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Properties

Label 2-126-63.41-c3-0-13
Degree $2$
Conductor $126$
Sign $-0.975 + 0.222i$
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 + (−5.10 − 0.957i)3-s + (1.99 − 3.46i)4-s + (0.0529 − 0.0917i)5-s + (9.80 − 3.44i)6-s + (18.4 + 1.91i)7-s + 7.99i·8-s + (25.1 + 9.77i)9-s + 0.211i·10-s + (−23.9 + 13.8i)11-s + (−13.5 + 15.7i)12-s + (−54.2 − 31.3i)13-s + (−33.8 + 15.1i)14-s + (−0.358 + 0.418i)15-s + (−8 − 13.8i)16-s − 57.7·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.982 − 0.184i)3-s + (0.249 − 0.433i)4-s + (0.00473 − 0.00820i)5-s + (0.667 − 0.234i)6-s + (0.994 + 0.103i)7-s + 0.353i·8-s + (0.932 + 0.362i)9-s + 0.00670i·10-s + (−0.655 + 0.378i)11-s + (−0.325 + 0.379i)12-s + (−1.15 − 0.668i)13-s + (−0.645 + 0.288i)14-s + (−0.00617 + 0.00719i)15-s + (−0.125 − 0.216i)16-s − 0.824·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.222i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.975 + 0.222i$
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: \((2,\ 126,\ (\ :3/2),\ -0.975 + 0.222i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.000221516 - 0.00196914i\)
\(L(\frac12)\) \(\approx\) \(0.000221516 - 0.00196914i\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.10 + 0.957i)T \)
7 \( 1 + (-18.4 - 1.91i)T \)
good5 \( 1 + (-0.0529 + 0.0917i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (23.9 - 13.8i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (54.2 + 31.3i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 57.7T + 4.91e3T^{2} \)
19 \( 1 + 26.1iT - 6.85e3T^{2} \)
23 \( 1 + (82.4 + 47.6i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (242. - 139. i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (196. + 113. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 104.T + 5.06e4T^{2} \)
41 \( 1 + (130. - 225. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (220. + 381. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-134. - 233. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 26.5iT - 1.48e5T^{2} \)
59 \( 1 + (97.1 - 168. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (418. - 241. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-225. + 391. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 979. iT - 3.57e5T^{2} \)
73 \( 1 + 30.5iT - 3.89e5T^{2} \)
79 \( 1 + (-227. - 394. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (345. + 598. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 145.T + 7.04e5T^{2} \)
97 \( 1 + (909. - 525. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.28396363508566998734096134123, −11.16477716457907762754693937045, −10.51533742097167590685141224947, −9.288864103034585425635170151328, −7.81648643572139007235992603685, −7.13328894843382169988952457890, −5.59679601269723947808071617510, −4.77098203759455030424163318076, −1.95275820610893156486851298333, −0.00134111492593259733313704385, 1.95563602408904032373824068861, 4.24546564759779715697574885309, 5.44805697841541550462718074497, 6.96566843340152911867056562860, 8.013311842875223729995295798763, 9.394705723421969580951676367089, 10.43216642819074082033858398702, 11.27113582639911376734419828041, 11.97130136934105951601605201117, 13.06754585757372983529168888492

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