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Properties

Label 2-126-21.5-c9-0-17
Degree $2$
Conductor $126$
Sign $-0.0477 + 0.998i$
Analytic cond. $64.8945$
Root an. cond. $8.05571$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−13.8 − 8i)2-s + (127. + 221. i)4-s + (−356. + 618. i)5-s + (3.24e3 − 5.45e3i)7-s − 4.09e3i·8-s + (9.89e3 − 5.71e3i)10-s + (6.38e4 − 3.68e4i)11-s − 5.57e4i·13-s + (−8.86e4 + 4.96e4i)14-s + (−3.27e4 + 5.67e4i)16-s + (1.69e5 + 2.92e5i)17-s + (−3.83e5 − 2.21e5i)19-s − 1.82e5·20-s − 1.17e6·22-s + (−3.35e3 − 1.93e3i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.255 + 0.442i)5-s + (0.511 − 0.859i)7-s − 0.353i·8-s + (0.312 − 0.180i)10-s + (1.31 − 0.759i)11-s − 0.541i·13-s + (−0.616 + 0.345i)14-s + (−0.125 + 0.216i)16-s + (0.491 + 0.850i)17-s + (−0.675 − 0.389i)19-s − 0.255·20-s − 1.07·22-s + (−0.00249 − 0.00144i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0477 + 0.998i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.0477 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.0477 + 0.998i$
Analytic conductor: \(64.8945\)
Root analytic conductor: \(8.05571\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :9/2),\ -0.0477 + 0.998i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.494315225\)
\(L(\frac12)\) \(\approx\) \(1.494315225\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (13.8 + 8i)T \)
3 \( 1 \)
7 \( 1 + (-3.24e3 + 5.45e3i)T \)
good5 \( 1 + (356. - 618. i)T + (-9.76e5 - 1.69e6i)T^{2} \)
11 \( 1 + (-6.38e4 + 3.68e4i)T + (1.17e9 - 2.04e9i)T^{2} \)
13 \( 1 + 5.57e4iT - 1.06e10T^{2} \)
17 \( 1 + (-1.69e5 - 2.92e5i)T + (-5.92e10 + 1.02e11i)T^{2} \)
19 \( 1 + (3.83e5 + 2.21e5i)T + (1.61e11 + 2.79e11i)T^{2} \)
23 \( 1 + (3.35e3 + 1.93e3i)T + (9.00e11 + 1.55e12i)T^{2} \)
29 \( 1 + 5.37e6iT - 1.45e13T^{2} \)
31 \( 1 + (5.16e6 - 2.98e6i)T + (1.32e13 - 2.28e13i)T^{2} \)
37 \( 1 + (5.29e6 - 9.17e6i)T + (-6.49e13 - 1.12e14i)T^{2} \)
41 \( 1 - 2.43e7T + 3.27e14T^{2} \)
43 \( 1 - 2.01e7T + 5.02e14T^{2} \)
47 \( 1 + (7.28e5 - 1.26e6i)T + (-5.59e14 - 9.69e14i)T^{2} \)
53 \( 1 + (-1.32e7 + 7.64e6i)T + (1.64e15 - 2.85e15i)T^{2} \)
59 \( 1 + (-6.10e7 - 1.05e8i)T + (-4.33e15 + 7.50e15i)T^{2} \)
61 \( 1 + (1.46e8 + 8.44e7i)T + (5.84e15 + 1.01e16i)T^{2} \)
67 \( 1 + (2.65e7 + 4.60e7i)T + (-1.36e16 + 2.35e16i)T^{2} \)
71 \( 1 + 2.21e8iT - 4.58e16T^{2} \)
73 \( 1 + (1.41e8 - 8.19e7i)T + (2.94e16 - 5.09e16i)T^{2} \)
79 \( 1 + (-3.21e8 + 5.56e8i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 - 5.56e8T + 1.86e17T^{2} \)
89 \( 1 + (8.45e6 - 1.46e7i)T + (-1.75e17 - 3.03e17i)T^{2} \)
97 \( 1 + 1.52e9iT - 7.60e17T^{2} \)
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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

−11.07722184239936625574998151066, −10.56227585118578590541723788811, −9.253180038090365323179027298557, −8.197736192186943057162239202437, −7.23406680250551855439634543545, −6.08085101331000348088854073520, −4.21035995821218406521971650472, −3.26566126555142565436828340175, −1.58452962740310399697174781878, −0.53198227057995894604677555833, 1.09323090694802973490351043661, 2.20231331586340518462699285002, 4.14141891298560083896770362182, 5.33849136185239968272977354571, 6.60769239189017022843675018553, 7.69085685921855576842021815452, 8.936984533504470578169238264415, 9.341672947629908264630806370996, 10.84714067770458367109058972462, 11.92530449831911857043959825735

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