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Properties

Label 2-126-7.5-c8-0-15
Degree $2$
Conductor $126$
Sign $0.377 + 0.926i$
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 + (−254. − 146. i)5-s + (−1.18e3 − 2.09e3i)7-s + 1.44e3·8-s + (2.87e3 − 1.65e3i)10-s + (9.84e3 + 1.70e4i)11-s + 2.73e4i·13-s + (2.71e4 + 265. i)14-s + (−8.19e3 + 1.41e4i)16-s + (−4.14e4 + 2.39e4i)17-s + (7.22e4 + 4.17e4i)19-s + 3.75e4i·20-s − 2.22e5·22-s + (6.46e4 − 1.11e5i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.406 − 0.234i)5-s + (−0.491 − 0.870i)7-s + 0.353·8-s + (0.287 − 0.165i)10-s + (0.672 + 1.16i)11-s + 0.958i·13-s + (0.707 + 0.00691i)14-s + (−0.125 + 0.216i)16-s + (−0.495 + 0.286i)17-s + (0.554 + 0.320i)19-s + 0.234i·20-s − 0.951·22-s + (0.231 − 0.400i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.926i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.377 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.377 + 0.926i$
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: \((2,\ 126,\ (\ :4),\ 0.377 + 0.926i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.8074865188\)
\(L(\frac12)\) \(\approx\) \(0.8074865188\)
\(L(5)\) 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 \( 1 + (1.18e3 + 2.09e3i)T \)
good5 \( 1 + (254. + 146. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (-9.84e3 - 1.70e4i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 2.73e4iT - 8.15e8T^{2} \)
17 \( 1 + (4.14e4 - 2.39e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (-7.22e4 - 4.17e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (-6.46e4 + 1.11e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 - 7.10e5T + 5.00e11T^{2} \)
31 \( 1 + (3.84e5 - 2.21e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-1.76e5 + 3.06e5i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 + 3.55e6iT - 7.98e12T^{2} \)
43 \( 1 + 6.52e6T + 1.16e13T^{2} \)
47 \( 1 + (-9.65e5 - 5.57e5i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (5.98e6 + 1.03e7i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (-1.01e7 + 5.85e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (-1.13e7 - 6.57e6i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (7.88e6 + 1.36e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 - 2.41e7T + 6.45e14T^{2} \)
73 \( 1 + (-1.51e7 + 8.76e6i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (-3.57e6 + 6.18e6i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + 6.34e7iT - 2.25e15T^{2} \)
89 \( 1 + (7.67e7 + 4.43e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 - 6.52e7iT - 7.83e15T^{2} \)
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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

−11.64642942591546925817464937184, −10.29676461027065757885142927676, −9.493201451262913767206487954792, −8.367185228735499917801516994983, −7.12298331264489373094760773199, −6.53646214396524345681763948434, −4.75588755050754188613478301783, −3.84892066417393845394132406558, −1.72612253862406969939232333168, −0.29208571149584557792184630295, 1.01174602153322238773382265100, 2.75818209649305801200891538947, 3.56020307870057045840707030324, 5.30792159117313871353108160954, 6.58374519411588757997410407179, 8.020719271104891398955897539949, 8.936495359167889526345694450200, 9.886115111541086667489997496225, 11.20365113062231078484398489370, 11.73064127720768082323449356403

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