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Properties

Label 2-147-21.2-c2-0-19
Degree $2$
Conductor $147$
Sign $-0.497 + 0.867i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.497 + 0.867i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.497 + 0.867i$
Analytic conductor: \(4.00545\)
Root analytic conductor: \(2.00136\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1),\ -0.497 + 0.867i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.210814 - 0.363998i\)
\(L(\frac12)\) \(\approx\) \(0.210814 - 0.363998i\)
\(L(2)\) 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

−12.44564857543828181700893956880, −11.56373122341594893168612158892, −10.48755564796679565423797467999, −9.406612010864407135692404655258, −8.423608580369172197292470964200, −6.91713035785233333106367120596, −5.43479645372207640435397649691, −4.61601780482125125187979042501, −3.74792616938683709740828108132, −0.23315745401031353204927232322, 2.62296100195433506339357526511, 3.89518291239947958195651970092, 5.31233951198924826124874073044, 6.96624437547137554169088520467, 7.69241715058324807584559274031, 8.643222877003455838320843634469, 10.61942175369651429954533217441, 11.60824263621023728092808712065, 12.02106966815400496208459494264, 12.99603890415384898606946004340

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