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Properties

Label 2-126-7.4-c7-0-17
Degree $2$
Conductor $126$
Sign $0.670 + 0.741i$
Analytic cond. $39.3605$
Root an. cond. $6.27379$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4 + 6.92i)2-s + (−31.9 + 55.4i)4-s + (6.21 + 10.7i)5-s + (−879. − 223. i)7-s − 511.·8-s + (−49.7 + 86.1i)10-s + (−3.36e3 + 5.82e3i)11-s + 7.23e3·13-s + (−1.97e3 − 6.98e3i)14-s + (−2.04e3 − 3.54e3i)16-s + (7.66e3 − 1.32e4i)17-s + (−9.45e3 − 1.63e4i)19-s − 795.·20-s − 5.37e4·22-s + (−2.38e4 − 4.13e4i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.0222 + 0.0385i)5-s + (−0.969 − 0.246i)7-s − 0.353·8-s + (−0.0157 + 0.0272i)10-s + (−0.761 + 1.31i)11-s + 0.912·13-s + (−0.192 − 0.680i)14-s + (−0.125 − 0.216i)16-s + (0.378 − 0.655i)17-s + (−0.316 − 0.547i)19-s − 0.0222·20-s − 1.07·22-s + (−0.408 − 0.708i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 + 0.741i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.670 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.670 + 0.741i$
Analytic conductor: \(39.3605\)
Root analytic conductor: \(6.27379\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :7/2),\ 0.670 + 0.741i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.186696445\)
\(L(\frac12)\) \(\approx\) \(1.186696445\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4 - 6.92i)T \)
3 \( 1 \)
7 \( 1 + (879. + 223. i)T \)
good5 \( 1 + (-6.21 - 10.7i)T + (-3.90e4 + 6.76e4i)T^{2} \)
11 \( 1 + (3.36e3 - 5.82e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 - 7.23e3T + 6.27e7T^{2} \)
17 \( 1 + (-7.66e3 + 1.32e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (9.45e3 + 1.63e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (2.38e4 + 4.13e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 - 6.74e4T + 1.72e10T^{2} \)
31 \( 1 + (5.48e4 - 9.50e4i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + (3.04e5 + 5.27e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 - 8.08e5T + 1.94e11T^{2} \)
43 \( 1 + 3.34e5T + 2.71e11T^{2} \)
47 \( 1 + (3.82e5 + 6.62e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (-6.22e5 + 1.07e6i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-1.02e6 + 1.77e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (7.04e4 + 1.22e5i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (1.46e6 - 2.54e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 - 6.15e5T + 9.09e12T^{2} \)
73 \( 1 + (-7.11e5 + 1.23e6i)T + (-5.52e12 - 9.56e12i)T^{2} \)
79 \( 1 + (-1.19e6 - 2.06e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 - 7.53e6T + 2.71e13T^{2} \)
89 \( 1 + (2.67e6 + 4.63e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + 1.58e7T + 8.07e13T^{2} \)
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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

−12.30665520652206147876362653969, −10.71342787997836072509877445621, −9.758395186201560120257430963016, −8.579144232023946942690905852111, −7.28313314102042112858361151718, −6.49295223040701401976594658871, −5.15923355520160377862754706903, −3.92258283357970668948444383882, −2.50663626104104335141945779327, −0.32789551687188016673700767168, 1.17863234279105396416378251299, 2.91090923151674030626595992115, 3.76698029037631391237237910718, 5.54522711731269340978223923840, 6.28588746930884417183695159703, 8.073049441851985282610288569864, 9.116918904163055335268624398201, 10.29304193510279593027703988216, 11.08352740811714192422778439265, 12.23565675591143381061381952356

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