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Properties

Label 2-126-63.41-c3-0-16
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: $\chi_{126} (41, \cdot )$
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

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

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