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Properties

Label 2-126-21.11-c6-0-11
Degree $2$
Conductor $126$
Sign $0.488 + 0.872i$
Analytic cond. $28.9868$
Root an. cond. $5.38394$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.89 − 2.82i)2-s + (15.9 − 27.7i)4-s + (123. − 71.0i)5-s + (339. − 48.4i)7-s − 181. i·8-s + (402. − 696. i)10-s + (1.08e3 + 627. i)11-s − 472.·13-s + (1.52e3 − 1.19e3i)14-s + (−512. − 886. i)16-s + (1.47e3 + 852. i)17-s + (187. + 325. i)19-s − 4.54e3i·20-s + 7.10e3·22-s + (−2.89e3 + 1.67e3i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.984 − 0.568i)5-s + (0.989 − 0.141i)7-s − 0.353i·8-s + (0.402 − 0.696i)10-s + (0.816 + 0.471i)11-s − 0.214·13-s + (0.556 − 0.436i)14-s + (−0.125 − 0.216i)16-s + (0.300 + 0.173i)17-s + (0.0273 + 0.0473i)19-s − 0.568i·20-s + 0.667·22-s + (−0.238 + 0.137i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.488 + 0.872i$
Analytic conductor: \(28.9868\)
Root analytic conductor: \(5.38394\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3),\ 0.488 + 0.872i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.935669328\)
\(L(\frac12)\) \(\approx\) \(3.935669328\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.89 + 2.82i)T \)
3 \( 1 \)
7 \( 1 + (-339. + 48.4i)T \)
good5 \( 1 + (-123. + 71.0i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (-1.08e3 - 627. i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + 472.T + 4.82e6T^{2} \)
17 \( 1 + (-1.47e3 - 852. i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-187. - 325. i)T + (-2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (2.89e3 - 1.67e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + 3.88e3iT - 5.94e8T^{2} \)
31 \( 1 + (-1.94e4 + 3.36e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (1.00e4 + 1.73e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 5.76e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.44e5T + 6.32e9T^{2} \)
47 \( 1 + (9.23e4 - 5.33e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (1.78e5 + 1.03e5i)T + (1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (3.11e5 + 1.80e5i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (-1.34e5 - 2.32e5i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-1.24e5 + 2.15e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 3.69e5iT - 1.28e11T^{2} \)
73 \( 1 + (2.22e5 - 3.84e5i)T + (-7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (2.46e5 + 4.27e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 - 3.90e5iT - 3.26e11T^{2} \)
89 \( 1 + (-1.02e5 + 5.94e4i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 7.20e5T + 8.32e11T^{2} \)
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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

−12.13912496452497113467193229264, −11.19102344234496610124071904123, −9.990552491246312821292903320463, −9.114976031624338625564026711740, −7.68493183567989902882241379953, −6.19362377152883278292953265515, −5.15019857403209088644911280679, −4.09758996919274931876194126305, −2.17433611992956519479544036389, −1.19161201799800758693643366032, 1.55063040305676053119371406949, 2.94419068844775866333497467834, 4.56472558906809492432439838304, 5.75902101322471437831305707937, 6.67766571097240878905115912835, 7.972153740851565275905485822568, 9.187062871529137597967652069925, 10.46723721406770957056273984799, 11.47024824240186396799847756764, 12.44275895141009366206141153964

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