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Properties

Label 2-126-7.2-c5-0-9
Degree $2$
Conductor $126$
Sign $0.701 + 0.712i$
Analytic cond. $20.2083$
Root an. cond. $4.49537$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 3.46i)2-s + (−7.99 − 13.8i)4-s + (43 − 74.4i)5-s + (24.5 + 127. i)7-s + 63.9·8-s + (172 + 297. i)10-s + (17 + 29.4i)11-s − 3·13-s + (−490 − 169. i)14-s + (−128 + 221. i)16-s + (−952 − 1.64e3i)17-s + (744.5 − 1.28e3i)19-s − 1.37e3·20-s − 136·22-s + (−112 + 193. i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.769 − 1.33i)5-s + (0.188 + 0.981i)7-s + 0.353·8-s + (0.543 + 0.942i)10-s + (0.0423 + 0.0733i)11-s − 0.00492·13-s + (−0.668 − 0.231i)14-s + (−0.125 + 0.216i)16-s + (−0.798 − 1.38i)17-s + (0.473 − 0.819i)19-s − 0.769·20-s − 0.0599·22-s + (−0.0441 + 0.0764i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.701 + 0.712i$
Analytic conductor: \(20.2083\)
Root analytic conductor: \(4.49537\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :5/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.575203374\)
\(L(\frac12)\) \(\approx\) \(1.575203374\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 - 3.46i)T \)
3 \( 1 \)
7 \( 1 + (-24.5 - 127. i)T \)
good5 \( 1 + (-43 + 74.4i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-17 - 29.4i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 3T + 3.71e5T^{2} \)
17 \( 1 + (952 + 1.64e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-744.5 + 1.28e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (112 - 193. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 6.50e3T + 2.05e7T^{2} \)
31 \( 1 + (865.5 + 1.49e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-3.81e3 + 6.61e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 1.54e4T + 1.15e8T^{2} \)
43 \( 1 - 1.84e4T + 1.47e8T^{2} \)
47 \( 1 + (-9.23e3 + 1.59e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (9.97e3 + 1.72e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.59e4 + 2.75e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-2.88e4 + 4.99e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-3.02e4 - 5.24e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 4.48e4T + 1.80e9T^{2} \)
73 \( 1 + (1.04e4 + 1.80e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-1.52e4 + 2.64e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 1.10e5T + 3.93e9T^{2} \)
89 \( 1 + (2.94e4 - 5.10e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 1.19e5T + 8.58e9T^{2} \)
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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

−12.43962766554333170932045131223, −11.36026901215058977324883575385, −9.685611258196233211697745416918, −9.104236082713579592205068801502, −8.287070765314431019593228174159, −6.75741454429977318695320174693, −5.44230902473452813092550765238, −4.79996547767837381629470601268, −2.26218463432017689930234043840, −0.67595369847524546068737375325, 1.44460658467957459808430116037, 2.86239199192186352278225345257, 4.16386758640112654404793520164, 6.11395645778833663265538543906, 7.13178686974369896815202397204, 8.359306321789112939515752766080, 9.871633778189003266884067843141, 10.49288002165596559377141457118, 11.15066630278907109007839729394, 12.52730758874145941623508495239

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