663115713577705289924731

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