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Properties

Label 2-126-63.20-c3-0-2
Degree $2$
Conductor $126$
Sign $-0.538 - 0.842i$
Analytic cond. $7.43424$
Root an. cond. $2.72658$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.73 − i)2-s + (−0.111 − 5.19i)3-s + (1.99 + 3.46i)4-s + (8.24 + 14.2i)5-s + (−5.00 + 9.10i)6-s + (−17.4 − 6.29i)7-s − 7.99i·8-s + (−26.9 + 1.15i)9-s − 32.9i·10-s + (−36.5 − 21.0i)11-s + (17.7 − 10.7i)12-s + (−39.5 + 22.8i)13-s + (23.8 + 28.3i)14-s + (73.3 − 44.4i)15-s + (−8 + 13.8i)16-s − 65.3·17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.0214 − 0.999i)3-s + (0.249 + 0.433i)4-s + (0.737 + 1.27i)5-s + (−0.340 + 0.619i)6-s + (−0.940 − 0.339i)7-s − 0.353i·8-s + (−0.999 + 0.0428i)9-s − 1.04i·10-s + (−1.00 − 0.578i)11-s + (0.427 − 0.259i)12-s + (−0.844 + 0.487i)13-s + (0.455 + 0.540i)14-s + (1.26 − 0.765i)15-s + (−0.125 + 0.216i)16-s − 0.932·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.538 - 0.842i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ -0.538 - 0.842i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0841695 + 0.153750i\)
\(L(\frac12)\) \(\approx\) \(0.0841695 + 0.153750i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73 + i)T \)
3 \( 1 + (0.111 + 5.19i)T \)
7 \( 1 + (17.4 + 6.29i)T \)
good5 \( 1 + (-8.24 - 14.2i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (36.5 + 21.0i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (39.5 - 22.8i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 65.3T + 4.91e3T^{2} \)
19 \( 1 - 161. iT - 6.85e3T^{2} \)
23 \( 1 + (-147. + 85.2i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (171. + 99.2i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (48.4 - 27.9i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 133.T + 5.06e4T^{2} \)
41 \( 1 + (9.23 + 15.9i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-12.1 + 21.1i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-61.6 + 106. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 400. iT - 1.48e5T^{2} \)
59 \( 1 + (-114. - 197. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (212. + 122. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (466. + 808. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 808. iT - 3.57e5T^{2} \)
73 \( 1 - 98.0iT - 3.89e5T^{2} \)
79 \( 1 + (450. - 779. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-605. + 1.04e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 678.T + 7.04e5T^{2} \)
97 \( 1 + (-749. - 432. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−13.20208907228560914650821033707, −12.27123099629614243639859002727, −10.92509586476049077500300570578, −10.32090185675314172163639707948, −9.133396567016100010927537681447, −7.67511389902993212432526917919, −6.82739243247715961293506460834, −5.92831248378167421663622440198, −3.16189246988404133668701943110, −2.14931460895713072949684972401, 0.099903085326077211826745091254, 2.61849849353973646666584256830, 4.89679420271571187833508132453, 5.46480543118132475293427296768, 7.10963722011681044872456484767, 8.818296197826014475089344918649, 9.269628956182702242890850674221, 10.06407448036988275059806234083, 11.21955860838216728756901384319, 12.80998336157574644543340346168

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