Properties

Label 2-126-7.5-c8-0-17
Degree $2$
Conductor $126$
Sign $-0.746 + 0.665i$
Analytic cond. $51.3297$
Root an. cond. $7.16447$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.65 − 9.79i)2-s + (−63.9 − 110. i)4-s + (268. + 155. i)5-s + (−1.57e3 + 1.81e3i)7-s − 1.44e3·8-s + (3.04e3 − 1.75e3i)10-s + (3.66e3 + 6.34e3i)11-s − 1.06e4i·13-s + (8.82e3 + 2.56e4i)14-s + (−8.19e3 + 1.41e4i)16-s + (2.45e4 − 1.41e4i)17-s + (−1.06e5 − 6.16e4i)19-s − 3.97e4i·20-s + 8.28e4·22-s + (1.61e5 − 2.80e5i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.430 + 0.248i)5-s + (−0.656 + 0.754i)7-s − 0.353·8-s + (0.304 − 0.175i)10-s + (0.250 + 0.433i)11-s − 0.372i·13-s + (0.229 + 0.668i)14-s + (−0.125 + 0.216i)16-s + (0.293 − 0.169i)17-s + (−0.819 − 0.473i)19-s − 0.248i·20-s + 0.353·22-s + (0.578 − 1.00i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.746 + 0.665i$
Analytic conductor: \(51.3297\)
Root analytic conductor: \(7.16447\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :4),\ -0.746 + 0.665i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.485851618\)
\(L(\frac12)\) \(\approx\) \(1.485851618\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.65 + 9.79i)T \)
3 \( 1 \)
7 \( 1 + (1.57e3 - 1.81e3i)T \)
good5 \( 1 + (-268. - 155. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (-3.66e3 - 6.34e3i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 + 1.06e4iT - 8.15e8T^{2} \)
17 \( 1 + (-2.45e4 + 1.41e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (1.06e5 + 6.16e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (-1.61e5 + 2.80e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 6.93e4T + 5.00e11T^{2} \)
31 \( 1 + (-5.86e5 + 3.38e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-6.69e5 + 1.15e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 - 1.00e5iT - 7.98e12T^{2} \)
43 \( 1 - 1.01e6T + 1.16e13T^{2} \)
47 \( 1 + (3.43e6 + 1.98e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (6.75e6 + 1.16e7i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (-6.99e4 + 4.03e4i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (-3.34e6 - 1.93e6i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (6.54e5 + 1.13e6i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 + 3.90e7T + 6.45e14T^{2} \)
73 \( 1 + (-1.19e7 + 6.87e6i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (2.54e7 - 4.41e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + 6.90e7iT - 2.25e15T^{2} \)
89 \( 1 + (9.62e7 + 5.55e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 + 3.18e7iT - 7.83e15T^{2} \)
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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

−11.51643850523171751634965520040, −10.34773598982332703582613191412, −9.559752311081013752267863761867, −8.475972020745347380959500016080, −6.74641239678478663902253454626, −5.81110533235583017430551039872, −4.49821739543531424693740410420, −3.00732913885826270926808378469, −2.07681331599009112548243076086, −0.35034238092255017129011595333, 1.29854301114577017363233160375, 3.20880827987505772310184250102, 4.34485662099257166525133480662, 5.73682923575496203948484001870, 6.65033003970338114954131365829, 7.77061669539425863728516083429, 9.032091902787607445182493135777, 9.970854837749345143659856557878, 11.22340592995755564533541481805, 12.51281382943977864124438885550

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